Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Date: Thursday, 11/Jul/2019
8:25am - 8:30amAnnouncements
vonRoll, Fabrikstr. 6, 001 
8:30am - 9:30amIP05: Alicia Dickenstein: Algebra and geometry in the study of enzymatic cascades
vonRoll, Fabrikstr. 6, 001 
 
8:30am - 9:30am

Algebra and geometry in the study of enzymatic cascades

Alicia Dickenstein

Universidad de Buenos Aires, Argentine Republic

In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible.
I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.

 
8:30am - 9:30amIP05-streamed from 001: Alicia Dickenstein: Algebra and geometry in the study of enzymatic cascades
vonRoll, Fabrikstr. 6, 004 
9:30am - 10:00amCoffee break
Unitobler, F wing, floors 0 and -1 
10:00am - 12:00pmMS137, part 1: Symbolic Combinatorics
Unitobler, F005 
 
10:00am - 12:00pm

Symbolic Combinatorics

Chair(s): Shaoshi Chen (Chinese Academy of Sciences), Manuel Kauers (Johannes Kepler University, Linz, Austria), Stephen Melczer (University of Pennsylvania)

In recent years algorithms and software have been developed that allow researchers to discover and verify combinatorial identities as well as understand analytic and algebraic properties of generating functions. The interaction of combinatorics and symbolic computation has had a beneficial impact on both fields. This minisymposium will feature 12 speakers describing recent research combining these areas.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Enumeration of walks in three quarters of the plane

Axel Bacher
University Paris 13

Walks in the quarter plane with a prescribed set of steps have, in the past decades, attracted a lot of attention. One of the main question is to find the nature of the generating function (algebraicity, D-finiteness, etc.) as a function of the step set. Since these walks are now rather well understood, we turn our attention to a related problem, walks in three quarters of the plane, for which we present the latest results.

Work in common with Alin Bostan and Killian Raschel.

 

On the growth of algebras

Jason Bell
University of Waterloo

The growth function of an algebra is a combinatorial invariant that gives insight into the underlying algebraic structure. There are several well-known combinatorial constraints that growth functions must have. We show that in fact all constraints are now known: given a function satisfying the previously known constraints, we show that there is an algebra having this function as its growth function. This is joint work with Efim Zelmanov.

 

A Gessel way to the diagonal theorem on D-finite power series

Shaoshi Chen
Chinese Academy of Sciences

Special functions that satisfy linear differential equations with polynomial coefficients appear ubiquitously in combinatorics and mathematical physics. Such special functions are said to be D-finite by Stanley. In the early 1980's, Gessel and Zeilberger independently proved that the diagonal of D-finite power series in several variables is D-finite. However their proofs were not complete and later Lipshitz gave a complete proof using an elimination lemma. Zeilberger completed his proof with the theory of holonomic D-modules. We follow the spirit of Gessel's proof strategy and fix the gap in his proof directly. This is a joint work with Chaochao Zhu.

 

Inhomogeneous Lattice Walks

Manfred Buchacher
Johannes Kepler University Linz

We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. For the models in the quarter plane, we have carried out an experimental classification of all models with small steps. We discovered many (apparently) D-finite cases for most of which we have no explanation yet.

Joint work with Manuel Kauers. To appear in the proceedings of FPSAC 2019.

 
10:00am - 12:00pmMS146, part 1: Random geometry and topology
Unitobler, F006 
 
10:00am - 12:00pm

Random geometry and topology

Chair(s): Paul Breiding (Max-Planck Institute for Mathematics in the Sciences, Germany), Lerario Antonio (SISSA), Lundberg Erik (Florida Atlantic University), Kozhasov Khazhgali (Max-Planck Institute for Mathematics in the Sciences, Germany)

This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications.

More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Zero-sets of 3D random waves

Federico Dalmao
Universidad de la Republica de Uruguay

We study Berry's model in the three-dimensional case. This model contains as particular cases the monochromatic random waves and the black-body radiation, which are isotropic Gaussian fields that a.s. solve the Helmholtz equation. We generalize it to include more general features as anisotropy. We are interested in the zero-sets of the random waves as they represent lines of darkness, threads of silence, etc. We compute moments and find limit distributions under mild hypothesis and compare it with the well studied 2D-case. This is a joint work with Anne Estrade and José R. León.

 

Curvature and randomness

Emil Horobet
Sapientia Hungarian University

The Euclidean Distance degree measures the algebraic complexity of writing the optimal solution to the best approximation problem to an algebraic variety as a function of the coordinates of the data point. The number of real-valued critical points of the distance function can be different for different data points. For randomly sampled data the expected number of real valued critical points is of high interest and it is called the average ED degree. In this talk we will see connections between the average ED degree, the ED discriminant and different curvatures of the underlying variety.

 

Random sections of line bundles over real Riemann surfaces

Michele Ancona
Univ. Claude Bernard Lyon 1

We will explain how to compute the higher moments of the random variable ”number of real zeros of a random polynomial”. More generally, given a line bundle L over a real Riemann surface, we explain how to compute all the moments of the random variable ”number of real zeros of a random section of L”.

 

On the topology of real components of real sections of vector bundles

Chris Peterson
Colorado State University

This talk will present joint work with Tanner Strunk. At present, the talk will consist of a collection of methods and numerical data concerning the probability of various topologies that arise as the real zeros of real sections of vector bundles. Some of the methods utilized for collecting such data are interesting and might be useful in a general context. However, the main focus of the talk will be on the number of different topologies that can arise in various settings and conjectural relationships between their probabilities. While some initial results are rather intriguing, we are currently only able to provide statistical data rather than theory to support the data. It is the hope that additional insight into the results is obtained by the time of the conference. At the very least, perhaps the data will lead to some interesting discussions.

 
10:00am - 12:00pmMS181, part 1: Integral and algebraic geometric methods in the study of Gaussian random fields
Unitobler, F007 
 
10:00am - 12:00pm

Integral and algebraic geometric methods in the study of Gaussian random fields

Chair(s): David Ginsbourger (Idiap Research Institute and University of Bern, Switzerland), Jean-Marc Azaïs (Institut de Mathématiques de Toulouse)

Integral and algebraic geometry are at the heart of a number of contributions pertaining to the study of Gaussian random fields and related topics, not only from probabilistic and statistical viewpoints but also from the realm of interpolation and function approximation. This minisymposium will gather a team of junior researchers and established experts presenting original research results reflecting diverse challenges of geometrical and applied geometrical nature primarily involving Gaussian fields.

These encompass the study of geometrical and topological properties of sets implicitly defined by random fields such as zeros of random polynomials, excursion sets, as well as integral curves stemming for instance from filament estimation. Also, Gaussian field approximations dedicated to the estimation of excursion probabilities and more general geometric questions will be tackled, as well as algebraic methods in sparse grids for polynomial and Gaussian process interpolation.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Asymptotic normality for the Volume of the nodal set for Kostlan-Shub-Smale polynomial systems

Jean-Marc Azaïs
Institut de Mathématiques de Toulouse

We study the asymptotic variance and the CLT, as the degree goes to infinity, of the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system.

 

Euler characteristic and bicovariogram of random excursions

Raphaël Lachieze-Rey
Université Paris Descartes

The Euler characteristic of a planar compact smooth set is a privileged tool of geometric analysis as it is a local quantity carrying information on various macroscopic features of the set topology. We indicate here how to compute it directly from the covariograms of the set, i.e. the volumes of the intersection of the set with translated copies of itself. This approach works under hypotheses of $mathcal{C}^{1,1}$ regularity, and can be applied to excursions of $mathcal{C}^{1,1}$ bivariate functions. In the realm of random sets or fields, this identity gives the mean Euler characteristic in terms of the third order marginals.

 

Bayesian approach to filament estimation with a latent Gaussian random field model

Wolfgang Polonik1, Johannes Krebs2
1UC David, 2UC Davis

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process. We survey existing results and present expansions of these results to larger classes of underlying sampling schemes and to the multivariate case.

 

On the universality of roots of random polynomials

Guillaume Poly
Université de Rennes I

A classic question in random polynomial theory is to determine whether the distribution of the roots depends on the distribution of the random coefficients. We will explore this question both for algebraic and trigonometric models and point out some important differences between the two regimes.

 
10:00am - 12:00pmMS126, part 1: Euclidean distance geometry and its applications
Unitobler, F011 
 
10:00am - 12:00pm

Euclidean distance geometry and its applications

Chair(s): Kaie Kubjas (Sorbonne Université)

Given a natural number d and a weighted graph G=(V,E), the fundamental problem in Euclidean distance geometry is to determine whether there exists a realization of the graph G in Rd such that distances between pairs of points are equal to the corresponding edge weights. This problem naturally arises in many applications that require recovering locations of objects from the distances between these objects. Usually, measurements of the distances are noisy and there can be missing data. Examples of applications are sensor network localization, molecular conformation, genome reconstruction, robotics and data visualization. Algebraic varieties and semialgebraic sets naturally come up in Euclidean distance geometry, since distances between objects are given by polynomials. Hence questions about uniqueness and finiteness of realizations are often algebraic in nature, whereas realizations are found using semidefinite or nonconvex optimization methods. The goal of this minisymposium is to present theory and applications of Euclidean distance geometry, and connect researchers working in Euclidean distance geometry with applied algebraic geometers.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Isometries in Euclidean, Homogeneous, and Conformal Spaces

Carlile Lavor
University of Campinas, Brazil

It is known, from linear algebra, that isometries in Euclidean Spaces are described by orthogonal transformations up to translations. We will discuss what happens when isometries are considered in Homogeneous and Conformal Spaces.
 

Auxetic deformations of triply periodic minimal surfaces

Ciprian S. Borcea
Rider University, USA

The notion of one-parameter auxetic deformation, introduced previously for periodic frameworks, can be used in the context of triply periodic minimal surfaces. We exhibit auxetic paths in several new families of triply periodic minimal surfaces of low genus.

 

Voronoi Cells of Varieties

Maddie Weinstein
University of California, Berkeley, USA

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.

 

Critical points of the Hamming and taxicab distance functions

Jonathan Hauenstein
University of Notre Dame, USA

Minimizing the Euclidean distance from a given point to the solution set of a given system of polynomial equations can be accomplished via critical point techniques. This talk will explore extending critical point techniques to minimization with respect to the Hamming distance and taxicab distance. Numerical algebraic geometric techniques are derived for computing a finite set of real points satisfying the polynomial equations which contains a global minimizer. Several examples will be used to demonstrate the new techniques. This is joint work with D. Brake, N. Daleo, and S. Sherman.

 
10:00am - 12:00pmMS173, part 1: Numerical methods in algebraic geometry
Unitobler, F012 
 
10:00am - 12:00pm

Numerical methods in algebraic geometry

Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)

This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Minimal problems in multiview 3D reconstruction via homotopy continuation

Anton Leykin
Georgia Tech

This will be a short survey of a class of problems in computer vision, for which it is plausible to construct efficient solvers based on polynomial homotopy continuation. For some of these problems alternative solvers do not exist at the moment.
The problems we consider concern relative camera pose recovery from points and lines in more than 2 views. In addition to classical point correspondences and line correspondences, we use incidence correspondences, which result from points lying on lines. We show how to build a solver based on a parameter homotopy coming from this framework.
(This survey is based on collaboration and discussions with many people involved in the Nonlinear Algebra semester at ICERM in Fall 2018.)

 

Computing the real CANDECOMP/PARAFAC decomposition of real tensors

Tsung-Lin Lee
National Sun Yat-sen University

The real Candecomp/Parafac decomposition (CPD) has many applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Several methods have been provided for computing the CPD such as alternating least squares (ALS), nonlinear least squares (NLS) and unconstrained nonlinear optimization. Those methods may take many iterations to converge and are not guar-anteed to converge to the solution. Recently, homotopy continuation techniques have been applied in computing tensor decomposition. In this talk, the real CPD of a real unbalanced tensor will be considered.

 

Computing transcendental invariants of hypersurfaces via homotopy

Emre Sertoz
Max-Planck-Institute MiS, Leipzig

Deep geometric properties of each projective variety is encoded in a matrix of complex numbers, called its periods. Knowing the periods of a variety, one can often say quite a lot about the type of subvarieties it contains using LLL methods, without resorting to symbolic elimination. However, numerical computation of periods have been previously confined to curves in the plane and to varieties enjoying many symmetries. We will demonstrate how periods of hypersurfaces can be computed using a form of homotopy and how they can be studied to reveal the geometry of the hypersurface.

 

On the nonlinearity interval in parametric semidefinite optimization

Tingting Tang
University of Notre Dame

We consider the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective vector along a fixed direction. We characterize the so-called transition point of the optimal partition where the ranks of a maximally complementary optimal solution suddenly change, and the nonlinearity interval of the optimal partition where the ranks of maximally complementary optimal solutions stay constant. The continuity of the optimal set mapping on the basis of Painleve-Kuratowski set convergence in a nonlinearity interval is investigated. We show that not only the continuity might fail, even the sequence of maximally complementary optimal solutions might jump in the interior of a nonlinearity interval. Finally, we present a procedure stemming from numerical algebraic geometry to efficiently compute nonlinearity intervals.

 
10:00am - 12:00pmMS144: Tropical geometry in machine learning
Unitobler, F013 
 
10:00am - 12:00pm

Tropical geometry in machine learning

Chair(s): Gregory Naisat (The University of Chicago, United States of America)

A connection between tropical polynomials and neural networks has been recently established. This connection remains to be explored in full. Currently, most basic notions from tropical geometry are used to quantify the number of linear regions in a neural network. Purpose of this session is to present what is currently know about the relationship between tropical polynomials and neural networks and promote further exploration of tropical algebra in the context of machine learning at neural networks.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Tropical geometry of deep neural networks

Gregory Naisat
The University of Chicago, United States of America

Abstract: We exlore connections between feedforward neural networks with ReLU activation and tropical geometry — we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network.

 

Tropical geometry and weighted lattices

Petros Maragos
FNational Technical University of Athens

We present advances on extending the max-plus or min-plus algebraic structure of tropical geometry by using weighted lattices and a max-* algebra with an arbitrary binary operation * that distributes over max or min. The envisioned application areas include geometry, image analysis, optimization and learning. Further, we generalize some tropical geometrical objects using weighted lattices. For example, we outline the optimal solution of max-* equations using weighted lattice adjunctions, and apply it to optimal regression for fitting max-* tropical curves on arbitrary data.

 

A Tropical Approach to Neural Networks with Piecewise Linear Activations

Vasileios Charisopoulos
Cornell University

This talk revisits the problem of counting the regions of linearity of piecewise linear neural networks. We treat layers of neural networks with piecewise linear activations as tropical signomials, which generalize polynomials in the so-called (max, +) or "tropical" algebra to the case of real-valued exponents. Motivated by the discussion in (Montufar et. al, 2014), this approach enables us to recover tight bounds on linear regions of layers with ReLU / leaky ReLU activations, as well as bounds for layers with arbitrary convex, piecewise linear activations.

Our approach crucially relies on exploiting a correspondence between regions of linearity and vertices of Newton polytopes, which also enables us to design a randomized method for counting linear regions in practice. This algorithm relies on sampling vertices and places no restrictions on the range of inputs of the neural network, avoiding the overhead of existing exact approaches which rely on solving a large number of linear or mixed-integer programs. Moreover, it extends beyond rectifier networks.

The results presented in the talk are joint work with Petros Maragos.

 
10:00am - 12:00pmMS153, part 1: Symmetry in algorithmic questions of real algebraic geometry
Unitobler, F021 
 
10:00am - 12:00pm

Symmetry in algorithmic questions of real algebraic geometry

Chair(s): Cordian Riener (UiT - The Arctic University of Norway, Norway), Philippe Moustrou (UiT - The Arctic University of Norway, Norway)

Symmetry arises quite naturally in many computational problems and from a computational perspective, it allows to reduce the complexity of problems. The mini-symposium aims to presents various instances of computational problems in real algebraic geometry, where symmetry playes an important role.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Complete positivity and distance-avoiding sets

Fernando de Oliveira Filho
Technical University of Delft

The completely positive matrices form a subcone of the cone of positive semidefinite matrices over which optimization is NP-hard, but which allows us to provide exact formulations for many combinatorial optimization problems: if, for instance, one replaces positive semidefinite matrices by completely positive matrices in the Lovász theta number, the resulting parameter is equal to the independence number instead of being just an upper bound for it.

The Lovász theta number can be extended to distance graphs on infinite spaces, like the unit distance graph of R^n, whose vertices are all points of R^n and in which two vertices are adjacent if they are at distance 1 from each other. In the case of the unit distance graph, the Lovász theta number provides an upper bound for the maximum density m_1(R^n) that a measurable subset of R^n not containing points at distance 1 can have. I will define the cone of completely positive operators on a Hilbert space and show how it can be used to give an exact formulation for parameters such as m_1(R^n), in analogy with the situation of finite graphs.

This exact formulation can then be used to compute better upper bounds and also to give alternative proofs of results such as the Turing computability of m_1(R^n).

(Joint work with Evan DeCorte and Frank Vallentin.)

 

Kissing number of the hemisphere in dimension 8

Maria Dostert
EPFL Lausanne

The kissing number of spherical caps asks for the maximal number of pairwise non-overlapping unit spheres that can simultaneously touch a central spherical cap in n-dimensional Euclidean space. We consider especially the kissing number of the hemisphere in dimension 8. The kissing number of hemispheres provides less symmetries than the kissing number of unit spheres, which makes the problem more difficult.

The kissing number problem of spheres coincides with the problem of finding a maximal spherical code with minimal angular distance pi/3. The famous configuration of 240 points of unit spheres in dimension 8 given by the root lattice E8, which is an optimal spherical code of minimal angular distance pi/3, is unique up to isometry. From these 240 points we get a configuration on the hemisphere with 183 pairwise non-overlapping unit spheres. Bachoc and Vallentin determined an upper bound of 183.012 using semidefinite optimization, hence the kissing number of the hemisphere in dimension 8 is 183.

Using the semidefinite program of Bachoc and Vallentin we obtained a sharp numerical bound of 183. Based on this floating point solution and the configuration given by the E8 lattice we study uniqueness. This is a joint work with David de Laat and Philippe Moustrou.

 

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

David de Laat
MIT

In this talk I will explain how sum-of-squares characterizations and semidefinite programming can be used to obtain improved bounds for quantities related to zeros of the Riemann zeta function. This is based on Montgomery's pair correlation approach. I will show how this connects to the sphere packing problem, and speculate about future improvements.

Joint work with Andrés Chirre and Felipe Gonçalves.

 

Cut polytopes and minors in graphs

Tim Römer
Universität Osnabrück

The study of cuts in graphs is a very interesting topic in discrete mathematics with relations and applications to many other fields like algebraic geometry, algebraic statistics, commutative algebra and combinatorial optimization. Here we focus on cut polytopes and associated algebraic objects. Special attention is given to the (non-)existence of minors of interest in the underlying graph and algebraic as well as geometric consequences of such a fact.

 
10:00am - 12:00pmMS130, part 2: Polynomial optimization and its applications
Unitobler, F022 
 
10:00am - 12:00pm

Polynomial optimization and its applications

Chair(s): Timo de Wolff (Technische Universität Berlin, Germany), Simone Naldi (Université de Limoges, France), João Gouveia (Universidade de Coimbra, Portugal)

The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Moments and convex optimization for analysis and control of nonlinear partial differential equations

Milan Korda, Didier Henrion, Jean-Bernard Lasserre
CNRS-LAAS, Toulouse, France

This talk will present a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is then approximated by the classical Lasserre hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the solutions to these SDPs provide bounds on a specified, possibly nonlinear, functional of the solutions to the PDE; in the case of PDE control, the solutions to these SDPs provide bounds on the optimal value of a given optimal control problem as well as suboptimal feedback controllers. The entire approach is based solely on convex optimization with no reliance on spatio-temporal gridding. The approach is applicable to a very broad class of fully nonlinear PDEs with polynomial data.

 

Two-player games between polynomial optimizers and semidefinite solvers.

Victor Magron1, Mohab Safey El Din2, Jean-Bernard Lasserre1
1CNRS-LAAS, Toulouse, France, 2Sorbonne Université, Paris, France

We interpret some wrong results, due to numerical inaccuracies, already observed when solving semidefinite programming (SDP) relaxations for polynomial optimization, on a double precision floating point solver. It turns out that this behavior can be explained and justified satisfactorily by a relatively simple paradigm. In such a situation, the SDP solver - and not the user - performs some "robust optimization" without being told to do so. In other words the resulting procedure can be viewed as a "max-min" robust optimization problem with two players: the solver which maximizes on a ball of arbitrary small radius, centered at the input polynomial, and the user who minimizes over the original decision variables.

Next, we consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of polynomials, while relying on arbitrary-precision SDP solvers. We provide a perturbation-compensation algorithm computing exact decompositions for polynomials lying in the interior of the SOS cone. First, the user perturbates the input polynomial to obtain an approximate SOS decomposition. Then, one obtains an exact SOS decomposition after compensating the numerical error with the perturbation terms. We prove that this algorithm runs in boolean time, which is polynomial in the degree of the input and simply exponential in the number of variables. We apply this algorithm to compute exact Polya and Putinar's representations, respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets.

 

A Generalization of SAGE Certificates for Constrained Optimization

Riley Murray, Venkat Chandrasekaran
Caltech, Los Angeles, CA, USA

We describe a generalization of the SAGE relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that SAGE conveniently and transparently blends with convex duality, in a manner that SOS does not. Key properties of traditional SAGE relaxations (e.g. sparsity preservation) are retained by this more general approach. We illustrate the utility of this methodology with a range of examples from the global optimization literature, along with a publicly available software package.

 

On positive duality gaps in semidefinite programming

Gabor Pataki
University of North Carolina at Chapel Hill, NC, USA

We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e., different optimal values in the primal and dual problems. These SDPs are considered extremely pathological, they are often unsolvable, and they also serve as models of more general pathological convex programs. We first characterize two variable SDPs with positive gaps: we transform them into a standard form which makes the positive gap easy to recognize. The transformation is very simple, as it mostly uses elementary row operations coming from Gaussian elimination.

We next show that the two variable case sheds light on larger SDPs with positive gaps: we present SDPs in any dimension in which the positive gap is certified by the same structure as in the two variable case. We analyze an important parameter, the singularity degree of the duals of our SDPs and show that it is the largest that can result in a positive gap. We complete the paper by generating a library of difficult SDPs with positive gaps (some of these SDPs have only two variables), and a computational study.

 
10:00am - 12:00pmMS124, part 1: The algebra and geometry of tensors 1: general tensors
Unitobler, F023 
 
10:00am - 12:00pm

The algebra and geometry of tensors 1: general tensors

Chair(s): Yang Qi (University of Chicago, United States of America), Nick Vannieuwenhoven (KU Leuven)

Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. These topics raise challenging computational problems, but also the theory behind them is far from fully understood. Algebraic geometry has already played an important role in the study of tensors. It has shed light on: the ill-posedness of tensor approximation problems, the generic number of decompositions of a rank-r tensor, the number and structure of tensor eigen- and singular tuples, the number and structure of the critical points of tensor approximation problems, and on the sensitivity of tensor decompositions among many others. This minisymposium focuses on recent developments on the geometry of tensors and their decompositions, their applications, and mathematical tools for studying them, and is a sister minisymposium to "The algebra and geometry of tensors 2: structured tensors" organized by E. Angelini, E. Carlini, and A. Oneto.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

The distance function from a real algebraic variety

Giorgio Ottaviani
Università di Firenze

The distance function from a real algebraic variety X is a algebraic function, its degree is twice the Euclidean distance degree of X. Its constant term describes the points at zero Euclidean distance; while on real numbers these are just the points of X, there are additional points with complex entries. When X is projective, the constant term vanishes on the variety dual to X^vee cap Q, where Q is the isotropic quadric. The leading term of the distance function is a scalar when X is transversal to Q, according to a Whitney stratification of X. The important case when X is the variety of rank one tensors is exposed in another talk by Sodomaco, who is coauthor of the above results.

 

Algorithms for rank, tangential and cactus decompositions of polynomials

Alessandra Bernardi
University of Trento

I will present a review of the famous algorithm for symmetric tensor decomposition due to J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas. I will also present a generalization to decompositions of polynomials involving points on the tangential variety of a Veronese variety. I will conclude by showing how the same technique allows to compute the cactus rank and decomposition of any polynomial. This is the outcome of a joint work together with D. Taufer.

 

Pencil-based algorithms for tensor rank decomposition are not stable

Paul Breiding
Max-Planck-Institute for Mathematics in the Sciences

I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositionsis numerically unstable. Algorithm of this class are based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition. The analysis shows that the instability is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1×n2×2 tensors than for the n1×n2×n3 input tensor. Joint work with Carlos Beltran and Nick Vannieuwenhoven.

 

Identifiability of a general polynomial

Francesco Galuppi
Max-Planck-Institute for Mathematics in the Sciences

The study of tensor decompositions is a wonderful topic with powerful applications and a lovely geometric interpretation. One of the most interesting scenarios is identifiability, that is the existence of a unique decomposition for the tensor. Identifiability was conjectured to be a very rare phenomenon. In this joint work with Massimiliano Mella we look at it from a geometric viewpoint and we use birational techniques to completely classify all pairs (n,d) such that the general degree d polynomial in n+1 variables admits a unique decomposition.

 
10:00am - 12:00pmMS174, part 1: Algebraic aspects of biochemical reaction networks
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic aspects of biochemical reaction networks

Chair(s): Alicia Dickenstein (Universidad de Buenos Aires), Georg Regensburger (Johannes Kepler University Linz)

ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Network models and polynomial positivity

Murad Banaji
Middlesex University, London

A number of problems in chemical reaction network theory - and in the study of other networks with model structure - can be restated as questions about the positivity of polynomials, or more generally the emptiness or otherwise of semialgebraic sets. In particular, positivity claims can be used to deduce the absence of particular bifurcations in networks with certain structural features. Sometimes positivity can be determined trivially, but it is not uncommon for polynomials to arise which are positive but in nontrivial ways, namely they do not belong to the smallest "natural" cone of polynomials positive on the set in question. Some problems and examples will be surveyed, and a few results using techniques from real algebraic geometry and exterior algebra will be outlined.

 

Some approaches to understand the parameter region of multistationarity

Elisenda Feliu
University of Copenhagen

In the context of chemical reaction networks, the dynamics of the concentrations in time are modelled by a system of parameter-dependent ordinary differential equations, which typically admit invariant linear subspaces, called stoichiometric compatibility classes. Multistationarity refers to the existence of two positive equilibrium points in some stoichiometric compatibility class. Numerous approaches exist to address the qualitative question of whether a network exhibits multistationarity for at least one choice of parameter values. However, tools and strategies to determine 'when' this is the case, that is, to determine for which parameter values the network is multistationary, have only recently emerged.

In this talk I will focus on conditions on the reaction rate constants that guarantee, or preclude, multistastionarity. I will discuss a result, joint with de Wolff, Kaihnsa, Sturmfels and Yürük, based on the use of Sums of Nonnegative Circuits (SONC). Our benchmark example is the n-site phosphorylation system.

 

On the bijectivity of families of exponential maps

Stefan Müller
University of Vienna

In the setting of generalized mass-action systems, uniqueness and existence of complex-balanced equilibria (in every compatibility class and for all rate constants) are equivalent to injectivity and surjectivity of a certain family of exponential maps. In previous work, we have shown that injectivity can be characterized in terms of sign vectors of the stoichiometric and kinetic-order subspaces, that is, of the coefficient and exponent subspaces given by the family of maps. The negation of the sign-vector condition is equivalent to the existence of multiple complex-balanced equilibria (in some compatibility class and for some rate constant). In this work, we characterize the existence of a unique complex-balanced equilibrium, that is, the bijectivity of the family of exponential maps. As it turns out, the conditions for bijectivity do not only involve sign vectors, but also the exponent subspace itself. Further, we provide sufficient conditions involving only sign vectors or the Newton polytope. In terms of generalized mass-action systems, we provide an extension of the classical deficiency zero theorem.

(Joint work with Josef Hofbauer and Georg Regensburger)

 

An algebraic approach to detecting bistability in chemical reaction networks

Angélica Torres
University of Copenhagen

In recent years, algebraic parameterizations of varieties have been used to study parameter regions where a Chemical Reaction Network has multistationarity. In this work we combine these algebraic parameterizations, the Hurwitz criterion for stability and structural reduction techniques for chemical reaction networks to additionally explore the existence of bistability. The procedure can be used to find parameter regions where bistability arises. In this talk I will present our approach, how to detect bistability in a special case and some examples from cell signaling where our procedure was successfully applied.

 
10:00am - 12:00pmMS164, part 1: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
10:00am - 12:00pm

Algebra, geometry, and combinatorics of subspace packings

Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.
The theme of the first session is "Systems with non-abelian group symmetry."

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Algebra, Geometry, and Combinatorics of Subspace Packings: Gabor-Steiner Equiangular Tight Frames

Emily King
University of Bremen

Desirable traits of subspace arrangements in applications like signal processing and quantum information theory include having large geometric spread between any two subspaces and yielding a resolution of the identity. Methods from algebraic graph theory, real algebraic geometry, symplectic geometry, combinatorial design theory, semidefinite programming, and more can be used to design and characterize such subspace packings. This talk will serve as an introduction to the minisymposium “Algebra, Geometry, and Combinatorics of Subspace Packings.” Gabor-Steiner equiangular tight frames, which are covariant under the Weyl-Heisenberg group and have properties described by different combinatorial designs will also be discussed.

 

Group frames, full spark, and other topics

Romanos-Diogenes Malikiosis
Aristotle University of Thessaloniki

A group frame is the orbit of a vector in a vector space of dimension N under the action of a (projective) linear representation of a finite group. Such a frame satisfies the full spark property, if every selection of N vectors from the frame constitutes a basis.

We will examine whether certain families of group frames satisfy the full spark property, extending the results by the speaker for the Weyl-Heisenberg group (i.e. Gabor frames) and by Oussa-Sheehan for the dihedral case. If time allows, we will also mention other topics as well, for example equiangularity.

This is joint work with Vignon Oussa.

 

Equiangular tight frames from nonabeilan groups

John Jasper
South Dakota State University

Several applications in signal processing require lines through the origin of a finite-dimensional Hilbert space with the property that the smallest interior angle is as large as possible. Packings that achieve equality in the Welch bound are known as equiangular tight frames (ETFs). Since optimal packings often exhibit symmetry, it is natural to expect such packings to be related to groups. Indeed, a popular type of ETFs are the so-called harmonic ETFs, that is, ETFs that arise from the action of an abelian group on a single vector. On the other hand, perhaps the most famous open problem in this area is Zauner's conjecture, which asks for an ETF from the action of the Heisenberg group, which is nonabelian. The theory of harmonic ETFs is fairly well understood as it is equivalent to well-studied objects known as difference sets. The theory of ETFs generated by nonabelian groups is much more mysterious. In this talk we will discuss this theory and present a construction of the first infinite family of ETFs arising from nonabelian groups.

 

SIC-POVM existence and the Stark conjectures

Gene Kopp
University of Bristol

The existence of a configuration of equiangular lines in d-dimensional complex Hilbert space of cardinality achieving the theoretical upper bound of d^2 is known only for finitely many dimensions d. Such configurations have been studied extensively in the context of quantum information theory, in which they are known as symmetric informationally complete positive operator-valued measures (SIC-POVMs).

We give an explicit conjectural construction of SIC-POVMs in an infinite family of dimensions. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. Moreover, in the same family, we prove a conditional result stating that SIC-POVMs exist under a strong algebraic hypothesis about units in a certain number field. The talk will include a worked example in dimension d=5 and an overview of some number-theoretic background necessary to understand the main results.

 
10:00am - 12:00pmMS140, part 3: Multivariate spline approximation and algebraic geometry
Unitobler, F-107 
 
10:00am - 12:00pm

Multivariate spline approximation and algebraic geometry

Chair(s): Michael DiPasquale (Colorado State University, United States of America), Nelly Villamizar (Swansea University)

The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Bivariate Semialgebraic Splines

Frank Sottile1, Michael DiPasquale2
1Texas A&M University, 2Colorado State University

We consider bivariate splines over partitions defined by arcs of irreducible algebraic curves. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the forms defining the edges span a three-dimensional space of forms of degree $n$, then we show that the dimensions can be reduced to the linear case. If the partition is sufficiently generic, we give a formula for the dimension of the spline space in large degree and bound how large the degree must be for the formula to be correct. We also study the dimension of the spline space in some examples where the curves do not satisfy either extreme. The results are derived using commutative and homological algebra. This is joint work with Michael DiPasquale.

 

Geometrically smooth spline bases for geometric modeling

Ahmed Blidia, Bernard Mourrain
Inria

Given a topological complex M with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth splines that satisfy differentiability properties across shared edges. We present new constructions of basis functions of the space of $G^1$- spline functions on quadrangular meshes, which are tensor product bspline functions on each quadrangle and with b-spline transition maps across the shared edges. By analysing the syzygy equation induced by the $G^1$ constraints over a single edge, we show that the separability of the space of $G^1$ splines across an edge allows to determine the dimension and a bases of the space of $G^1$ splines on M. This leads to new explicit construction of basis functions attached to the vertices, edges and faces of M.

The construction of smooth basis functions attached to a topological structure has important applications in geometric modeling. We illustrate it on the fitting of point clouds by $G^1$ splines on quadrangular meshes of complex topology. The ingredients are detailled and experimentation results showing the behavior of the method are presented.

 

Splines, Stable Bundles, and PDE’s

Peter Stiller
Texas A&M University

We will explain a number of connections between certain local and global problems in approximation theory related to spaces of splines and certain stable or semi-stable vector bundles/reflexive sheaves on complex projective spaces. These connections lead to an interesting relationship between the spaces of solutions of certain systems of constant coefficient partial differential equations and the first cohomology group of those vector bundles/reflexive sheaves. Using results of Grothendieck and Shatz, the case of two variables and the projective plane is analyzed. We will also discuss extensions to vector bundles on higher-dimensional projective spaces as they relate to splines and PDE’s in three or more variables.

 

Computing the dimension of spline spaces using homological techniques

Andrea Bressan
University of Oslo

Homological techniques have been successfully employed for computing the dimensions of piecewise polynomial spaces on triangulations and quad meshes in the plane. Examples of applications to other spline spaces will be presented.

 
10:00am - 12:00pmMS149, part 3: Stability of moment problems and super-resolution imaging
Unitobler, F-111 
 
10:00am - 12:00pm

Stability of moment problems and super-resolution imaging

Chair(s): Stefan Kunis (University Osnabrueck, Germany), Dmitry Batenkov (MIT Boston)

Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model.

Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Learning algebraic decompositions using Prony structures

Ulrich v. d. Ohe
University Genova

We propose a framework encompassing variants of Prony's reconstructiong method and clarifying their relations. This includes multivariate Hankel as well as Toeplitz variants for several classes of functions and relative versions.

 

Multidimensional Superresolution in Sonar and Radar Imaging

Annie Cuyt1, Wen-shin Lee2
1University Antwerpen, 2University of Stirling

Some sonar and radar imaging are essentially multidimensional exponential analysis techniques, consisting in identifying the linear coefficients αi and the distinct nonlinear parameters φi, i=1,...,n, in f(x)=Σi αi exp(<φi,x>) from samples f(xj) taken at regularly distributed points xj in the d-dimensional space. Exponential analysis is itself again connected to sparse interpolation from computer algebra, Padé approximation from rational approximation theory and tensor decomposition from numerical multilinear algebra.

We present a multidimensional generalization of a one-dimensional exponential analysis algorithm that: (1) requires the minimal number of (d + 1)n samples (through its connection with sparse interpolation), (2) validates the computed output for the φi (through its connection with Padé approximation), and (3) is robust against outliers. In addition, the samples may be collected at a rate below the classical Nyquist rate and the algorithm is easy to parallellize. We illustrate the working of the algorithm on some simulated examples taken from the engineering literature. The latter is joint work with Ferre Knaepkens and Yuan Hou from the Universiteit Antwerpen.

 

Recovery of surfaces and inference on surfaces: theory & applications to image recovery

Mathews Jacob, Qing Zou
University of Iowa

We introduce a sampling theoretic framework for the recovery of smooth surfaces and functions living on smooth surfaces from few samples. The proposed approach is as a nonlinear generalization of union of subspace models widely used in signal processing. This scheme relies on an exponential lifting of the original data points to feature space, where the features live on union of subspaces. The low-rank property of the features are used to recover the surfaces as well as to determine the number of measurements needed to recover the surface. The low-rank property of the features also provides an efficient approach which resembles a neural network for the local representation of multidimensional functions on the surface; the significantly reduced number of parameters make the computational structure attractive for learning inference from limited labeled training data.

 

Looking beyond Pixels: Continuous-domain Sparse Recovery with an Application to Radioastronomy

Martin Vetterli, Pan Hanjie
EPFL

We propose a continuous-domain sparse recovery technique by generalizing the finite rate of innovation (FRI) sampling framework to cases with non-uniform measurements. We achieve this by identifying a set of unknown uniform sinusoidal samples (which are related to the sparse signal parameters to be estimated) and the linear transformation that links the uniform samples of sinusoids to the measurements. It is shown that the continuous-domain sparsity constraint can be equivalently enforced with a discrete convolution equation of these sinusoidal samples. Then, the sparse signal is reconstructed by minimizing the fitting error between the given and the re-synthesized measurements (based on the estimated sparse signal parameters) subject to the sparsity constraint. Further, we develop a multi-dimensional sampling framework for Diracs in two or higher dimensions with linear sample complexity. This is a significant improvement over previous methods, which have a complexity that increases exponentially with space dimension. An efficient algorithm has been proposed to find a valid solution to the continuous-domain sparse recovery problem such that the reconstruction (i) satisfies the sparsity constraint; and (ii) fits the given measurements (up to the noise level). We validate the flexibility and robustness of the FRI-based continuous-domain sparse recovery in both simulations and experiments with real data in radioastronomy.

 
10:00am - 12:00pmMS150, part 1: Fitness landscapes and epistasis
Unitobler, F-112 
 
10:00am - 12:00pm

Fitness landscapes and epistasis

Chair(s): Kristina Crona (American University, Washington, USA), Joachim Krug (Uni Koeln, Germany), Lisa Lamberti (ETHZ, Switzerland)

Studying relations, effects and properties of modified genes or organisms is an important topic in biology with implications in evolution, drug resistance and targeting, and much more. Biological data can many times be represented in digital form, a mutation has occurred or not, a species is present in an ecological system, or not. A fitness landscape is a function from such bit strings to some measured quality.
A property of fitness landscapes is epistasis, which is a phenomenon describing dependency relations among effects of combinations of modified genes. Polyhedral decompositions, such as cube triangulations induced by fitness landscapes, provide a systematic approach to epistasis.
In this session, we aim at bringing researches of various areas of science together to discuss contact points between applied polyhedral geometry, statistics and biology, and present recent developments in the field.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Introduction to fitness landscapes and epistasis

Lisa Lamberti
ETHZ, Switzerland

How are fitness landscapes used in biology? What type of data is analyzed? Which methods are applicable to this theory?

These are some of the issues we will touch upon in this introductory talk and elaborate further throughout the session. The starting point of this presentation will be given by reviewing the geometric framework to study epistasis proposed by Beerenwinkel et al.(2007). In particular, we will define various forms of genetic interactions of current interest in evolutionary biology and demonstrate their utility. Computational limitations of these approaches will also be discussed.

 

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Holger Eble1, Michael Joswig1, Lisa Lamberti2, William Ludington3
1TU Berlin, Germany, 2ETHZ, Switzerland, 3Carnegie Institution for Science, Baltimore, USA

Beerenwinkel et al.(2007) suggested studying fitness landscapes via regular subdivisions of convex polytopes.Building on their approach we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool. In this way, we provide a concise combinatorial way of processing metric information from epistatic interactions. Using existing Drosophila microbiome data, we demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information where the previous approach is less conclusive.

 

A mechanistic approach to understanding multi-way interactions between mutations

Michael Harms
University of Oregon, USA

An important goal for biologists is construction of quantitative, predictive models relating the genome sequence of an organism (its genotype) to its observed traits (its phenotype). This is a challenging problem. If mutations behave independently, the difference in phenotype between two genotypes that differ at L positions can be described as the sum of the individual effects of all L mutations. Mutations, however, rarely act independently: the effect of a mutation can change depending on the presence or absence of another mutation. We, and others, have even documented extensive three-way and even higher-ordered interactions between mutations. In my talk, I will discuss the evidence for these multi-way interactions, as well as how such interactions undermine models that sum the effects of mutations to predict phenotype from genotype. I will then discuss an alternative, mechanistically informed model, and describe experimental work done in my lab testing predictions of this model. Our work demonstrates a powerful, workable alternative to linear models and points to other classes of models that may be better suited for describing the map between genotype and phenotype.

 

Understanding the biophysics of molecules from large functional assays

Jakub Otwinowski
MPI for Dynamics and Self-Organization, Germany

Quantifying the relationship between a biomolecule's genetic sequence and its biological function is a fundamental problem that addresses how complex molecules can evolve. With statistical models inferred from large numbers of sequence-function pairs I show how a small number of intermediate molecular phenotypes can explain many aspects of sequence-function relationship. From a library of mutated promoter sequences and expression measurements in e.coli lac operon I infer detailed physical interactions between two regulatory proteins. Using a different heuristic approach, I infer fold stability from an thousands of mutated variants of green fluorescent protein. Finally, with a thermodynamic model I infer a detailed energy landscape of a small bacterial protein and separate the effects of mutations on binding and folding stability from 500k variants.

 
10:00am - 12:00pmMS180, part 1: Network coding and subspace designs
Unitobler, F-113 
 
10:00am - 12:00pm

Network coding and subspace designs

Chair(s): Daniele Bartoli (University of Perugia), Anna-Lena Horlemann-Trautmann (University of St. Gallen, Switzerland)

This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

More on exceptional scattered polynomials

Daniele Bartoli
University of Perugia

Let f be an F_q-linear function over F_(q^n). If the F_q-subspace U= { (x^(q^t), f(x)) : x in F_(q^n) } defines a maximum scattered linear set, then we call f a scattered polynomial of index t. We say a function f is an exceptional scattered polynomial of index t if the subspace U associated with f defines a maximum scattered linear set in PG(1, q^(mn)) for infinitely many m. There is a very interesting link between maximum scattered linear sets and the so called maximum rank distance (MRD for short) codes. In particular, a scattered polynomial over F_(q^n) defines an MRD code in (F_q)^(nxn) of minimum distance n-1. Exceptional scattered monic polynomials of index 0 (for q>5) and of index 1 have been already classified. In this work, we investigate the case t>1.

 

The size of linear sets on a finite projective line

Jan de Beule
University of Brussels

A linear set in a finite projective space, i.e. a finite dimensional projective space over a finite field, is a set of points whose defining vectors belong to an additive subgroup of the underlying vector space of the projective space.

Linear sets were introduced by G. Lunardon in the context of the construction of small blocking sets of finite Desarguesian projective planes. Meanwhile, linear sets have been used to construct and/or characterize many substructures of finite projective spaces. Recently, J. Sheekey described a correspondence between certain MRD codes and scattered linear sets on the finite projective line.

After giving briefly some properties of linear sets, and connections with other objects like blocking sets and MRD codes, we will report on joint work with Geertrui Van de Voorde in which we showed a lower bound on the number of points of a linear set on a finite projective line.

 

Rank Metric Codes and Subspace Codes in a Convolutional Setting

Joachim Rosenthal
University of Zurich

Subspace codes have been introduced by Koetter and Kschischang in order to tackle coding problems in the area of random linear network coding. Subspace codes are subsets of a fixed Grassmannian defined over a finite field. This class of codes is also closely related to the class of ``rank metric codes''.

In a first part we will show how rank metric codes induce in a natural way so called rank metric convolutional codes.

We will then report about some basic properties of rank metric convolutional codes.

In a second part we will show how rank metric convolutional codes can be lifted to subspace convolutional codes.

 

Partitions of Matrix Spaces and q-Rook Polynomials

Alberto Ravagnani
University College Dublin

I will describe the row-space and the pivot partition on the space of n x m matrices over GF(q). Both these partitions are Fourier-reflexive and yield invertible MacWilliams identities for matrix codes endowed with the row-space and the pivot enumerators, respectively. Moreover, they naturally give rise to notions of extremality. Codes that are extremal with respect to any of these notions satisfy strong rigidity properties, analogous to those of MRD codes.

The Krawtchouk coefficients of both the row-space and the pivot partition can be explicitly computed using combinatorial methods. For the pivot partition, the computation relies on the properties of the q-rook polynomials associated with Ferrers diagrams, introduced by Garsia/Remmel and Haglund in the 80's. I will describe this connection between codes and rook theory, and present a closed formula for the q-rook polynomial (of any degree) associated to an arbitrary Ferrers board.

The new results in this talk are joint work with Heide Gluesing-Luerssen.

 
10:00am - 12:00pmMS194: Latent graphical models
Unitobler, F-121 
 
10:00am - 12:00pm

Latent graphical models

Chair(s): Piotr Zwiernik (Universitat Pompeu Fabra, Spain)

Algebro-geometric methods have been extensively applied to study probabilistic graphical models. They became particularly useful in the context of graphical models with hidden variables (latent graphical models). Latent variables appear in graphical models in several important contexts: to represent processes that cannot be observed or measured (e.g. economic activity in business cycle dating, ancestral species in phylogenetics), in causal modelling (confounders), and in machine learning (deep learning, dimension reduction).

Graphical models with latent variables lead to sophisticated geometry problems. The simplest examples, like the latent class model, link directly to secant varieties of the Segre variety and low rank tensors. Understanding the underlying geometry proved to be the driving force behind designing new learning algorithms and was essential to understand fundamental limits of these models. This minisession features three speakers who have been leading this research in the last couple of years.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Latent-variable graphical modeling with generalized linear models

Venkat Chandrasekaran
California Institute of Technology

We describe a convex optimization framework for fitting latent-variable graphical models in the class of generalized linear models. We discuss scaling laws under which our framework succeeds in identifying a population model with high probability as well as experimental results with real data. We also highlight natural tradeoffs in our setup between computational resources and sample size. (Joint with Armeen Taeb and Parikshit Shah)

 

Representation of Markov kernels with deep graphical models

Guido Montúfar
University of California Los Angeles

We revisit the topic representational power of deep probabilistic graphical models. We consider directed and undirected models with multiple layers of finite valued hidden variables. We discuss relations between directed and undirected models, as well as relations between deep and shallow models, in relation to the number of layers and variables within layers that are needed and sufficient to express any Markov kernel.

 

Conditional independence statements with hidden variables

Fatemeh Mohammadi
Bristol University

Conditional independence is an important tool in statistical modeling, as, for example, it gives a statistical interpretation to graphical models. In causal reasoning, it is important to know what constraints on the observed variables are caused by hidden variables. In general, given a sub-family of random variables satisfying a list of conditional independence (CI) statements, it is difficult to say which constraints are implied by the CI statements on this sub-family. However, the CI statements correspond to some determinantal conditions on the tensor of joint probabilities of the observed random variables. Hence, the algebraic analogue of this question relates to determinantal varieties and their irreducible decompositions. In a joint project with Ollie Clarke and Johannes Rauh, we generalize the intersection axiom for CI statements, and we study a family of CI statements whose corresponding variety and its irreducible components are all determinantal varieties.

 
10:00am - 12:00pmMS185, part 1: Algebraic Geometry Codes
Unitobler, F-122 
 
10:00am - 12:00pm

Algebraic Geometry Codes

Chair(s): Daniele Bartoli (Univerity of Perugia, Italy), Anna-Lena Horlemann (University of St. Gallen)

The problem of finding good codes is central to the theory of error correcting codes. For many years coding theorists have addressed this problem by adding algebraic and combinatorial structure to C.

In the early 80s Goppa used algebraic curves to construct linear error correcting codes, the socalled algebraic geometric codes (AG codes). The construction of an AG code with alphabet a finite field Fq requires that the underlying curve is Fq-rational and involves two Fq-rational divisors D and G on the curve.

In this minisymposium we will present results on Algebraic Geometry codes and their performances.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Weierstrass semigroups on, and a generalization of the Giulietti-Korchmáros curve

Maria Montanucci
University of Padua

The Giulietti-Korchmáros (GK) curve C is a maximal curve over GF(q^6) that was discovered in 2009. The first topic that is addressed in this talk, concerns the structure of the Weierstrass semigroups of points of this curve. It turns out that there are three possibilities for these semigroups and that the Weierstrass points of the GK curve are exactly the GF(q^6)-rational points. A description of these three possible Weierstrass semigroups will be presented. The GK curve was generalized by Garcia, Stichtenoth and Xing in 2010 in the construction of the so called GGS maximal curves. More precisely they found for each odd n>2 a curve Cn, maximal over GF(q2n). The curve C3 equals the GK curve C. In the second part of this talk a different generalization of the GK curve will be presented. Similarities and differences with the GGS curves will be discussed, especially their genera and automorphism groups. This is a joint work with Peter Beelen.

 

Codes from the GGS maximal curves

Giovanni Zini
University of Milan

For any prime power q and odd integer n≥5, we consider the Fq^2n-maximal curve Xq,n : Z(q^n+1)/(q+1)=Yq^2-Y, Yq+1=Xq+X introduced by Garcia, Güneri, and Stichtenoth, and we construct over Fq^2n dual one-point AG codes C from an Fq^2-rational affine point P of Xq,n.

We study the automorphism group of C starting from the automorphism group of the curve. We determine the Weierstrass semigroup at any affine Fq^2-rational point P of Xq,n and apply this result to the parameters of C; in particular, we compute the Feng-Rao minimum distance of C when q=2. Finally, we apply some constructions known in the literature to our codes, in order to produce families of quantum codes and convolutional codes.

 

An Open Source Environment for Research on AG Codes

Kwankyu Lee
Chosun University

Algebraic Geometry codes are studied for applications as error-correcting codes, to code-based post-quantum cryptosystems and ramp secret-sharing schemes, and so on. Thus they are mathematical objects not only described on papers but to be computed explicitly on computers. As originally defined by Goppa, AG codes are based on algebraic curves and their function fields. Therefore a computing environment for AG codes should allow computations with them as well. Such a computing environment is de facto unique, and it is Magma. Though powerful, the closed nature of the software is an obstacle in spreading the achievements of researchers in this field to other researchers and students. Here we demonstrate the current status of the endeavors to provide an open source computing environment on Sage for algebraic curves, function fields, and AG codes.

 

Multi-point Codes from the GGS Curves

Shudi Yang
Qufu Normal University

This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters [216,190,>= 18] over GF(64) yields a new record.

 
10:00am - 12:00pmMS145, part 2: Isogenies in Cryptography
Unitobler, F-123 
 
10:00am - 12:00pm

Isogenies in Cryptography

Chair(s): Tanja Lange (Eindhoven University of Technology, Netherlands, The), Chloe Martindale (Eindhoven University of Technology, Netherlands, The), Lorenz Panny (Eindhoven University of Technology, Netherlands, The)

The isogeny graph of elliptic curves over finite fields has long been a subject of study in algebraic geometry and number theory. During the past 10 years several authors have shown multiple applications in cryptology. One interesting feature is that systems built on isogenies seem to resist attacks by quantum computers, making them the most recent family of cryptosystems studied in post-quantum cryptography.

This mini-symposium brings together presentations on cryptosystems built on top of isogenies, their use in applications, and different approaches to the cryptanalysis, including quantum cryptanalysis.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Constant-time isogeny implementations

David Jao
University of Waterloo

We discuss recent progress in implementing isogeny-based cryptosystems in constant time to resist side-channel attacks. We propose an implementation of supersingular isogeny Diffie-Hellman (SIDH) for complete Edwards curves. While the use of Edwards curves does not actually provide a faster implementation of SIDH, it does provide some security benefits against side-channel attacks. In addition, we present an optimized, constant-time software library for the Commutative supersingular isogeny Diffie-Hellman key exchange (CSIDH) scheme proposed by Castryck et al., targeting 64-bit ARM processors, and designed to offer resistance against SPA and DPA side-channel attacks.

SIDH results are joint work of Reza Azarderakhsh, Elena Bakos Lang, David Jao, and Brian Koziel.

CSIDH results are joint work of Amir Jalali, Reza Azarderakhsh, Mehran Mozaffari Kermani, and David Jao.

 

Isogeny-based cryptography: a cryptanalysis perspective

Christophe Petit
Birmingham University

In this talk I will survey known results on the security of isogeny-based protocols.
 

Fast isogeny-based signatures

Frederik Vercauteren
KU Leuven

Although several isogeny based signature schemes have been proposed, none of them can be considered really practical. In this talk I will describe a signature scheme based on CSIDH that has moderate public key sizes and is very efficient, in particular, signing a message only requires a couple of hundreds of milliseconds.

 

Orienting supersingular isogeny graphs

David Kohel
University of Marseilles

Supersingular isogeny graphs have been used in the Charles–Goren–Lauter cryptographic hash function and the supersingular isogeny Diffie–Hellman (SIDH) protocol of De Feo and Jao. A recently proposed alternative to SIDH is the commutative supersingular isogeny Diffie–Hellman (CSIDH) protocol, which in which the isogeny graph is first restricted to Fp-rational curves E and Fp-rational isogenies then oriented by the quadratic subring Z[π] ⊂ End(E) generated by the Frobenius endomorphism π : E → E. We introduce a general notion of orienting supersingular elliptic curves and their isogenies, and use this as the basis to construct a general oriented supersingular isogeny Diffie-Hellman (OSIDH) protocol.

 
1:30pm - 2:30pmIP06: Jonas Peters: Data Science and Causality
vonRoll, Fabrikstr. 6, 001 
 
1:30pm - 2:30pm

Data Science and Causality

Jonas Peters

University of Copenhagen, Denmark

In data science, we are used to infer models that predict the observed data as well as possible. In causality, we try to understand how a system reacts under interventions, e.g., in gene knock-out experiments. Bringing together data science and causality may yield two benefits. (1) One may try to learn causal models from observations, and (2) enhancing standard regression or classification techniques with causal ideas may yield models that generalize better to unseen experiments. In this talk, we introduce the concept of causality, discuss ideas for addressing the above goals, and mention open problems that could benefit from an algebraic geometry point of view. No prior knowledge about causality is required.

 
1:30pm - 2:30pmIP06-streamed from 001: Jonas Peters: Data Science and Causality
vonRoll, Fabrikstr. 6, 004 
2:30pm - 3:00pmCoffee break
Unitobler, F wing, floors 0 and -1 
3:00pm - 5:00pmMS188: Probability and randomness in commutative algebra and algebraic geometry
Unitobler, F005 
 
3:00pm - 5:00pm

Probability and randomness in commutative algebra and algebraic geometry

Chair(s): Dane Wilburne (Brown University, United States of America), Christopher O'Neill (San Diego State University)

Randomness has long been used to study polynomials. Several classical instances include Lit- tlewood and Offord’s examination of the expected number of real roots of an algebraic equation defined by random coefficients, as well as work of Kac and Kouchnirenko on varieties defined by random coefficients on a fixed Newton polytope support. Additionally, the use of smooth analysis, which measures the expected performance of an algorithm under slight random perturbations of worst-case inputs, has been used in the context of algebraic geometry. The aim of this minisymposium is to highlight a recent surge of interactions between the fields of probability and commutative algebra/algebraic geometry, in which questions of expected (average, typical) or unlikely (rare, non-generic) behavior of ideals and varieties are studied formally using probability distributions. Recent work has seen the successful application of techniques from statistics and probabilistic combinatorics in this setting. Our goal is to bring researchers working in this intersection together to share their work and form potential new collaborations.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

What can be predicted in algebraic geometry?

Lily Silverstein
UC Davis

This talk explores how supervised machine learning can be used to predict the algebraic and combinatorial properties of polynomial ideals prior to doing a computation. This is practical for fast approximations of algebraic invariants, and also for the problem of "algorithm selection." When several algorithms exist for performing an exact computation, we train a neural network to automatically select the fastest algorithm, on a case-by-case basis, by learning features of the input that are predictive of algorithm performance.

Joint work with Jesus De Loera, Robert Krone, and Zekai Zhao.

 

Degree of Random Monomial Ideals

Jay Yang
University of Minnesota

In joint work with Lily Silverstein and Dane Wilburne, we investigate the behavior of the standard pairs of a random monomial ideal. We then use this to explore the degree and arithmetic degree of random monomial ideals.

 

Stochastic Exploration of Real Varieties

David Kahle
Baylor University

Nonlinear systems of polynomial equations arise naturally in many applied settings. The solution sets to these systems over the reals are often positive dimensional spaces that in general may be very complicated yet have very nice local behavior almost everywhere. Standard methods in real algebraic geometry for describing positive dimensional real solution sets include cylindrical algebraic decomposition and numerical cell decomposition, both of which can be costly to compute in many practical applications. In this talk, we communicate recent progress towards a Monte Carlo framework that provides a probabilistic method for exploring such real solution sets. After describing how to construct probability distributions whose mass focuses on a variety of interest, we show how state-of-the-art Hamiltonian Monte Carlo methods can be used to sample points near the variety that may then be magnetized to the variety using endgames. We conclude by showcasing trial experiments using practical implementations of the method in the probabilistic programming language Stan.

 

Random numerical semigroups

Christopher O'Neill
San Diego State University

A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in the process.

 
3:00pm - 5:00pmMS189, part 1: Geometry and topology in applications.
Unitobler, F006 
 
3:00pm - 5:00pm

Geometry and topology in applications.

Chair(s): Jacek Brodzki (University of Southampton, United Kingdom), Heather Harrington (University of Oxford)

This symppsium will bring together leading practitioners, mid-carreer scientists as well as PhD students and postdoctoral fellows who are interested in the theory and practice of the applications of geometry and topology in real life problems. The spectrum of possible applications is very wide, and covers the sciences, biology, medicine, materials science, and many others. The talks will address the theoretical foundations of the methodology as well as the state of the art of geometric and topological modelling.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Topological data analysis in materials science

Yasu Hiraoka
Kyoto University

Topological data analysis (TDA) is an emerging concept in applied mathematics in which we characterize “shape of data” using topological methods. In particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In my talk, I will explain our recent activity of TDA on materials science, e.g. glass, polymer, granular system, iron ore sinter etc. By developing several new mathematical tools based on quiver representations, inverse analysis, and machine learnings, we can explicitly characterize significant geometric and topological (hierarchical) features embedded in those materials, which are practically important for materials properties.

 

Optimal transport in tropical geometric phylogenetic tree space

Anthea Monod
Columbia University

Recent results by Monod et al. (2018) establish that palm tree space---that is, the space of phylogenetic trees in the tropical geometric construction, and endowed with the tropical metric---is a metric measure space with well-defined properties for probability and statistics on sets of phylogenetic trees. With the tropical metric as ground metric, we construct foundations for optimal transport theory on palm tree space. In particular, we build the Wasserstein-p metric which allows for the comparison of probability distributions of different random variables on palm tree space. We study the cases where p = 1, which gives an efficient way to compute geodesics, and p = 2, which gives deeper insight into the geometry of palm tree space.

This is joint work with Wuchen Li (UCLA) and Bo Lin (Georgia Tech).
 

Primary distance for multipersistence

Ezra Miller
Duke

When persistent homology is used to summarize data objects, distances between the resulting persistence modules serve as proxies for distances between the data objects themselves. In the presence of more than one parameter, module distances are complicated by the rich algebraic structure of multipersistence. In particular, unboundedness of the set of parameters presents problems with integration, interleaving, and other measures. Primary decomposition and algebraic operations related to it provide canonical (functorial) ways to extract bounded parameter sets, yielding convergence for existing measures that are based on integration. In addition, primary distances isolate from mixtures of multipersistence types pure contributions that would, in many existing measures, otherwise introduce bias when truncation or enforced decay are used without taking into account the algebraic structure.

 

Outlier robust subsampling techniques for persistent homology

Bernadette Stolz
Oxford

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

 
3:00pm - 5:00pmMS200, part 3: From algebraic geometry to geometric topology: Crossroads on applications
Unitobler, F007 
 
3:00pm - 5:00pm

From algebraic geometry to geometric topology: crossroads on applications

Chair(s): Jose Carlos Gomez Larrañaga (CIMAT), Renzo Ricca (University of Milano-Bicocca), De Witt Sumners (Florida State University)

The purpose of the Minisymposium "From Algebraic Geometry to Geometric Topology: Crossroads on Applications" is to bring together researchers who use algebraic, combinatorial and geometric topology in industrial and applied mathematics. These methods have already seen applications in: biology, physics, chemistry, fluid dynamics, distributed computing, robotics, neural networks and data analysis.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Time-reversal homotopical properties of concurrent systems

Eric Goubault
École Polytechnique

Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this talk we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal (as first noticed by e.g. K. Hess and L. Fajstrup). We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects.These refined invariants, natural systems with a composition pairing, enjoy lots of interesting properties, as first noticed by Timothy Porter. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.Joint work with Philippe Malbos and Cameron Calk.

 

Efficient computation of multiparameter persistent homology

Abraham Martín del Campo Sánchez
CONACYT-CIMAT

We will present an efficient implementation of an algorithm to compute multiparametric persistent homology. The algorithm uses algebraic techniques and was originally proposed by Chacholski, Scolamiero, and Vaccarino. During the talk, we will explain the different reformulations of the definition of multiparametric persistence that give rise to the algorithm we corrected and implemented. This is joint work with Oliver Gäfvert (KTH) and Nina Otter (UCLA).

 

Classification of Streamline Topologies for Hamiltonian vector fields and its applications to Topological Flow Data Analysis

Takashi Sakajo
Kyoto University

We have developed a classification theory for structurally stable Hamiltonian vector fields on multiply connected planar domains in the presence of a uniform flow, which is a model of two-dimensional incompressible fluid flows. The theory enables us to assign a unique sequence of letters and a tree structure, called maximal words, and Reeb graphs, to every topological streamline structure of the Hamiltonian vector fields. They are intuitively interpretable to those who are not familiar with mathematics. An automatic conversion algorithm is now available as a computer software, and it is thus applicable to massive flow pattern data obtained by numerical simulations and/or physical measurements in fluid science, engineering and medical studies. By extracting global topological information from flow data, one is expected to figure out latent knowledge that are not recognized by experts in those fields so far. In addition, we have also developed a mathematical theory describing all possible global transitions of streamline topologies, without exceptions, through marginal structurally unstable Hamiltonian vector fields in terms of the changes of the sequence of letters. By simply comparing them, we predict the change of global flow patterns that could possibly happen in future.
Based on the classification theory, we introduce a new way of topological data analysis, called Topological Flow Data Analysis (TFDA). Owing to TFDA, long-time evolutions of flows (or Hamiltonian vector fields), whose data size often exceeds more than giga-bytes, is drastically compressed into a small size of text data expressing the change of streamline topologies, which is amenable to statistical and/or time-series analysis, and machine learning for global topological information with ease. We show some applications to medical images of heart flows and flow patterns in meteorology. We also show another example illustrating that TFDA is available to create a data-driven model predicting a complex flow phenomenon.

 

Robot motion planning and equivariant cohomology

Michael Farber
Queen Mary, University of London

The task of creating software AI for autonomous robots has an interesting topological aspect. A motion planning algorithm for a mechanical system can be represented by a section of a specific fibration and the complexity of such a section is measured by numerical invariants TC(X) and TC_r(X). Here X denotes the configuration space of the system and r>1 is an integer. I will describe recent results about computing the numbers TC(X) and TC_r(X) in the case when the space X is aspherical, i.e. has vanishing higher homotopy groups. The talk will include some results obtained jointly with S. Mescher, M. Grant, J.Oprea and G. Lupton.

 
3:00pm - 5:00pmMS166, part 2: Computational aspects of finite groups and their representations
Unitobler, F011 
 
3:00pm - 5:00pm

Computational aspects of finite groups and their representations

Chair(s): Armin Jamshidpey (University of Waterloo, Canada), Eric Schost (University of Waterloo, Canada), Mark Giesbrecht (University of Waterloo, Canada)

The theory of finite groups and their representations is not only an interesting topic for mathematicians but also provides powerful tools in solving problems in science. New computational tools are making this even more feasible. To name a few, one may find applications in physics, coding theory and cryptography. On the other hand representation theory is useful in different areas of mathematics such as algebraic geometry and algebraic topology. Due to this wide range of applications, new algorithmic methods are being developed to study finite groups and their representations from a computational perspective.

Recent developments in computer algebra systems and more specifically computational linear algebra, provide tools for developments in computational aspects of finite groups and their representations. The aim of this minisymposium is to gather experts in the area to discuss the recent achievements and potential new directions.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Calculations with Symplectic Hypergeometric Groups

Alexander Hulpke
Colorado State University

We study monodromy groups of hypergeometric equations that can be considered as matrix groups generated by the companion matrices of pimitive pairs of integral polynomials. It is known that these groups preseve a nondegenerate symplectic form and are Zariski-dense in the respective symplectic group, and much recent work has concentrated on the question whether particular groups are arithmetic, that is have finite index in the symplectic group.
Using matrix group algorithms on congruence images, we are able to calculate indices of the arithmetic closures of these groups, The information obtained also enables us to prove arithmeticity in some cases through a coset enumeration.
This is joint work with Dane Flannery (Galway) and Alla Detinko (St Andrews).

 

Algorithmic factorization of noncommutative polynomials

Viktor Levandovskyy
RWTH Acchen University

We are interested in factorizing polynomials over non-commutative rings. Let us start with a field K and a finitely presented associative K-algebra A, which is a domain.

There are at least two distinct notions of a factorization of polynomials over A. One of them originates from the ring theory (N. Jacobson, P.M. Cohn) and uses a weak notion of association relation (called left or right similarity), what is at the same time hard to approach algorithmically. On the contrary, in applications we'd like to use the classical association relation, i.e. when two elements differ by a factor, which is nonzero central unit.

I will present long-seeked conditions for a given algebra A to be a finite factorization domain, i.e. a domain, where every nonunit has at most finite number of factorizations. Over such domains a factorization procedure thus becomes into an algorithm. Examples, bounds and counterexamples will be given. Over the well-known class of ubiquitous G-algebras (a.k.a. PBW a.k.a. algebras of solvable type), we provide a factorization algorithm, its' smarter graded-driven version for graded algebras and a factorizing Groebner algorithm. All of these are implemented in Singular:Plural (www.singular.uni-kl.de). We view the factorizing Groebner algorithm as the only general possibility to obtain a weaker analogon to the primary decomposition from the commutative algebra.
Recent complexity results and applications of the mentioned algorithms will be presented.

 

Finite groups of Lie type and computer algebra

Meinolf Geck
Universität Stuttgart

The classification of finite simple groups highlights the importance of studying the class of groups in the title. These are defined in terms of algebraic groups over algebraically closed fields of positive characteristic. We discuss a few recent examples where computer algebra methods have played a significant role in developing and establishing new results.

 

Classification of regular parametrized one-relation operads

Murray Bremner
University of Saskatchewan

I will discuss an application of representation theory of symmetric groups to algebraic operads and nonassociative algebra. J.-L. Loday introduced parametrized one-relation operads (POROs): symmetric operads generated by one binary operation subject to one relation showing how to reassociate a left-normed product into to a linear combination of right-normed products:

(ab)c =Σσ in S_3 xσ aσ ( bσ cσ ) ( xσ in Q ).

For some values of xσ, the operad is regular: for all n its homogeneous component of degree n is isomorphic to the regular representation of Sn. (Equivalently, the corresponding free algebra on a vector space V is isomorphic as a graded vector space to the tensor algebra of V.) The familiar examples of regular POROs are those governing associative, Poisson, Leibniz, and Zinbiel algebras. We use computer algebra based on a constructive version of representation theory of symmetric groups to classify all regular POROs. We show that in addition to the above four operads, the only other example is the nilpotent operad. This is joint work with Vladimir Dotsenko of Trinity College Dublin.

 
3:00pm - 5:00pmMS160, part 3: Numerical methods for structured polynomial system solving
Unitobler, F012 
 
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

Improvements in the understanding of numerical methods for dense polynomial system solving led to the complete solution of Smale's 17th problem. At this point, it remains an open challenge to achieve the same success in the solution of structured polynomial systems: explain the typical behavior of current algorithms and devise polynomial-time algorithms for computing roots of polynomial systems. In this minisymposium, researchers will present the current progress on applying numerical methods to structured polynomial systems.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Certifying solutions to a square system involving analytic functions

Michael Burr1, Kisun Lee2, Anton Leykin2
1Clemson University, 2Georgia Institute of Technology

In this talk, we introduce two different approaches for the certification of solutions to a square system built from univariate analytic functions. The first approach is based on alpha-theory, and the second is based on interval Newton's Method. The certification methods are based on the existence of oracles from input analytic functions. We show that these two oracles exist for D-finite functions. Finally, we compare these two approaches using our SageMath implementation for D-finite function cases.

 

Toric witness sets for sampling positive dimensional solution sets of polynomial systems

Tianran Chen
Auburn University at Montgomery

The linear slicing technique and the construction of "witness sets" proposed by Sommese and Wampler in 1996 form the foundation for numerical algebraic geometry. They are the indispensable tools for numerically finding and studying positive dimensional (non-isolated) solution sets defined by systems of polynomial equations. Combining the strength of the polyhedral homotopy method and the toric approach for studying algebraic sets, we propose a general framework for efficiently sampling positive dimensional solution sets that can potentially take advantage of the sparsity in the system. The practical usefulness of this approach is demonstrated through an application to the "power-flow equation" from electric engineering.

 

Farewell to Weyl: Condition-based analysis with a Banach norm in numerical algebraic geometry

Josue Tonelli-Cueto1, Felipe Cucker2, Alperen Ergür1
1TU Berlin, 2City University of Hong Kong

Condition-based complexity analyses of numerical algorithms in algebraic geometry seem to rely heavily on inner product norms, such as the Weyl norm. This contrasts with the situation in numerical linear algebra where it is common to use plenty of Banach norm that do not come from an inner product. We show that similar advantages can be obtained in numerical algebraic geometry by showing how such an analysis can be carried out with respect a Banach norm in various settings, obtaining substancial improvements over the known complexity estimates for linear homotopy continuation and grid based methods. This is on going work with Felipe Cucker and Alperen A. Ergür.

 

Singular polynomial eigenvalue problems are not ill-conditioned

Martin Lotz1, Vanni Noferini2
1Warwick University, 2Aalto University

Numerical methods are not supposed to work on ill-posed inputs, and even less so if the function to be computed is discontinuous. Yet, there are examples where arbitrary small perturbations in the input can lead to literally any function value, but where standard numerical algorithms that are oblivious to the special structure of the problem work perfectly fine. Neither the classical nor the stochastic theories of conditioning are adequate to predict the typical forward accuracy in such cases. Motivated by this limitation, and using singular polynomial eigenvalue problems as running example, we define and study weak worst-case and a weak stochastic condition numbers. This new theory can be a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input are not finite. We illustrate how such condition number can be estimated and used in practice.

 
3:00pm - 5:00pmMS167, part 2: Computational tropical geometry
Unitobler, F013 
 
3:00pm - 5:00pm

Computational tropical geometry

Chair(s): Kalina Mincheva (Yale University), Yue Ren (Max Planck Institute for Mathematics in the Sciences, Germany)

This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Connectivity of tropical varieties

Diane Maclagan1, Josephine Yu2
1Warwick University, 2Georgia Tech

The standard algorithm to compute tropical varieties makes crucial use of the fact that the tropicalization of an irreducible variety is connected. I will discuss joint work with Josephine Yu showing that the tropicalization of a d-dimensional irreducible variety satisfies a stronger d-connectedness property.

 

Tropical convex hull of polytopes

Cvetelina Hill1, Sara Lamboglia2, Faye Pasley Simon3
1Georgia Tech, 2Goethe Universität Frankfurt, 3North Carolina State University

Tropical convexity has been mostly focused on tropical convex hull of finitely many points, i.e., tropical polytopes. Moreover there has been some work on polytropes which are convex tropical polytopes. In this talk I will consider the tropical convex hull of polytopes and polyhedra. I will show that these are convex sets and that in some cases tconv(conv(S))=conv(tconv(S)) and tconv(pos (S))=pos(tconv(0,S)) for a finite set S. This will lead the way to compute the tropical convex hull of a tropical variety.

 

Algorithmic questions around tropical Carathéodory

Georg Peter Loho
London School of Economics

Since Imre Bárány found the colourful version of Carathéodory's theorem in 1982, many combinatorial generalizations and algorithmic variations have been considered. This ranges from variations of the colour classes to different notions of convexity. We take a closer look at the tropical convexity version of this theorem. We provide new insights on colourful linear programming and matroid generalizations from a tropical point of view, by considering additional sign informations. We focus on explicit constructions for 'colourful simplices'. The difficulty of the arising algorithmic questions ranges from greedily solvable to NP-hard.

 

Convergent Puiseux series and tropical geometry of higher rank


Ben Smith
Queen Mary University of London

Tropical hypersurfaces arising from polynomials over the Puiseux series are well studied and well understood objects. The picture becomes less clear when considering Puiseux series in multiple indeterminates. Unlike their rank one counterparts, these higher rank tropical hypersurfaces are not ordinary polyhedral complexes, but we shall see they still have a large amount of structure. Moreover, by restricting to convergent Puiseux series we show how one can describe them via the rank one tropical hypersurfaces arising from substitution of indeterminates. We will also consider a couple of applications of this framework, including a new viewpoint for stable intersection in the vein of symbolic perturbation.

 
3:00pm - 5:00pmMS158, part 1: Structured sums of squares
Unitobler, F021 
 
3:00pm - 5:00pm

Structured sums of squares

Chair(s): James Saunderson (Monash University, Australia), Mauricio Velasco (Universidad de los Andes)

A description of a nonnegative polynomial as a sum of squares gives a concise proof of its nonnegativity. Computationally, such sum-of-squares decompositions are appealing because we can search for them by solving a semidefinite feasibility problem. This connection means that optimization and decision problems arising in a range of areas, from robotics to extremal combinatorics, can be reformulated as, or approximated with, semidefinite optimization problems.

This minisymposium highlights the roles of various kinds of additional structures, including symmetry and sparsity, in understanding when (structured) sum of squares decompositions do and do not exist. It will also showcase interesting connections between sums of squares and a range of areas, such as extremal combinatorics, dynamical systems and control, and algorithms and complexity theory.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Learning dynamical systems with side information

Amir Ali Ahmadi, Bachir El Khadir
Princeton University

In several safety-critical applications, one has to learn the behavior of an unknown dynamical system from noisy observations of a very limited number of trajectories. For example, to autonomously land an airplane that has just gone through engine failure, limited time is available to learn the modified dynamics of the plane before appropriate control action can be taken. Similarly, when a new infectious disease breaks out, few observations are initially available to understand the dynamics of contagion.

In situations of this type where data is limited, it is essential to exploit “side information”---e.g. physical laws or contextual information---to assist the task of learning. We present a mathematical formalism of the problem of learning a dynamical system with side information, where side information can mean a concrete collection of local or global properties of the dynamical system. We show that sum of squares optimization is particularly suited for learning a dynamical system that best agrees with the observations and respects the side information.

Based on joint work with Bachir El Khadir (Princeton).

 

Convergence analysis of measure-based bounds for polynomial optimization on compact sets

Lucas Slot, Monique Laurent
CWI Amsterdam

We investigate the convergence rate of a hierarchy of measure-based upper bounds introduced by Lasserre (2011) for the minimization of a polynomial f over a compact set K. These bounds are obtained by searching for a degree 2r sum-of-squares density function h minimizing the expected value of f over K.

The convergence rate to the global minimum of f over K is known to be in O(1/r^2) for special sets K like the box, ball and sphere, so we consider here general compact sets. We show a convergence rate in O((log r)/r) when K satisfies a minor geometric condition and a rate in O(((log r)/r)^2) when K is a convex body, improving on the current best known bounds for these cases. These results can be refined when making assumptions on the order of f at a global minimizer.

Our analysis relies on combining tools from convex geometry and approximation theory, making use in particular of approximations of the Dirac delta function by fast-decreasing polynomials.

This is based on joint work with Monique Laurent.

 

Sums-of-squares for extremal discrete geometry on the unit sphere

Frank Vallentin
Universität zu Köln

In this talk I will show how one can apply sum-of-squares techniques for various extremal geometric problems on the unit sphere, especially finding thinnest coverings or spherical designs.

 

Computing spectral bounds for geometric graphs via polynomial optimization

Philippe Moustrou
UiT - The Arctic University of Norway

A powerful lower bound on the chromatic number of a finite graph is the spectral bound due to Hoffman, which is related to the eigenvalues of the adjacency matrix of the graph. This bound has been generalized by Bachoc, DeCorte, Oliveira and Vallentin to infinite graphs. In this talk we describe how this bound can be adapted to two particular problems arising from geometry:

- Given a norm what is the largest density of a subset of the n-dimensional real space that does not contain any pair of points such that the norm of their difference is 1? This problem is closely related to the determination of the famous chromatic number of the plane.

- What is the least number of colors needed to color the interiors of the cells of the tessellation associated with the Voronoi cell of a given lattice, in such a way that two cells sharing a facet do not receive the same color?

After introducing these two problems, we show how to compute the spectral bound by solving a polynomial optimization problem using sums of squares.

 
3:00pm - 5:00pmMS195, part 3: Algebraic methods for convex sets
Unitobler, F022 
 
3:00pm - 5:00pm

Algebraic methods for convex sets

Chair(s): Rainer Sinn (Freie Universität Berlin, Germany), Greg Blekherman (Georgia Institute of Technology), Daniel Plaumann (Technische Universität Dortmund), Yong Sheng Soh (Institute of High Performance Computing, Singapore), Dogyoon Song (Massachusetts Institute of Technology)

Convex relaxations are extensively used to solve intractable optimization instances in a wide range of applications. For example, convex relaxations are prominently utilized to find solutions of combinatorial problems that are computationally hard. In addition, convexity-based regularization functions are employed in (potentially ill-posed) inverse problems, e.g., regression, to impose certain desirable structure on the solution.

In this mini-symposium, we discuss the use of convex relaxations and the study of convex sets from an algebraic perspective. In particular, the goal of this minisymposium is to bring together experts from algebraic geometry (real and classical), commutative algebra, optimization, statistics, functional analysis and control theory, as well as discrete geometry to discuss recent connections and discoveries at the interfaces of these fields.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Average-Case Algorithm Design Using Sum-of-Squares

Pravesh Kothari
Princeton University

Finding planted "signals" in random "noise" is a theme that captures problems arising in several different areas such as machine learning (compressed sensing, matrix completion, sparse principal component analysis, regression, recovering planted clusters), average-case complexity (stochastic block model, planted clique, random constraint satisfaction), and cryptography (attacking security of pseudorandom generators). For some of these problems (e.g. variants of compressed sensing and matrix completion), influential works in the past two decades identified the right convex relaxation and techniques for analyzing it that guarantee nearly optimal (w.r.t. to the information theoretic threshold) recovery guarantees. However these methods are problem specific and do not immediately generalize to other problems/variants.

 

Fitting Semidefinite-Representable Sets to Support Function Evaluations

Yong Sheng Soh
Institute of High Performance Computing, Singapore

The geometric problem of estimating an unknown convex set from its support function evaluations arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets. These methods, however, do not allow for the incorporation of prior structural information about the underlying set and the resulting estimates become increasingly more complicated to describe as the number of measurements available grows. We address these shortcomings by describing and analyzing a framework based on searching over structured families of convex sets that are specified as linear images of the free spectrahedron. Our results highlight the utility of our framework in settings where the number of measurements available is limited and where the underlying set to be reconstructed is non-polyhedral. A by-product of our framework that arises from taking the appropriate dual perspective is a numerical tool for computing the optimal approximation of a given convex set as a spectrahedra of fixed size.

 

Measuring Optimality Gap in Conic Programming Approximations with Gaussian Width

Dogyoon Song
Massachusetts Institute of Technology

It is a common practice to approximate hard optimization problems with simpler convex programs for the purpose of computational efficiency. However, this often introduces a nontrivial optimality gap between the true optimum and the approximate values. We evaluate the quality of approximations by studying the Gaussian width of the underlying convex cones as a generic measure to evaluate the optimality gap. Specifically, we consider two canonical examples: (a) approximation of the positive semidefinite (PSD) cone by the (scaled) diagonally dominant (DD) cone ($$DD^n, SDD^n$$); and (b) the sequence of hyperbolic cones, $$mathbb{R}^{n,(k)}$$, which are the derivative relaxations of the nonnegative orthant. We show that there is a significant gap between the width of PSD cone and (S)DD cone ($$Theta(n^2)$$ vs $$Theta(n log n)$$). On the other hand, (perhaps, surprisingly) the width of the hyperbolic cones remains almost invariant in the linear regime of relaxation ($$k = alpha n$$for $$alpha < 1$$).

 

False discovery and its control for low rank estimation

Armeen Taeb
California Institute of Technology

Cross-Validation (CV) is a commonly employed procedure that selects a model based on predictive evaluations. Despite its widespread use, empirical and theoretical studies have shown that CV produces overly complex models that consist of many false detections. As such, decades of research in statistics has lead to model selection techniques that assess the extent to which the estimated model signifies discoveries about an underlying phenomena. However, existing approaches rely on the discrete structure of the decision space and are not applicable in settings where the underlying model exhibits a more complicated structure such as low-rank estimation problems. We address this challenge via a geometric reformulation of the concept of true/false discovery, which then enables a natural definition in the low-rank case. We describe and analyze a generalization of the Stability Selection method of Meinshausen and Buehlmann to control for false discoveries in low-rank estimation, and we demonstrate its utility via numerical experiments. Concepts from algebraic geometry (e.g. tangent spaces to determinantal varieties) play a central role in the proposed framework.

 
3:00pm - 5:00pmMS124, part 2: The algebra and geometry of tensors 1: general tensors
Unitobler, F023 
 
3:00pm - 5:00pm

The algebra and geometry of tensors 1: general tensors

Chair(s): Yang Qi (University of Chicago, United States of America), Nick Vannieuwenhoven (KU Leuven)

Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. These topics raise challenging computational problems, but also the theory behind them is far from fully understood. Algebraic geometry has already played an important role in the study of tensors. It has shed light on: the ill-posedness of tensor approximation problems, the generic number of decompositions of a rank-r tensor, the number and structure of tensor eigen- and singular tuples, the number and structure of the critical points of tensor approximation problems, and on the sensitivity of tensor decompositions among many others. This minisymposium focuses on recent developments on the geometry of tensors and their decompositions, their applications, and mathematical tools for studying them, and is a sister minisymposium to "The algebra and geometry of tensors 2: structured tensors" organized by E. Angelini, E. Carlini, and A. Oneto.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Bounds on the rank general and special results

Enrico Carlini
Politecnico di Torino

We will see how algebra and geometry can lead to intetesting bounds on the rank with some special focus on the symmetric case.

 

On the identifiability of ternary forms beyond the Kruskal's bound

Elena Angelini
Universita di Siena

I will describe a new method to determine the minimality and identifiability of a Waring decomposition A of a specific ternary form T, even beyond the range of applicability of Kruskal's criterion. This method is based on the study of the Hilbert function and Cayley-Bacharach of A. As an application, we will see the cases of ternary optics and nonics. (joint works with Luca Chiantini).

 

Variants of Comon's problem via simultaneous ranks

Alessandro Oneto
Universitat Politècnica de Catalunya

The rank of a tensor is the smallest length of an additive decomposition as sum of decomposable tensors. Whenever the tensor has symmetries, it can be useful to consider additive decompositions whose summands respect the same symmetries. A symmetric tensor can be regarded as an element of the space of partially symmetric tensors for different choices of partial symmetries and one can ask what are the relations among the different (partially symmetric) ranks which arise in this way. This was the object of a famous question raised by Comon, who asked whether the tensor rank of a symmetric tensor equals its symmetric rank. This problem received a great deal of attention in the last few years. Affirmative answers were derived under certain assumptions, but recently Shitov provided an example where Comon’s question has negative answer. In a joint work with Fulvio Gesmundo and Emanuele Ventura (arXiv:1810.07679), we approached a partially symmetricversion of Comon’s question investigating relations among the partially symmetric ranks of a symmetric tensor. In particular, by exploiting algebraic tools as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. In this way, we try to understand to what extent the symmetries of a tensor affect its rank. In this communication, after a brief introduction of the topic, I will present the main tools and results of our work.

 

Complex best r-term approximations almost always exist in finite dimensions

Lek-Heng Lim
University of Chicago

We show that in finite-dimensional nonlinear approximations, the best r-term approximant of a function almost always exists over complex numbers but that the same is not true over the reals. Our result extends to functions that possess special properties like symmetry or skew-symmetry under permutations of arguments. For the case where we use separable functions for approximations, the problem becomes that of best rank-r tensor approximations. We show that over the complex numbers, any tensor almost always has a unique best rank-r approximation. This extends to other notions of tensor ranks such as symmetric rank and alternating rank, to best r-block-terms approximations, and to best approximations by tensor networks. When applied to sparse-plus-low-rank approximations, we obtain that for any given r and k, a general tensor has a unique best approximation by a sum of a rank-r tensor and a k-sparse tensor with a fixed sparsity pattern. The existential (but not the uniqueness) part of our result also applies to best approximations by a sum of a rank-r tensor and a k-sparse tensor with no fixed sparsity pattern, as well as to tensor completion problems.

 
3:00pm - 5:00pmMS183, part 2: Polyhedral geometry methods for biochemical reaction networks
Unitobler, F-105 
 
3:00pm - 5:00pm

Polyhedral geometry methods for biochemical reaction networks

Chair(s): Elisenda Feliu (University of Copenhagen, Denmark), Stefan Müller (University of Vienna)

This minisymposium focuses on geometric objects arising in the study of parametrized polynomial ODEs given by biochemical reaction networks. In particular, we consider recent work that employs techniques from convex, polyhedral, and tropical geometry in order to extract properties of interest from the ODE system and to relate them to the choice of parameter values.

Specific problems covered in the minisimposium include the analysis of forward-invariant regions of the ODE system, the determination of parameter regions for multistationarity or oscillations, the performance of model reduction close to metastable regimes, and the characterization of unique existence of equilibria using oriented matroids.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Algorithmic Aspects of Computing Tropical Prevarieties Parametrically

Andreas Weber
University of Bonn

Tropical prevarieties of polynomial vector fields arising from chemical reaction networks have been found useful in the analysis of such systems. For fixed parameter values it has been shown that even for systems having 30 dimensions or more tropical prevarieties can be computed. As for biochemical reaction networks there is typically high parameter uncertainty (and the qualitative dependency of the system on parameters is also of high interest) parametric computations of tropical prevarieties are very desireable. Based on experiments with the PtCut software we compare the outcome (also in terms of needed computational ressources) of simple grid sampling strategies, for which the curse of dimensionality fully applies, against computations in a polyhedral setting including parameters. The presented results are joint work with Christoph Lüders

 

Empiric investigations on the number and structure of solution polytopes for tropical equilibration problems arising from biological networks

Christoph Lüders
University of Bonn

A quite recent approach to assist in solving ODE systems (with polynomial vector fields) lies in methods of tropical geometry. Tropical geometry transforms polynomials into piece-wise linear functions and still preserves some structure of the original polynomial (like the number of roots). The polynomial is transformed into a set of polyhedra and multiple of such sets can be intersected to find common roots. Thus tropical geometry problems are combinatorial problems.

We have developed the "PtCut" program to compute the tropical prevariety resp. tropical equilibrium of a polynomial system. Details of use and implementation of PtCut are presented. Large models can cause a lot of polyhedra to be created and calculating their intersection can be very slow. Several methods of remedy are shown.

Since ODE systems often arise in biology, we wrote a free SBML-parser and used PtCut to compute tropical solutions of the curated models listed in the BioModels database. Statistics about the number of solutions, their dimension, their number of connected components and run-time are presented.

 

Perturbations of exponents of exponential maps: robustness of bijectivity

Georg Regensburger
Johannes Kepler University Linz

For generalized mass-action systems, uniqueness and existence of complex-balanced equilibria (in every compatibility class and for all rate constants) are equivalent to injectivity and surjectivity of a certain family of exponential maps. In this talk, we discuss when the existence of a unique solution is robust with respect to small perturbations of the exponents. In particular, we give a characterization in terms of sign vectors of the stoichiometric and kinetic-order subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. This characterization allows to formulate a robust deficiency zero theorem for generalized mass-action systems. As a corollary, we show that the classical deficiency theorem for mass-action kinetics by Horn, Jackson, and Feinberg is robust with respect to small perturbations of the kinetic orders (from the stoichiometric coefficients).


(Joint work with Josef Hofbauer and Stefan Müller)

 

Weakly reversible mass-action systems with infinitely many positive steady states

Balázs Boros
University of Vienna

In 2011, Deng, Feinberg, Jones, and Nachman investigated the number of positive steady states of weakly reversible mass-action systems. They proposed a proof of the existence of a positive steady state in each positive stoichiometric class. Furthermore, they claimed that there can only be finitely many positive steady states in each positive stoichiometric class. Recently, I provided a complete and clearer proof of the existence part. Moreover, together with Craciun and Yu, I constructed examples with infinitely many positive steady states within a positive stoichiometric class, thereby disproving the finiteness part. In this talk, I will sketch our method that produces such mass-action systems.

 
3:00pm - 5:00pmMS154, part 3: New developments in matroid theory
Unitobler, F-106 
 
3:00pm - 5:00pm

New developments in matroid theory

Chair(s): Alex FInk (Queen Mary), Ivan Martino (Northeastern University, United States of America), Luca Moci (Bologna)

The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015).

The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Characterizing quotients of positroids

Anastasia Chavez
UC Berkeley

We characterize quotients of specific families of positroids. Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this paper, we make use of two of these objects to combinatorially characterize when certain positroids are quotients. Furthermore, we conjecture a general rule for quotients among arbitrary positroids on the same ground set.

 

Algebraic matroids and flocks

Rudi Pendavingh
TU Eindhoven

Let $K$ be a field, $E$ a finite set, and $Xsubseteq K^E$ an algebraic variety. Then $M(X)$ is the matroid on ground set $E$ in which a set $Fsubseteq E$ is independent if and only if the projection ${(x_i: i in F): xin X}$ is a dominant subset of $K^F$. In general it is difficult to decide if a given matroid $M$ is algebraic over $K$, that is if $M=M(X)$ for some variety $Xsubseteq K^E$.

We have recently found that if the field $K$ has positive characteristic $p$, then the variety $X$ determines further structure on $M(X)$ which comprises information on the generic tangent spaces of $X$ as well as a family of closely related varieties. This additional structure can either be cast as a matroid over a hyperfield, or as a {em flock}, which is essentially a labelling of the cells of a tropical linear space by linear subspaces of $K^E$.

We show how this gives useful necessary conditions on the algebraicity of matroids.

This is joint work with Guus Bollen and Jan Draisma.

 

Tropical Ideals

Jeffrey Herschel Giansiracusa
Swansea

The scheme-theoretic approach to tropical geometry has motivated the study of tropical ideals, which are sequences of (valuated) matroids $M_i$ on the monomials of a polynomial ring that form an ideal in the sense that $x_j M_i subset M_{i+1}$. While the class of arbitrary ideals can behave very badly, tropical ideals exhibit many nice properties, while also presenting many new features, challenges, and mysteries. There are realizable tropical ideals, meaning that they are formed by tropicalizing classical ideals, and there are non-realizable tropical ideals. Three interesting questions are:

1. What invariants of a classical ideal are encoded in its associated tropical ideal?
2. How does the tropicalization of an ideal change as the ideal changes (moving within the Hilbert scheme)?
3. How can one construct non-realizable tropical ideals?

In this talk I will discuss examples, progress on each of these questions.

 
3:00pm - 5:00pmMS136, part 1: Syzygies and applications to geometry
Unitobler, F-107 
 
3:00pm - 5:00pm

Syzygies and applications to geometry

Chair(s): Laurent Busé (INRIA Sophia Antipolis), Yairon Cid Ruiz (Universitat de Barcelona), Carlos D'Andrea (Universitat de Barcelona)

In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces

Fatmanur Yildrim
INRIA Sophia Antipolis, France

I will present a new algebraic approach for computing the orthogonal projection of a point onto a rational algebraic surface embedded in the three dimensional projective space, which is a joint work with Nicolás Botbol, Laurent Busé and Marc Chardin. Our approach amounts to turn this problem into the computation of the finite fibers of a generically finite trivariate rational map whose source space is either bi-graded or tri-graded and which has one dimensional base locus: the congruence of normal lines to the rational surface. This latter problem is solved by using certain syzygies associated to this rational map for building matrices that depend linearly in the variables of the three dimensional ambient space. In fact, these matrices have the property that their cokernels at a given point p in three dimensional space are related to the pre-images of the p via the rational map. Thus, they are also related to the orthogonal projections of p onto the rational surface. Then, the orthogonal projections of a point are approximately computed by means of eigenvalues and eigenvectors numerical computations.

 

Complete intersection points in product of projective spaces

Navid Nemati
Université Pierre et Marie Curie

I will report on ongoing project with Marc Chardin. we study the bigraded Hilbert function of complete intersection sets of points in $mathbb{P}^n times mathbb{P}^m$. We give a sharp lower bound for the stabilization of the bigraded Hilbert function. In addition, we show that, in a specific and pretty large region, the bigraded Hilbert Function only depends upon the degree of the forms defining the points. Finally, we consider the case where the forms defining the points are chosen generically. In this case we show that the natural projections to $mathbb{P}^n$ and $mathbb{P}^m$ are one-to-one.

 

Fibers of rational maps and Jacobian matrices

Marc Chardin
Université Pierre et Marie Curie

A rational map $varphi$ from a projective space of dimension m to another is defined by homogeneous polynomials of a common degree d. We establish a linear bound in terms of d for the number of (m 1)-dimensional fibers of $varphi$, by using ideals of minors of the Jacobian matrix. This is joint work with S. Dale Cutkosky and Tran Quang Hoa.

 

Syzygies and the geometry of rational maps (introductory talk)

Laurent Busé
INRIA Sophia Antipolis

During the past years, the analysis of the syzygies (i.e. the algebraic relations of first order) between the equations defining a geometric object leaded to important advances on many problems lying at the interface of commutative algebra and algebraic geometry, motivated in large part by computer aided assisted computations. In this introductory talk, I will provide an overview of a range of methods and results on the study of the geometric properties of rational maps by means of the syzygies of their defining polynomials, in particular on the understanding of their image and fibers. The computational aspects and the relevance of these results in the field of geometric modeling will also be discussed.

 
3:00pm - 5:00pmMS175, part 2: Algebraic geometry and combinatorics of jammed structures
Unitobler, F-111 
 
3:00pm - 5:00pm

Algebraic geometry and combinatorics of jammed structures

Chair(s): Anthony Nixon (Lancaster), Louis Theran (St Andrews)

The minisymposium will combine the classical rigidity theory of linkages in discrete and computational geometry with the theory of circle packing, and patterns, on surfaces that arose from the study of 2- and 3-manifolds in geometry and topology. The aim being to facilitate interaction between these two areas. The classical theory of rigidity goes back to work by Euler and Cauchy on triangulated Euclidean polyhedra. The general area is concerned with the problem of determining the nature of the configuration space of geometric objects. In the modern theory the objects are geometric graphs (bar-joint structures) and the graph is rigid if the configuration space is finite (up to isometries). More generally one can consider tensegrity structures where distance constraints between points can be replaced by inequality constraints. The theory of (circle, disk and sphere) packings is vast and well known, with numerous practical applications. Of particular relevance here are conditions that result in the packing being non-deformable (jammed) as well as recent work on inversive distance packings. These inversive distance circle packings generalised the much studied tangency and overlapping packings by allowing ``adjacent'' circles to be disjoint, but with the control of an inversive distance parameter that measures the separation of the circles. The potential for overlap between these areas can be easily seen by modelling a packing of disks in the plane by a tensegrity structure where each disk is replaced by a point at its centre and the constraint that the disks cannot overlap becomes the constraint that the points cannot get closer together.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Rigid realizations of planar graphs with few locations in the plane

Csaba Király
Eotvos Lorand

A d-dimensional framework is a pair (G, p), where G=(V, E) is a graph and p is a map from V to the d-dimensional Euclidean space. An infinitesimal motion of (G, p) is another map from V to R^d such that moving each point of the framework in that direction does not change the distances corresponding to edges in the first order. The framework is infinitesimally rigid if all of its infinitesimal motions correspond to some isometries of R^d.

Laman (1970) characterized the infinitesimal rigidity of bar-joint frameworks in the plane when the framework is in generic position, that is, when the coordinates of the points are algebraically independent over the field of rationals. Adiprasito and Nevo (2018) recently asked the following question: Which graph classes have infinitesimally rigid realizations for each of its members on a fixed constant number of points in R^d. They showed that triangulated planar graphs have such realizations on 76 points in R^3, however, for each constant c and for d>1, there always exists a graph in the class of generically rigid graphs in R^d that cannot be realized as an infinitesimally rigid bar-joint framework on any c points in R^d.

Based on the above results, it is a natural question whether planar graphs which are generically rigid in the plane have an infinitesimally rigid realization on a constant number of points of the plane. The main result of my talk is that every planar graph which is generically rigid in the plane has an infinitesimally rigid realization on 26 points of the plane. Moreover, given any set of 26 points in the plane such that the coordinates of the points are algebraically independent over the field of rationals, one can find an infinitesimally rigid realization of any rigid planar graph on that set.

 

Global rigidity of linearly constrained frameworks

Anthony Nixon
Lancaster

A (bar-joint) framework (G,p) in R^d is the combination of a graph G and a map p assigning positions to the vertices of G. A framework is rigid if the only edge-length-preserving continuous motions of the vertices arise from isometries of R^d. The framework is globally rigid if every other framework with the same edge lengths arises from isometries of R^d. Both rigidity and global rigidity, generically, are well understood when d=2. A linearly constrained framework in R^d is a generalisation of framework in which some vertices are constrained to lie on one or more given hyperplanes. Streinu and Theran characterised rigid linearly constrained generic frameworks in R^2 in 2010. In this talk I will discuss an analogous result for the global rigidity of linearly constrained generic frameworks. This is joint work with Hakan Guler and Bill Jackson.

 

Hyperbolic polyhedra and discrete uniformization

Boris Springborn
TU Berlin

We will explain how Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric is equivalent to a discrete conformal uniformization theorem for spheres, and how both can be proved in a constructive way using a convex variational principle.

 

Symmetric frameworks in normed spaces

Derek Kitson
Lancaster

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all lp spaces with pnot=2). Complete combinatorial characterisations are obtained for half-turn rotation in the l1 and l-infinity plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight graphs. This is joint work with Anthony Nixon and Bernd Schulze.

 
3:00pm - 5:00pmMS150, part 2: Fitness landscapes and epistasis
Unitobler, F-112 
 
3:00pm - 5:00pm

Fitness landscapes and epistasis

Chair(s): Kristina Crona (American University, Washington, USA), Joachim Krug (Uni Koeln, Germany), Lisa Lamberti (ETHZ, Switzerland)

Studying relations, effects and properties of modified genes or organisms is an important topic in biology with implications in evolution, drug resistance and targeting, and much more. Biological data can many times be represented in digital form, a mutation has occurred or not, a species is present in an ecological system, or not. A fitness landscape is a function from such bit strings to some measured quality.
A property of fitness landscapes is epistasis, which is a phenomenon describing dependency relations among effects of combinations of modified genes. Polyhedral decompositions, such as cube triangulations induced by fitness landscapes, provide a systematic approach to epistasis.
In this session, we aim at bringing researches of various areas of science together to discuss contact points between applied polyhedral geometry, statistics and biology, and present recent developments in the field.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Shape theory, landscape topography and evolutionary dynamics

Joachim Krug, Malvika Srivastava
Uni Koeln, Germany

The increasing availability of highthroughput data has lead to an upsurge of interest in the quantitative characterization of fitness landscapes and the epistatic interactions they encode. After a brief introduction to the fitness landscape concept and its empirical basis, the talk will focus on a comparison between the geometric shape theory and more conventional combinatorial and graphtheoretic approaches. We will explore to what extent the geometric shape of a landscape constrains its topography and guides the dynamics of populations evolving on it, addressing in particular the potential of shape theory to predict the relative performance of recombining and nonrecombining populations. The investigation is based on probabilistic ensembles of fitness landscapes, primarily the houseofcards model comprising uncorrelated random fitness values.

 

Graphs, polytopes, and unpredictable evolution

Kristina Crona
American University, Washington, USA

Fitness graphs (cube orientations) and triangulations capture different aspects of fitness landscapes, in some sense analogous to first and second derivatives. However, the graphs are informative also about higher order interactions. By using graphs one can relate unpredictable fitness and unpredictable evolution. Specifically, higher order epistasis is associated with many peaks in the fitness landscapes. This result was verified by an exhaustive search of 193,270,310 4cube graphs distributed on 511,863 isomorphism classes, and applications of graph theory (including Hall's marriage theorem and chromatic polynomials). Fitness graphs can provide some intuition for genetic recombination, in particular for a category of landscapes where the benefit of recombination is dramatic. However, the limitation of fitness graphs is apparent since the effect of recombination is highly sensitive for curvature. Related open problems will be discussed.

 

Computational complexity as an ultimate constraint on evolution

Artem Kaznatcheev
University of Oxford, UK

Experiments show that evolutionary fitness landscapes can have a rich combinatorial structure due to epistasis. For some landscapes, this structure can produce a computational constraint that prevents evolution from finding local fitness optima -- thus overturning the traditional assumption that local fitness peaks can always be reached quickly if no other evolutionary forces challenge natural selection. Here, I introduce a distinction between easy landscapes of traditional theory where local fitness peaks can be found in a moderate number of steps and hard landscapes where finding local optima requires an infeasible amount of time. Hard examples exist even among landscapes with no reciprocal sign epistasis; on these semi-smooth fitness landscapes, strong selection weak mutation dynamics cannot find the unique peak in polynomial time. More generally, on hard rugged fitness landscapes that include reciprocal sign epistasis, no evolutionary dynamics -- even ones that do not follow adaptive paths -- can find a local fitness optimum quickly. Moreover, on hard landscapes, the fitness advantage of nearby mutants cannot drop off exponentially fast but has to follow a power-law that long term evolution experiments have associated with unbounded growth in fitness. Thus, the constraint of computational complexity enables open-ended evolution on finite landscapes. Knowing this constraint allows us to use the tools of theoretical computer science and combinatorial optimization to characterize the fitness landscapes that we expect to see in nature. I present candidates for hard landscapes at scales from single genes, to microbes, to complex organisms with costly learning (Baldwin effect) or maintained cooperation (Hankshaw effect). Just how ubiquitous hard landscapes (and the corresponding ultimate constraint on evolution) are in nature becomes an open empirical question.

 

Tropical Principal Component Analysis and its Applications to Phylogenomics

Ruriko Yoshida1, Leon Zhang2, Xu Zhang3
1Naval Postgraduate School, USA, 2University of California, Berkeley, USA, 3University of Kentucky, USA

Principal component analysis is a widely-used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes. This is joint work with Leon Zhang and Xu Zhang.

 
3:00pm - 5:00pmMS155, part 1: Massively parallel computations in algebraic geometry
Unitobler, F-113 
 
3:00pm - 5:00pm

Massively parallel computations in algebraic geometry

Chair(s): Janko Böhm (TU Kaiserlautern, Germany), Anne Frühbis-Krüger (Leibniz Universität Hannover)

Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

GPI-Space - Fraunhofer’s integrated solution to solve big problems on ultra scale machines

Franz-Josef Pfreundt, Mirko Rahn, Alexandra Carpen-Amarie
Fraunhofer ITWM

High Performance Computing is an essential prerequisite for numerous modern scientific discoveries, which often rely on carefully tuned software stacks to harness computing power across thousands of resources and to handle massive amounts of data. One of the main challenges in this field is to enable domain scientists to take advantage of this huge computing power, while hiding the complexity of the efficient management of such resources.

This talk will introduce GPI-Space, a workflow-management system for parallel applications, designed to automatically coordinate scalable, parallel executions in large, complex environments. A key advantage of GPI-Space is the separation it provides between the automatic management of parallel executions and the description of the problem-specific computational tasks and inter-dependencies.

The talk will highlight the features that make GPI-Space a suitable run-time environment for algebraic geometry. More specifically, it will be discussed how parallel patterns in algorithms can be exploited by GPI-Space, with a focus on its description language (based on Petri nets), which allows scientists to model domain-specific applications independently of the execution environment. Finally, it will be presented how GPI-Space was used on top of the Singular computer algebra system to speed up the execution of algorithms for topics such
as algebraic geometry, or singularity theory.

 

Using Petri nets for parallelizing algorithms in algebraic geometry

Lukas Ristau
TU Kaiserslautern / Fraunhofer ITWM

In theory, smoothness of an algebraic variety is checked by the classical Jacobian criterion. In many practical contexts, however, a direct application of this criterion is infeasible, in particular, if the codimension of the variety in its ambient space is large. A new hybrid smoothness test was recently suggested by Böhm and Frühbis-Krüger, which is based on the termination criterion from Hironaka's proof of resolution of singularities. This algorithm creates a sufficiently fine covering with affine charts, such that a relative version of the Jacobian criterion can be applied in each chart. In this talk, a massively parallel version of the algorithm is presented which has been implemented using Singular and GPI-Space.

This is joint work with Janko Böhm, Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt and Mirko Rahn.

 

Parallel enumeration of triangulations

Lars Kastner
TU Berlin

Triangulations of point configurations are a key tool in many areas of mathematics. In tropical geometry, enumeration of all regular triangulations of certain point configurations gives rise to classification results. We will present the software mptopcom for computing triangulations of point configurations. Its core algorithm reverse search allows it to overcome the memory restraint that prevented previous software to succeed. Furthermore it is able to run in parallel.

If there is a group acting on our point configuration, then it is often sufficient to enumerate orbits of triangulations rather than all triangulations. To this end we need an effective method of comparing triangulations up to symmetry. The method introduced in mptopcom avoids computing full orbits and hence is able to exploit the group action to improve performance.

 

Module intersection method for multi-loop Feynman integral reduction

Yang Zhang
Max Planck Institute for Physics, Munich

We aim at a bottleneck problem of high energy physics, the linear reduction of multi-loop Feynman integrals. Our idea is to use module intersection computations in algebraic geometry to greatly simplify the Feynman integral linear relations, and then to apply a sparse linear algebra method for the reduction. With our method, the open problem of complete linear reduction of the 2-loop 5-point nonplanar hexagon-box Feynman integral was solved.

 
3:00pm - 5:00pmMS139, part 1: Combinatorics and algorithms in decision and reason
Unitobler, F-121 
 
3:00pm - 5:00pm

Combinatorics and algorithms in decision and reason

Chair(s): Liam Solus (KTH Royal Institute of Technology, Sweden), Svante Linusson (KTH Royal Institute of Technology)

Combinatorial, or discrete, structures are a fundamental tool for modeling decision-making processes in a wide variety of fields including machine learning, biology, economics, sociology, and causality. Within these various contexts, the goal of key problems can often be phrased in terms of learning or manipulating a combinatorial object, such as a network, permutation, or directed acyclic graph, that exhibits pre-specified optimal features. In recent decades, major break-throughs in each of these fields can be attributed to the development of effective algorithms for learning and analyzing combinatorial models. Many of these advancements are tied to new developments connecting combinatorics, algebra, geometry, and statistics, particularly through the introduction of geometric and algebraic techniques to the development of combinatorial algorithms. The goal of this session is to bring together researchers from each of these fields who are using combinatorial or discrete models in data science so as to encourage further breakthroughs in this important area of mathematical research.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

(Machine) Learning Non-Linear Algebra

Jesus De Loera
University of California, Davis

Machine Learning has been used successfully to improve algorithms in Optimization and Computational Logic. By training a neural network one can predict the shape of the answer or select the best parameters to run an algorithm. In this presentation I discuss some experience with applying machine learning tools to improve algorithms manipulating multivariable polynomial systems.

 

Network Flows in Semi-Supervised Learning via Total Variation Minimization

Alexander Jung
Aalto University

We study the connection between methods for semi-supervised learning for partially labeled network data and network flow optimization. Many methods for semi-supervised learning are based on minimizing the total variation of graph signals induced by labels of data points. These methods can be interpreted as flow optimization on the network-structure underlying the data. Using basic results from convex duality we show that the dual problem of network Lasso is equivalent to maximizing the network flow over the data network. Moreover, the accuracy of network Lasso depends crucially on the existence of sufficiently large network flows between labeled data points. We also provide an interpretation of a primal-dual implementation of network Lasso as distributed flow maximization which bears some similarity with the push–relabel maximum flow algorithm.

 

Scalably vertex-programmable ideological forests from certain political twitterverses around US (2016), UK(2017) and Swedish (2018) national elections

Raazesh Sainudiin
Uppsala University

Using customised experimental designs via Twitter Streaming and REST APIs, we collected status updates in Twitter around three national elections. Dynamic retweet networks were transformed into empirical geometrically weighted directed graphs, where every node is a user account and every edge accounts for the number of retweets of one use by another, with a natural probabilistic interpretation. Distributed vertex programs were then used to find the most retweeted paths from every user in the population to a given set of subsets of users (subpopulations of interest). Using a pairwise-distance induced by such paths we build a population retweet-based ideological forest. This statistic can be presented while preserving the privacy of the users and attempting to increase self-awareness about how one's own ideological profiles and social norms are formed and influenced in social media. Concrete hypothesis tests around the 2016 US presidential election, SPLC-defined US hate groups and interference by Russian political bots will be the driving empirical skeleton of the talk.

 

The Kingman Coalescent as a density on a space of trees

Lena Walter
Freie Universität Berlin

Randomly pick n individuals from a population and trace their genealogy backwards in time until you reach the most recent common ancestor. The Kingman n-Coalescent is a probabilistic model for the tree one obtains this way. The common definitions are stated from a stochastic point of view. The Kingman Coalescent can also be described by a probability density function on a space of certain trees. We describe this approach and extend it to the Multispecies Coalescent model by relating population genetics to polyhedral and algebraic geometry. This is joint work in progress with Christian Haase.

 
3:00pm - 5:00pmMS134, part 5: Coding theory and cryptography
Unitobler, F-122 
 
3:00pm - 5:00pm

Coding theory and cryptography

Chair(s): Alessio Caminata (University of Neuchâtel, Switzerland), Alberto Ravagnani (University College Dublin, Ireland)

The focus of this proposal is on coding theory and cryptography, with emphasis on the algebraic aspects of these two research fields. Error-correcting codes are mathematical objects that allow reliable communications over noisy/lossy/adversarial channels. Constructing good codes and designing efficient decoding algorithms for them often reduces to solving algebra problems, such as counting rational points on curves, solving equations, and classifying finite rings and modules. Cryptosystems can be roughly defined as functions that are easy to evaluate, but whose inverse is difficult to compute in practice. These functions are in general constructed using algebraic objects and tools, such as polynomials, algebraic varieties, and groups. The security of the resulting cryptosystem heavily relies on the mathematical properties of these. The sessions we propose feature experts of algebraic methods in coding theory and cryptography. All levels of experience are represented, from junior to very experienced researchers.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Classifications of some partial MDS codes

Anna-Lena Horlemann-Trautmann
University of St. Gallen

Partial MDS codes are optimal locally recoverable codes, used for distributed storage systems. We will present some classification results of these codes for certain parameter sets. One of these results gives a relation to classical MDS codes, from which we can derive results about the neccessary minimal field size for the existence of these codes. The second classification gives a relation to projective lines in general position and hence a geometric point of view for these codes.

 

Batch properties of Affine Cartesian Codes

Felice Manganiello
Clemson University

Batch codes were introduced by Ishai et al. in 2004 and are useful for information retrieval. In this seminar we examine the properties of affine cartesian codes as batch codes. Starting from their local properties, we deduce a partition of the evaluation points into buckets that allows multiple independent users to simultaneously retrieve information.

 

Improved quantum codes from the Hermitian curve

Olav Geil
Aalborg University

We apply the CSS construction and Steane's enlargement to construct quantum codes from the Hermitian curve. Using improved information on the classical minimum distances of the involved nested codes and employing improved code constructions we obtain quantum codes that are much better than what could be obtained by using only one-point algebraic geometric codes in combination with the Goppa bound. We construct both asymmetric and symmetric codes. Our work includes closed formula estimates on the dimension of order bound improved Hermitian codes. This is joint work with René Bødker Christensen.

 

Concatenated constructions of LCD and LCP of codes

Cem Güneri
Sabancı University

Linear complementary dual (LCD) codes are codes which intersect their dual trivially. These codes, and their generalizations called linear complementary pair (LCP) of codes, have drawn attention lately due to their applications in the context of side channel and fault injection attacks in cryptography. It is known that LCD codes have higher density in the family of all linear codes when the alphabet size is large. So, using such codes over large finite fields (extension field) to obtain similar codes over small finite fields (base field) is a reasonable strategy. In this respect, concatenation is a natural technique to try, although finding concatenations that preserve LCD or LCP properties of codes over an extension, when descending to the base field, is a nontrivial problem. The problem of interest in this talk is to find such suitable concatenations. Results we will present have been obtained in joint works with Carlet, Özbudak, Saçıkara and Solé.

 
3:00pm - 5:00pmMS132, part 4: Polynomial equations in coding theory and cryptography
Unitobler, F-123 
 
3:00pm - 5:00pm

Polynomial equations in coding theory and cryptography

Chair(s): Alessio Caminata (University of Neuchâtel, Switzerland), Alberto Ravagnani (University College Dublin, Ireland)

Polynomial equations are central in algebraic geometry, being algebraic varieties geometric manifestations of solutions of systems of polynomial equations. Actually, modern algebraic geometry is based on the use of techniques for studying and solving geometrical problems about these sets of zeros. At the same time, polynomial equations have found interesting applications in coding theory and cryptography. The interplay between algebraic geometry and coding theory is old and goes back to the first examples of algebraic codes defined with polynomials and codes coming from algebraic curves. More recently, polynomial equations have found important applications in cryptography as well. For example, in multivariate cryptography, one of the prominent candidates for post-quantum cryptosystems, the trapdoor one-way function takes the form of a multivariate quadratic polynomial map over a finite field. Furthermore, the efficiency of the index calculus attack to break an elliptic curve cryptosystem relies on the effectiveness of solving a system of multivariate polynomial equations. This session will feature recent progress in these and other applications of polynomial equations to coding theory and cryptography.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Linearized Polynomials in Finite Geometry and Rank-Metric Coding

John Sheekey
University College Dublin

Linearized polynomials arise naturally in various areas of finite geometry, coding theory, and cryptography. In particular, most known constructions for good codes in the rank metric arise from studying properties of linearized polynomials. In this talk we will give an overview of the applications of these polynomials, as well as recent results towards characterising their number of roots, and present some open problems.

 

Quantum Algorithms for Optimization over Finite Fields and Applications in Cryptanalysis

Xiao-Shan Gao
Academy of Mathematics and Systems Science, Chinese Academy of Sciences

In this talk, we present quantum algorithms for two fundamental computation problems: solving polynomial systems and optimization over finite fields. The quantum algorithms can solve these problems with any given success probability and have complexities polynomial in the size of the input and the condition number of certain polynomial system related to the problem. So, we achieved exponential speedup for these problems when their condition numbers are small. We apply the quantum algorithm to the cryptanalysis of the stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, the multivariate public key cryptosystems, the lattice based cipher NTRU, and show that they are secure under quantum algebraic attack only if the condition numbers of the corresponding equation systems are large.

 

On the Complexity of ``Superdetermined'' Minrank Instances

Daniel Cabarcas
Universidad Nacional de Colombia

The Minrank (MR) problem is a computational problem closely related to attacks on code- and multivariate-based schemes. The MR problem is, given m matrices and a target rank r, to determine whether there exists a linear combination of the matrices with rank at most r. The Kipnis-Shamir (KS) approach to MR is to solve the quadratic system of equations that arises from the observation that the dimension of the right kernel of a rank r matrix of size p times q is q-r by setting the entries of a kernel basis as variables. I will present some recent results on the complexity of the KS approach. I will focus on a particular set of instances that yield a very overdetermined system. I show how to construct non-trivial syzygies through the analysis of the Jacobian of the resulting system, with respect to a group of variables. The resulting complexity estimate for such instances is tighter than other approaches. For example, for the HFE cryptosystem, the speedup is roughly a square root. This talk is based on a paper by the same name with my coauthors Javier Verbel, John Baena, Ray Perlner and Daniel Smith-Tone, that appeared on PQCrypto 2019.

 

MinRank Problems Arising from Rank-based Cryptography

Ray Perlner
NIST

Rank-based cryptosystems such as the second round candidates for NIST's post-quantum standardization process, ROLLO and RQC, have a number of desirable features, such as good performance and key size while defending against all currently known classical and quantum attacks. Nonetheless, these cryptosystems, and the underlying Rank Syndrome Decoding(RSD) problem have been less studied in the literature than competing lattice and code-based cryptosystems and their underlying security assumptions. Parameters for rank-based cryptosystems are currently set using the support trapping attack of Gaborit, Ruatta, and Schrek. However, it is possible that approaches relating the Rank Syndrome decoding problem to polynomial-based approaches to solving the MinRank Problem, such as minors and Kipnis-Shamir modeling may give better cryptanalysis for some parameters. The polynomial systems arising in these cases have a number of interesting features that distinguish them from MinRank problems that arise in multivariate cryptography. In particular 1) The number of matrices is quadratic rather than linear in the dimension of the matrices, which generally results in a solving degree that is significantly higher than the degree of regularity when an algebraic approach is used and 2) There is extra structure in the MinRank instances arising from RSD due to the fact that the solution space exhibits a linear symmetry with respect to the extension field used to define the RSD problem. This allows some variables to be set for free, often reducing the complexity of the MinRank problem. This talk will explore the mathematical techniques that may be employed to give better estimates for the complexity of the RSD and related problems, and better security estimates for Rank-based cryptosystems.

 
5:15pm - 6:30pmSIAGA meeting for corresponding and associate editors
Unitobler, F011