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: Saturday, 13/Jul/2019
8:25am - 8:30amAnnouncements
vonRoll, Fabrikstr. 6, 001 
8:30am - 9:30amIP09: Mauricio Velasco: Extremal properties of 2-regular varieties
vonRoll, Fabrikstr. 6, 001 
 
8:30am - 9:30am

Extremal properties of 2-regular varieties

Mauricio Velasco

Universidad de los Andes, Colombia

A projective variety is called two regular if it is defined by quadrics and all matrices in the minimal free resolutions of its homogeneous coordinate ring have linear entries. In an objective sense these are "the simplest" projective varieties and perhaps for this very reason they are ubiquitous in algebraic geometry. In this talk I will explain several novel contexts of interest for the SIAGA community where these varieties play a prominent role. In the process we will describe other properties which characterize two-regular varieties highlighting the fruitful interplay between classical and convex algebraic geometry.

 
8:30am - 9:30amIP09-streamed from 001: Mauricio Velasco: Extremal properties of 2-regular varieties
vonRoll, Fabrikstr. 6, 004 
9:30am - 10:00amCoffee break
Unitobler, F wing, floors 0 and -1 
10:00am - 12:00pmRoom reserved (unless you reserved it, please don't enter)
Unitobler, F005 
10:00am - 12:00pmMS146, part 3: 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)

 

The integer homology threshold for random simplicial complexes

Andrew Newman
TU Berlin

The very first problem considered in the now-classic Linial-Meshulam model was to generalize the connectivity threshold from Erdős-Rényi random graphs to higher dimensions as homological connectivity. Early work by Linial, Meshulam, and Wallach had established this homology-vanishing threshold for finite field coefficients, however this a priori does not establish the threshold for integer coefficients. In joint work with Elliot Paquette discussed here, we establish this threshold for homology with integer coefficients to vanish.

 

The real tau-conjecture is true on average

Peter Bürgisser
TU Berlin

Koiran's real tau-conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is O(mkt), which is linear in the number of parameters.

 

Geometric limit theorems in topological data analysis

Christian Lehn
Universität Chemnitz

In a joint work with V. Limic and S. Kalisnik Verosek we generalize the notion of barcodes in topological data analysis in order to prove limit theorems for point clouds sampled from an unknown distribution as the number of points goes to infinity. We also investigate rate of convergence questions for these limiting processes.

 

Quantitative Singularity theory for Random Polynomials

Hanieh Keneshlou
MPI MiS Leipzig

In this talk, based on a joint work with A. Lerario and P. Breiding, I will present some probabilistic approximations of singularity type of a polynomial. The case of special interest is the zero set of a polynomial. We will show with an overwhelming probability, the set of real zeros of a polynomial of degree d can be realized as the zero set of a polynomial of degree sqrt{d log(d)}.

 
10:00am - 12:00pmMS171, part 1: Grassmann and flag manifolds in data analysis
Unitobler, F007 
 
10:00am - 12:00pm

Grassmann and flag manifolds in data analysis

Chair(s): Chris Peterson (Colorado State University, United States of America), Michael Kirby (Colorado State University), Javier Alvarez-Vizoso (Max-Planck Institute for Solar System Research in Göttingen)

A number of applications in large scale geometric data analysis can be expressed in terms of an optimization problem on a Grassmann or flag manifold.The solution of the optimization problem helps one to understand structure underlying a data set for the purposes such as classification, feature selection, and anomaly detection.

For example, given a collection of points on a Grassmann manifold, one could imagine finding a Schubert variety of best fit corresponds to minimizing some function on the flag variety parameterizing the given class of Schubert varieties.

A number of different algorithms that exist for points in a linear space have analogues for points in a Grassmann or flag manifold such as clustering, endmember detection, self organized mappings, etc.

The purpose of this minisymposium is to bring together researchers who share a common interest in algorithms and techniques involving Grassmann and Flag varieties applied to problems in 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)

 

PCA Integral Invariants for Manifold Learning

Javier Alvarez-Vizoso
Max-Planck Institute for Solar System Research in Göttingen

Local integral invariants at scale based on principal component analysis have recently been shown to provide estimators of curvature information at every point of a manifold. These can thus be applied to perform manifold learning from point clouds sampled from embedded Riemannian manifolds, and to inform optimization and geometry processing methods in arbitrary dimension, e.g. feature detection at scale. In particular, regular curves in Euclidean space are completely characterized up to rigid motion by the EVD of the PCA covariance matrix at every point, which reproduces the Frenet-Serret apparatus asymptotically with scale. We will also introduce the general result that establishes a dictionary in the limit between these statistical integral invariants and the classical differential-geometric curvature, in the form of a generalized Darboux-Ricci frame, providing an algorithm to estimate the Riemann curvature tensor for embedded manifolds of arbitrary dimension.

 

Subspace Averaging in Multi-Sensor Array Processing

Ignacio Santamaria1, Louis Scharf2, Vaibhav Garg1, David Ramirez3
1Universidad de Cantabria, 2Colorado Statew University, 3University Carlos III of Madrid

In this talk we address the problem of averaging on the Grassmann manifold, with special emphasis placed on the question of estimating the dimension of the average. The solution to this problem provides a simple order fitting rule based on thresholding the eigenvalues of the average projection matrix, and thus it is free of penalty terms or other tuning parameters commonly used by other information-theoretic criteria for model order estimation such as the minimum description length (MDL) criterion. The proposed rule appears to be particularly well suited to problems involving high-dimensional data and low sample support, such as the determination of the number of sources with a large array of sensors: the so-called source enumeration problem. The talk will discuss subspace averaging (SA) for source enumeration under the challenging conditions of:

i) large uniform arrays with few snapshots (the small sample regime), and

ii) non-white or spatially correlated noises with arbitrary correlation.

As illustrated by some simulation examples, SA provides a very robust method of enumerating sources in these challenging scenarios.

 

Variations on Multidimensional Scaling for non-Euclidean Distance Matrices

Mark Blumstein
Colorado State University

Classical multidimensional scaling takes as input a distance matrix and extracts a configuration of points in a low dimensional Euclidean space whose Euclidean distances best approximate the input data. In this talk we put a twist on the classical algorithm by changing the geometry of the embedding space. Specifically, we show that pseudo-Euclidean coordinates are the natural choice when working with non-Euclidean distance data. Examples are furnished by Lie groups and homogeneous manifolds, which display characteristic signatures in pseudo-Euclidean space.

 
10:00am - 12:00pmMS186, part 2: Algebraic vision
Unitobler, F011 
 
10:00am - 12:00pm

Algebraic vision

Chair(s): Max David Lieblich (University of Washington, United States of America), Tomas Pajdla (Czech Technical University in Prague), Matthew Trager (Courant Institute of Mathematical Sciences at NYU)

There has been a burst of recent activity focused on the applications of modern abstract and numerical algebraic geometry to problems in computer vision, ranging from highly-optimized Gröbner-basis techniques, to homotopy continuation methods, to Ulrich sheaves and Chow forms, to functorial moduli theory. We will discuss this recent progress, with a focus on multiview geometry, both in theory and in practice.

 

(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)

 

Solving for camera configurations from pairs

Brian Osserman
University of California, Davis

We study the question of when a configuration of multiple cameras can be recovered when one has information about a subset of the pairs (given for instance as a collection of fundamental matrices). We find the minimal number of pairs which can suffice, and analyze more generally what sorts of conditions can ensure either that a given set of pairs does or does not suffice to determine the configuration. This is joint work with Matthew Trager.

 

Ideals of the Multiview Variety

Andrew Pryhuber
University of Washington

The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. While the ideals of these polynomials are all contained in the multiview ideal, we show that none of them coincide with the multiview ideal. We establish precise algebraic relationships between the multiview ideal and these various determinantal ideals. When the camera foci are non-coplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized to cut out the space of finite images.

 

Estimation under group action and fast polynomial solvers, with applications to cryo-EM

Joe Kileel
Princeton

In many applied contexts, the task is to estimate latent variables from noisy observations involving unknown rotations. One challenging example comes from cryo-electron microscopy (cryo-EM), recognized by the 2017 Nobel Prize in Chemistry, where the objective is to estimate a 3D molecule from highly noisy 2D projection images taken from unknown viewing directions.

In this talk, we introduce an abstract framework for statistical estimation under noisy group actions. We prove, for this class of problems, sample complexity relates to invariant rings and secant varieties, while method-of-moments is sample-efficient. In special cases, we find a computationally-efficient algorithm for inverting moments, using tensor decomposition and polynomial solving.

In particular, for one model of cryo-EM we present a polynomial solver for ~10000 variables running in ~2 minutes. Further, we develop a new variant of the power method for symmetric tensor decomposition, e.g. decomposing random 15^6 symmetric tensors of rank 450 in ~45 seconds. Our principled approach is validated on a real cryo-EM dataset, in the context of ab initio modeling.

Joint work with Amit Singer’s group and Afonso Bandeira’s group.

 
10:00am - 12:00pmMS173, part 3: 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)

 

Certification of approximate roots of exact ill-posed polynomial systems

Agnes Szanto
NCSU

In this talk, I will survey some of our recent work on certifying approximate roots of exact polynomial systems and will describe some applications. In particular, we will concentrate on systems with root multiplicity, and show ways to certify approximations to singular roots, as well as their multiplicity structure. The difficulty lies in the fact that having singular roots is not a continuous property, so traditional numerical certification techniques do not work. Our certification methods are based on hybrid symbolic-numeric techniques. This is joint work with Jonathan Hauenstein and Bernard Mourrain.

 

Numerical Implicitization

Justin Chen
Georgia Tech

It is increasingly important nowadays to perform explicit computations on varieties, even in the realm where symbolic (e.g. Grobner basis) methods are too slow. We give an overview of the Macaulay2 package NumericalImplicitization, which aims to provide numerical information about images of varieties, such as dimension, degree, and Hilbert function. We also discuss some changes and additions to the package, such as improvements to point sampling, completions of partial pseudo-witness sets, and parallelization. This is joint work with Joe Kileel.

 

The Distribution of Numbers of Operating Points of Power Networks

Julia Lindberg
Wisconsin Institute for Discovery

The operating points of an n-node power network are real solutions of the power flow equations, a system of 2n-2 quadratic polynomials in 2n-2 variables. Our work finds the distribution of the number of real solutions, which is important in determining the stability of the network. In general, the number of nontrivial operating points equals the number of real solutions of a single polynomial. We use this polynomial to visualize regions with a fixed number of solutions, finding that some cluster around hyperplanes.

 
10:00am - 12:00pmMS141, part 2: Chip-firing and tropical curves
Unitobler, F013 
 
10:00am - 12:00pm

Chip-firing and tropical curves

Chair(s): Chi Ho Yuen (University of Bern), Alejandro Vargas (University of Bern)

The chip-firing game on metric graphs is a simple combinatorial model that serves as a tropical analogue of divisor theory on algebraic curves, and it has been an active and fruitful research direction over the last decade. The behaviors of chip-firing resemble, but not always completely match, the classical situation in algebraic geometry. So on one hand, chip-firing can often be used to prove results (old and new) in algebraic geometry; while on the other hand, the combinatorics of chip-firing is interesting and surprising in its own right. We will focus on three main topics: (I) Tropical analogues (or failure thereof) of classical results of algebraic curves, (II) applications of chip-firing in algebraic geometry and other subjects, and (III) complexity issues of computational problems related to chip-firing.

 

(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)

 

Chip-firing and the tropical inverse problem

Dhruv Ranganathan
University of Cambridge

The chip-firing game on graphs gives strong combinatorial constraints on the types of meromorphic functions that can be defined on a Riemann surface. The framework provides useful combinatorial bounds for invariants of the Riemann surface: for instance, it can prove the generic non-existence of linear systems with prescribed numerical invariants. In general, the method is not set up to go the other way around: to prove the existence of geometric objects lifting the combinatorial ones. However, when even simple instances of these lifting statements can be proved, they have substantial applications. I will give a tour of the ideas surrounding this ``tropical inverse problem’’, and mention some applications along the way.

 

Tropical Prym Varieties

Yoav Len
Georgia Institute of Technology

My talk revolves around combinatorial aspects of Prym varieties with applications to Brill-Noether theory. Prym varieties are a class of Abelian varieties that appear in the presence of covers between Riemann surfaces, and have deep connections with 2-torsion points of Jacobians, bi-tangent lines, and spin structures on curves. In my talk, I will describe the tropical version of Prym varieties in terms of chip-ring, and discuss the relation with their algebraic counterpart. As a consequence of the tropical construction we obtain new results in the geometry of special algebraic curves. This is joint work with Martin Ulirsch.

 

Equidistribution of tropical Weierstrass points

Harry Richman
University of Michigan

The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on abstract tropical curves, and explain how their distribution can be described in terms of electrical networks.

 

Submodular functions in tropical geometry: the existence of semibreak divisors

Lilla Tóthmérész
Eötvös Loránd University

In the talk, I would like to show a situation where a statement in tropical geometry is proved using the submodular technique of combinatorial optimization.

Break divisors are a very useful concept in tropical geometry that were introduced by Mikhalkin and Zharkov. They give a system of representatives of divisor classes of degree g (where g is the genus). We introduce semibreak divisors, generalizing break divisors for degree less than the genus. We show that every effective divisor class of degree between 0 and g contains a semibreak representative. Semibreak divisors can be used to give elementary proofs for some properties of effective loci in tropical curves. Formerly, the only known proofs for these properties used the counterparts of these properties for algebraic curves.

To prove the existence of semibreak divisors in effective divisor classes of degree between 0 and g, we give a characterization of break divisors using a submodular function. Though the submodular function in our case is defined on an infinite set system (on those closed subsets of the metric graph that have finitely many path connected components), the problem has a quasi-discrete nature, which enables us to obtain a proof for the existence of semibreak divisors that very much resembles discrete arguments. We also obtain an algorithm that computes a semibreak representative for a given effective divisor, using a submodular minimization algorithm as subroutine. Joint work with Andreas Gross and Farbod Shokrieh.

 
10:00am - 12:00pmMS179, part 2: Algebraic methods for polynomial system solving
Unitobler, F021 
 
10:00am - 12:00pm

Algebraic methods for polynomial system solving solving

Chair(s): Mohab Safey El Din (Sorbonne Université, France), Éric Schost (University of Waterloo)

Polynomial system solving is at the heart of computational algebra and computational algebraic geometry. It arises in many applications ranging from computer security and coding theory (where computations must be done over finite fields) and engineering sciences such as chemistry, biology, signal theory or robotics among many others (here computations are done over inifinite domains such as complex or real numbers). The need of reliable algorithms for solving these problems is prominent because of the non-linear nature of the problems we have in hand.

Algebraic methods provide a nice framework for designing efficient and reliable algorithms solving polynoial systems. This mini-symposium will cover many aspects of this topic, including design of symbolic computation algorithms as well as the use of numerical methods in this framework with an emphasis on reliability.

 

(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)

 

On polynomial and regular images of Euclidean spaces

José Fernando Galvan
Univ. Madrid

Let $f:=(f_1,ldots,f_m):R^ntoR^m$ be a map. We say that $f$ is em polynomial em if its components $f_k$ are polynomials. The map $f$ is em regular em if its components can be represented as quotients $f_k=frac{g_k}{h_k}$ of two polynomials $g_k,h_k$ such that $h_k$ never vanishes on $R^n$. More generally, the map $f$ is em Nash em if each component $f_k$ is a Nash function, that is, an analytic function whose graph is a {sl semialgebraic set}. Recall that a subset $SssubsetR^n$ is em semialgebraic em if it has a description as a finite boolean combination of polynomial equalities and inequalities. By Tarski-Seidenberg's principle the image of a map whose graph is a semialgebraic set is a semialgebraic set. Consequently, the images of polynomial, regular and Nash maps are semialgebraic sets. In 1990 {em Oberwolfach reelle algebraische Geometrie} week Gamboa proposed a kind of converse problem: em To characterize the semialgebraic sets in $R^m$ that are either polynomial or regular images of some $R^n$em. In the same period Shiota formulated a conjecture that characterizes Nash images of $R^n$, which has been recently proved by the author. In this talk we collect some of our main contributions to these problems and announce some future work concerning polynomial images of the unit closed ball. We have approached our contributions along the last two decades in three directions:

(i) To construct explicitly polynomial and regular maps whose images are the members of large families of semialgebraic sets whose boundaries are piecewise linear.

(ii) To find obstructions to be polynomial/regular images of $R^n$.

(iii) To prove Shiota's conjecture and some relevant consequences.

 

Degree bounds for the sparse Nullstellensatz

Gabriela Jeronimo
Univ. Buenos Aires

We will present new upper bounds both the degrees in Hilbert's Nullstellensatz and for the Noether exponent of a polynomial ideal in terms of mixed volumes of convex sets associated with the supports of a finite family of given generators. Our main results are the first upper bounds valid for arbitrary polynomial systems that distinguish the different supports of the polynomials. In the mixed sparse setting, they can be considerably smaller than previously known bounds. This is joint work with María Isabel Herrero and Juan Sabia (Universidad de Buenos Aires and CONICET, Argentina).

 

Signature-based Möller's algorithm for strong Gröbner bases over PIDs

Thibaut Verron
Johannes Kepler Univ.

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial rings over fields, and recent work has focused on extending this technique to coefficients in rings.

Möller introduced in 1988 two algorithms for Gröbner bases over rings: one algorithm computing weak bases over any effective ring, and another computing strong bases if the coefficient ring is a Principal Ideal Domain (PID).

In 2018, we showed that it is possible to augment Möller's weak GB algorithm with signatures, in the case of PIDs. In this work, we show how the same technique can be used for Möller's strong GB algorithm. We prove that the resulting algorithm computes a strong Gröbner basis while ensuring that the signatures do not decrease, and in particular, that no signature drop occurs. Möller's strong GB algorithm requires special care compared to Möller's weak GB algorithm or to the fields case, because of its use of so-called $G$-polynomials whose signatures have to be controlled.

As in the case of fields or Möller's weak GB algorithm, it allows to use additional criteria such as the F5 criterion, which allows the algorithm to compute a Gröbner basis without a reduction to zero in the case of an ideal described by a regular sequence.

Furthermore, we show that Buchberger's coprime and chain criteria can be made compatible with signatures in Möller's strong GB algorithm. This makes use of the syzygy characterization of Gröbner bases, given by Möller's lifting theorem.

These results are supported by a toy implementation of the algorithms in Magma. In particular, Möller's strong GB algorithm does not suffer from the same combinatorial bottleneck as Möller's algorithm, which allowed us to gather experimental data regarding the number of $S$-polynomials computed, reduced and eliminated by each criterion.

(Joint work with Maria Francis)

 

Witness collections and a numerical algebraic geometry toolkit

Jose Rodriguez
Univ. of Wisconsin

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness set collection, whose structure is more involved. In this talk we build on recent work to give a complete treatment of witness set collections for multiprojective varieties, together with an algorithm for their numerical irreducible decomposition that exploits the structure of a witness set collection.

 
10:00am - 12:00pmMS130, part 4: 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)

 

Tighter bounds through rank-one convexification

Tillmann Weisser1, Sidhant Misra2, Hassan Hijzai2
1Los Alamos National Lab, NM, USA, 2Los Alamos National Laboratory, Los Alamos, NM, USA

To be completed.In classical polynomial optimization, point evaluation is relaxed to integration with respect to a measure. By optimizing over a measure instead of points, the search for the global optimum becomes a linear problem on a measure, and thanks to results from real algebraic geometry, a conic program on moments. The most common approach of this kind is the Lasserre (or SDP) hierarchy where linear constraints on the moments are combined with positive semi-definite constraints on the moment matrix. In case bounds on the optimization variables of the original problem are known, we propose to strengthen the SDP hierarchy by adding non-linear moment constraints that are derived from necessary condition on the moment matrix to be rank-one. Though non-linear, the additional constraints do not change the convex character of the relaxation. Hence, local non-linear solvers are able to solve the tightened relaxation to optimality.

 

Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

Yuzixuan Zhu, Pataki Gabor, Tran-Dinh Quoc
University of North Carolina at Chapel Hill, NC, USA

We introduce Sieve-SDP, a simple algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP belongs to the class of facial reduction algorithms. It inspects the constraints of the problem, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on more than seven hundred SDPs from the literature.

We conclude that Sieve-SDP is very fast: preprocessing by Sieve-SDP takes, on average, less than one percent of the time that it takes for a commercial solver to solve an SDP. Preprocessing by Sieve-SDP significantly improves accuracy and reduces the solution time.

 

Phaseless rank

António Goucha1, João Gouveia2
1Universidade de Coimbra, 2Universidade de Coimbra, Portugal

The phaseless rank of a nonnegative matrix M is defined to be the least k for which there exists a complex rank-k matrix N such that |N|=M, entrywise speaking. In optimization terms, it is the solution to the rank minimization of a matrix under complete phase uncertainty on the entries.

This concept has a strong connection not only with amoebas, since computing the phaseless rank amounts to solving the amoeba membership problem for the varieties of matrices with restricted rank, but also with positive semidefinite matrix factorizations, as the phaseless rank can be used to derive bounds on the psd rank.

In 1966, Camion and Hoffman proved a landmark characterization of what can be seen as the set of singular square matrices with respect to phaseless rank. In this talk we will revisit and extend these results, and show how they can be used to derive some new examples in both amoeba theory and psd factorizations.

 

Log-concave polynomials, entropy, and approximate counting

Cynthia Vinzant1, Nima Anari2, Kuikui Liu3, Shayan Oveis Gharan3
1North Carolina State University, NC, USA, 2Stanford University, CA, USA, 3University of Washington, Seattle, WA, USA

A polynomial is called completely log-concave if it is log-concave as a function on the nonnegative orthant and its directional derivatives have the same property. This class of polynomials includes homogeneous stable polynomials and the basis generating polynomials of matroids. I will introduce some of the basic properties of these polynomials and discuss a concave program that based on entropy maximization that can approximate the size of its support. An important application will be approximating the number of bases of a matroid. This is joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

 
10:00am - 12:00pmMS127, part 2: The algebra and geometry of tensors 2: structured tensors
Unitobler, F023 
 
10:00am - 12:00pm

The algebra and geometry of tensors 2: structured tensors

Chair(s): Elena Angelini (Università degli studi di Siena), Enrico Carlini (Politecnico di Torino), Alessandro Oneto (Humboldt Fundation, and Otto-von-Guericke-Universität Magdeburg)

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. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven.

 

(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 monic rank

Jan Draisma
Universität Bern

I will introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone X; and describe an algorithmic technique based on classical invariant theory to determine, in certain concrete situations, the maximal monic rank. Using this, we prove that each univariate complex polynomial of degree 6,9,12 is the sum of 3 cubes of polynomials of degrees 2,3,4, respectively, and similarly that each univariate octic is a sum of 4 fourth powers of quadrics---special cases of a question by Boris Shapiro. I will also raise the question whether for cones X over equivariantly embedded projective homogeneous varieties, and the hyperplane corresponding to a highest weight vector, the maximal (ordinary) rank and maximal monic rank coincide. This is true in several concrete examples. If true in general, it would yield sharper lower bounds to the maximal (ordinary) rank.

Based on joint work with Arthur Bik, Alessandro Oneto, and Emanuele Ventura.

 

The average condition number of tensor rank decomposition is infinite

Nick Vannieuwenhoven
KU Leuven

Tensor rank decomposition is the problem of computing a set of rank-1 tensors whose sum is a given tensor. We are interested in quantifying the sensitivity of real rank-1 summands when moving the tensor infinitesimally on the semialgebraic set of tensors of bounded real rank. For this purpose, the standard approach in numerical analysis consists of computing the condition number of this problem. If the condition number is infinite, then the problem is said to be ill-posed. In this talk, we present the condition number of tensor rank decomposition. For most ranks, we compute its average value over the semialgebraic set of real tensors of bounded rank, relative to a natural choice of probability distribution. The results show that the condition number blows up too fast in a neighborhood of ill-posed problems to result in a finite average value.

This is joint work with Carlos Beltrán and Paul Breiding.

 

Symmetry groups of tensors

Emanuele Ventura
Texas A&M

To analyze the complexity of the matrix multiplication tensor, Strassen introduced a class of tensors that vastly generalize it, the tight tensors. Tight tensors are essentially tensors with a ”good” positive dimensional symmetry group. Besides the motivation from algebraic complexity, the study of symmetry groups of vectors in a representation of an algebraic group is a classical topic in algebraic geometry and invariant theory. It is then natural to investigate tensors with large symmetry groups, under a genericity assumption (1-generic).

In this talk, we discuss some combinatorial consequences of tightness, and sketch the geometry behind the classification of 1-generic tensors with maximal symmetry groups.

This is based on joint works with A. Conner, F. Gesmundo, JM Landsberg, and Y. Wang.

 

On the rank preserving property of linear sections and its applications in tensors

Yang Qi
University of Chicago

This talk is motivated by several questions on tensor ranks arising from signal processing and complexity theory. In the talk, we will first translate these conjectures into the geometric language, and reduce the problems to the study of a particular property of a linear section of an irreducible nondegenerate projective variety, namely the rank preserving property. Then we will introduce several useful tools and show some results obtained via these tools.

This talk is based on a joint work with Lek-Heng Lim.

 
10:00am - 12:00pmMS174, part 3: 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)

 

Reduction of the number of parameters

János Tóth
Budapest University of Technology and Economics

We start from a kinetic differential equation and transform it using an extended positive diagonal transformation in such a way that we prescribe the values of some of the reaction rate coefficients in the transformed system. The problems to be studied are as follows.

1. Is it possible to prescribe the value of a given set of coefficients? If yes, does the transformation obeying the prescriptions uniquely determine the values of the other coefficients? In both cases: what are the new coefficients of the transformed equation?

2. Which are all the possible subsets of new coefficients that can be independently prescribed? What is the largest number of coefficient set(s) that can be prescribed?

3. Suppose we prescribe the values of some coefficients: they should be one. In this case we usually say we only have the remaining coefficient combinations as ''independent'' coefficients. Why?

We present some statements and examples as answers to the above questions and also we show a program to carry out the corresponding transformations. Furthermore, we mention some connections to the general problem of transforming kinetic differential equations.

 

"Good children" and "bad children"

Nicola Vassena
Free University Berlin

Equilibrium bifurcations arise from sign changes of Jacobian determinants, as parameters are varied. Therefore we address here the Jacobian determinant for metabolic networks with general reaction kinetics. Our approach is based on the concept of child selections: each (mother) metabolite is mapped, injectively, to one of those (child) reactions which it drives as an input.

Our analysis distinguishes reaction network Jacobians with constant sign from the bifurcation case, where that sign depends on specific reaction rates. In particular, we distinguish "good child" selections, which do not affect the sign, from more interesting / demanding / troublesome / mischievous "bad children", which gang up towards sign changes, instability, and bifurcations.

 

Tikhonov-Fenichel parameter values for chemical reaction networks

Sebastian Walcher
RWTH Aachen

The chemical reaction networks under consideration here are described by polynomial ordinary differential equations depending on positive parameters, with the positive orthant as a positively invariant set. In order to determine parameter regions where singular perturbation reduction (in the sense of Tikhonov and Fenichel) is possible, the notion of Tikhonov-Fenichel parameter values (TFPV) was introduced some time ago in Alexandra Goeke's dissertation and subsequent publications. A TFPV is characterized by the property that small perturbations give rise to a singular perturbation reduction. It is known that the TFPV of a given system form a semi-algebraic subset, and the defining equations may be determined algorithmically by standard elimination theory.

After reviewing the above notions and results, we discuss three types of questions arising for TFPV:

(i) Singular perturbation reduction versus "classical" quasi-steady state reduction.

(ii) The unreasonable simplicity of TFPV for CRN, and the unreasonable feasibility of their computation.

(iii) Nested TFPV for multiscale systems.

The talk reports on recent joint work with Elisenda Feliu, Niclas Kruff, Christian Lax and Carsten Wiuf.

 

Parameter geography

Jeremy Gunawardena
Department of Systems Biology, Harvard Medical School

I will discuss numerical observations of the parameteric region in which a two-site, post-translational modification system exhibits bi-stationarity. Aside from general features of connectedness and near convexity, we find a substantial difference in region volume between Michaelis-Menten and realistic enzymatic assumptions, which suggests that bi-stationarity may be rare under biological conditions. We uncover a previously unsuspected parameteric relationship underlying this phenomenon. We also find that boundary parameter points move back and forth between mono-stationarity and bi-stationarity (“blinking”), as conserved quantities increase, despite apparent monotonic increase in region volume. These results rely on combining the linear framework, which allows algebraic reduction of the steady state, with numerical algebraic geometry using Bertini, which permits sampling of approximately 10^9 points in parameter space.

 
10:00am - 12:00pmMS164, part 3: 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 third session is "Numerical methods in line configurations and spectral decompositions."

 

(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)

 

k-point semidefinite programming bounds for equiangular lines

Fabrício Machado
Universidade de São Paulo

The problem of equiangular lines asks for the maximum number of lines in the n-dimensional Euclidean space with a fixed common angle. By selecting a unit vector along each line we get a spherical code with inner-products a and -a for some fixed 0 < a < 1 and by making a graph with these vectors as vertices and defining an edge whenever the inner-product is -a, this problem also becomes interesting from the algebraic graph theory point of view.

Parallel to this, techniques from semidefinite programming have been successfully applied to many problems in discrete geometry. The problem is modeled with an infinite graph and methods such as the Lovász theta-number used to upper bound the independence number of a finite graph are extended to this infinite setting. A key step in this approach is the use of symmetries from the problem to simplify the formulation and make it solvable by computer.

In this work we define a hierarchy of semidefinite programs specially suited for symmetry reduction techniques. Let S^{n-1} denote the unit sphere in R^n. In the setting of spherical codes, the symmetry group is the orthogonal group O(n) and we consider kernels S^{n-1} x S^{n-1} -> R invariant with respect to the stabilizer subgroup of a set of k points. We show that these kernels can be represented with matrices with size depending on k and not on n and use this characterization to compute new bounds for the maximum number of equiangular lines with fixed common angle.

Joint work with D. de Laat (MIT), F.M. de Oliveira Filho (TU Delft), and F. Vallentin (Universität zu Köln).

 

Using quantum information techniques to find the number of mutually unbiased bases in any given dimension

Marcin Pawłoski
University of Gdansk

Quantum information has seen a very rapid development in the recent years mostly because it provides very abstract and, at the same time, intuitive view of the quantum theory. It makes it easier for us to find new applications in a wide range of fields from metrology, through cryptography to pure mathematics. In this talk I will start by briefly explaining why and how this happens and then spend the most of the time by presenting an example: our recent result in which we were able to link the number of mutually unbiased bases (MUBs) in any given dimension with a success probability of a certain information-processing task. Next I will present how to bound this probability with a hierarchy of semi-definite programmes (SDPs) and report on how we applied it in practice and where did it lead us.

 

Fourier expansions of discrepancy kernels

Martin Ehler
Universität Wien

Many geometrical and combinatorial problems can be formulated as the minimization of specific kernel functions over compact subsets. To derive efficient numerical schemes, we study the spectral decomposition of discrepancy kernels. In particular, we obtain the kernels' Fourier expansions for several distinct examples.

 

Detection of Ambiguities in Linear Arrays in Signal Processing

Frederic Matter
TU Damstadt

This talk considers the detection of ambiguities that arise in the reconstruction of signals that are measured by a linear array which is given by a set of sensors located on a line in the plane.

In this case, signals are related to measurements by a linear equation system, whose matrix depends on the directions-of-arrival (DOA) of the signals and the positions of the sensors.

It can happen that certain DOAs together with certain sensor positions yield a measurement matrix that does not have full rank. This means that for a corresponding measurement the underlying original signal cannot be uniquely identified, and an ambiguity is said to arise.

In order to detect which DOAs produce ambiguities, we consider quadratic submatrices of the measurement matrix. Using Young tableaux and roots of unity, determining rank-deficient quadratic submatrices can be reduced to a mixed-integer program, whose solutions correspond to roots of the so-called Schur polynomial and thus to ambiguities. We demonstrate this approach using examples.

 
10:00am - 12:00pmMS159: Intersections in practice
Unitobler, F-107 
 
10:00am - 12:00pm

Intersections in practice

Chair(s): Martin Helmer (Australian National University)

This mini-symposium will focus on practical computational methods in intersection theory and their applications. At its most basic, intersection theory gives a means to study the geometric and enumerative properties of intersections of two varieties within another. These questions are fundamental to both algebraic geometry and its applications. Fulton-MacPherson intersection theory provides a powerful tool-set with which to study these intersections; however, many mathematical objects which are needed in this framework have long been computationally inaccessible. This barrier has limited the use of these ideas in computations and applications. In recent years several new and computable expressions for Segre classes, Polar classes, Euler characteristics, Euler obstructions, and other fundamental objects in intersection theory have been developed. This has led to a variety of computationally effective symbolic and numeric algorithms and opened the way for ideas from intersection theory to be applied to solve both mathematical and scientific problems. Some of this recent work will be highlighted in this mini-symposium. The first talk in the session will be an introductory talk, which will demonstrate the natural relations between intersection theory and numerical algebraic geometry and will highlight how intersection theory can be applied to solve classical problems such as testing ideal membership (without computing a Groebner basis). Subsequent talks will explore computational aspects of intersection theory in more detail and will highlight their practical applications.

 

(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)

 

Segre-driven ideal membership testing

Martin Helmer
Australian National University

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties. These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This talk is based on joint work with Corey Harris (University of Oslo).

 

The bottleneck degree of a variety

Sandra Di Rocco
KTH Royal Institute of Technology in Stockholm

The talk is within the area of Algebraic Geometry of Data. Bottlenecks are pairs of points on a variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample of the variety. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and we call it the bottleneck degree. We show that it is determined by invariants of the variety, such as polar classes and Chern classes. The talk is based on joint work with D. Eklund and M. Weinstein.

 

Symbolic Computation of Invariants of Local Rings

Mahrud Sayrafi
University of Minnesota

For a local ring (A, m) and an ideal I such that A/I has finite length, the Hilbert-Samuel polynomial P(n) of I is a polynomial such that P(n)=length(A/I^n) for large n. The leading coefficient and degree of this polynomial are important invariants of the ideal and encode information about the singularities of its coordinate ring. We present methods for computing this polynomial for arbitrary ideals and give various geometric examples involving hypersurfaces, determinantal ideals, etc. Moreover, we will share progress on local computation of other invariants such as the Bernstein-Sato polynomial and multiplier ideals, which measure singularities of varieties.

 
10:00am - 12:00pmMS169, part 2: Applications of Algebraic geometry to quantum information
Unitobler, F-111 
 
10:00am - 12:00pm

Applications of Algebraic geometry to quantum information

Chair(s): Frédéric Holweck (University of Bourgogne Franche-Comté)

Quantum information science attempts to use quantum phenomena as non-classical resources to perform new communication protocols and develop new computational paradigms. The theoretical advantages of quantum communication and quantum algorithms were proved in the 80-90’s and nowadays experimentalists are working on making that technology available. One of the quantum phenomena responsible for the speed up of quantum algorithms and the security of quantum communication is entanglement. A system of m-particules (a multipartite quantum state) is said to be entangled when the state of a particle of the system cannot be described independently of the others. Entanglement is a consequence of the superposition principle in quantum physics which mathematically translates to the fact that the Hilbert space of a composite system is the tensor product of the Hilbert space of each part. Algebraic geometry entered the study of entanglement of multipartite systems when it was both noticed in the early 2000s that the rank of tensors could be interpreted as a measure of entanglement and also that invariant theory could be used to distinguish different classes of entanglement. Since then a large amount of research has been produced in the mathematical-physics literature to classify and/or measure entanglement using techniques from classical invariant theory, representation theory, and geometric invariant theory. Because of the exponential growth of the dimension of the multipartite Hilbert spaces, when the number of factors increases, only a few examples of explicit classifications are known. Therefore to study entanglement in larger Hilbert spaces, techniques from tensor decomposition and asymptotic geometry of tensors have been recently introduced. These techniques establish new connections between entanglement and algebraic complexity theory.

This minisymposium on applications of algebraic geometry to quantum information will propose talks by mathematicians and physicists who have been studying entanglement from a geometrical perspective with classical and more recent techniques.

 

(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)

 

Quantum entanglement from single particle perspective

Adam Sawicki
Center for Theoretical Physics Polish Academy of Sciences

Despite considerable interest in recent years, understanding of quantum correlations in multipartite finite dimensional quantum systems is still incomplete. I will consider a simple scenario in which we have access to the results of all one-particle measurements of such system. The aim is to understand how much information about quantum correlations is encoded in this data. It turns out that mathematically consistent way of studying this problem involves methods that are used in classical mechanics to describe phase spaces with symmetries, symplectic geometry and geometric invariant theory. In this talk I will discuss these methods and show their usefulness to our problem. This is a joint work with Marek Kus, Tomasz Maciazek and Michal Oszmaniec.

 

Entanglement indicators for mixed three-qubit states

Szilárd Szalay
Wigner Research Centre for Physics of the Hungarian Academy of Sciences

Although the convex geometric questions of multipartite entanglement of mixed states are not directly linked to the algebraic-geometric questions of multipartite entanglement of pure states in general, there are some exotic exceptions, such as in the case of systems of three qubits. In this talk, we briefly review the lattice of the entanglement classification of three-qubit mixed states, based on the SLOCC classification of state vectors. The latter can be given in terms of a Freudenthal Triple System: state vectors of different SLOCC-class are in one-to-one correspondence with FTS-elements of different rank, which can be characterized by vanishing conditions of a set of LSL(2) (Local Special Linear) covariants. From these covariants we construct a larger set of LU(2) (Local Unitary) invariant polynomials on pure states, showing the proper vanishing conditions to be indicator functions for all the 21 partial separability classes for mixed states.

 

Non-displacable manifolds, mutually coherent and mutually entangled states

Karol Zyczkowski
Jagiellonian University

A great circle $T_1$ on a sphere is {sl non-displacable}, as any two great circles on a sphere $S_2=CP^1$ do intersect. This statement can be generalized for higher dimensions: Cho showed (2004) that any great torus $T_K$ embedded in $CP^K$ is non displacable for any integer $K$. Above fact implies that for any choice of two orthogonal basis in $N=K+1$ dimensional space there exists a vector mutually coherent with respect to both bases, so that the sum of the entropies characterizing measurements in both bases is maximal and equal to 2log N. Bounds for the sum of entropies obtained for more than two ortogonal measurements are also discussed.

A related result by Tamarkin (2008) states that real projective space $RP^K$ embedded in $CP^K$ is non-displacable. Making use of this statement for $K=3$ we show that for any two-qubit unitary gate U in U(4) there exists a mutually entangled pure quantum state, which is maximally entangled with respect to the standard computational product basis, $B={|00>, |01>, |10>, |11>}$, and also with respect to the rotated basis $UB$.

 

Relating boundary entanglement to scattering data of the bulk in $AdS_3/CFT_2$

Péter Lévay
Budapest University of Technology and Economics

According to a recent idea bulk space-time is an emergent quantity coming from entanglement patterns of the boundary. By studying the space of geodesics in $AdS_3$, and quantizing a parametrized family of geodesic motion we show that scattering data is related to boundary entanglement of the $CFT_2$ vaccum. For the parametrized family of geodesics we calculate the Berry curvature living on the space of geomdesics. As a result we recover the Crofton form with a quantum coefficient related to the scattering energy. We argue that, by applying results coming from Algebraic Scattering Theory, this idea can be generalized for more general states and possibly for the general $AdS_{n+1}/CFT_n$ correspondence.

 
10:00am - 12:00pmMS128, part 2: Symbolic-numeric methods for non-linear equations: Algorithms and applications
Unitobler, F-112 
 
10:00am - 12:00pm

Symbolic-numeric methods for non-linear equations: Algorithms and applications

Chair(s): Angelos Mantzaflaris (Inria, France), Bernard Mourrain (Inria, France), Elias Tsigaridas (Inria, France)

Modeling real-world systems or processes in areas such as control theory, geometric modeling, biochemistry, coding theory, cryptology, and so on, almost certainly involves non-linear equations. Higher degree equations are the first step away from linear models. Available tools for recovering their solutions range from numerical methods such as Newton-Raphson, homotopy continuation algorithms, subdivision-based solvers, to symbolic tools such as Groebner bases, border bases, characteristic sets and multivariate resultants. There is continuous progress in combining symbolic methods and numerical solving, in order to devise new algorithms with varying blends of exactness, stability and robustness as well as computational complexity, that are tailored for different applications. Among the challenges which occur in the process is reliable root isolation, certification and approximation, treatment of singular solutions, the exploitation of structure coming from specific applications as well as efficient interpolation. The mini-symposium will host presentations related to state-of-the-art solution strategies for these problems, theoretical and algorithmic advances as well as emerging application 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)

 

On hybrid univariate polynomial root-finders

Victor Pan
Lehman College CUNY

We combine various known methods developed for nearly optimal root-finding for univariate polynomials with our new techniques and obtain new algorithms with improved efficiency for both complex and real root-finding.

 

A robust path tracking algorithm for polynomial homotopy continuation

Marc Van Barel1, Simon Telen1, Jan Verschelde2
1KU Leuven, 2University of Illinois at Chicago

Homotopy continuation is an important strategy for solving systems of polynomial equations and for tackling other problems in computational algebraic geometry. State of the art implementations suffer from `path jumping', which often causes the loss of some solutions. We propose a new algorithm that uses Padé approximants for detecting difficult regions along the path. This results in an adaptive stepsize path tracker which proves to be more robust than existing algorithms.

 

On the relationship of well conditioned polynomials and elliptic Fekete points

Jinsan Cheng, Junyi Wen
Chinese Academy of Mathematics and Systems Science

In this talk, we present a method for isolating real roots of a bivariate polynomial system in a box. Our method is a subdivision method and based on the real root isolation of univariate polynomials and the geometry properties of the given system. By using the upper and lower bound polynomials of the system, we get some candidate boxes. We give the uniqueness and existence conditions to check if the system has a unique simple real root in the box. The method is complete for the system containing only simple real zeros. The experimental results show the superiority of our method.

 

A sequence of polynomials with optimal condition number

Maria De Ujue Etayo Rodriguez, Carlos Beltrán, Jordi Marzo, Joaquim Ortega-Cerdà
University of Cantabria

During this talk we will solve a problem posed by Michael Shub and Stephen Smale in 1993, in their famous article "Complexity of Bezout’s theorem. III."

The problem ask to find an explicit sequence of univariate polynomials of degree N with normalized contidion number less or equal than N, using the definition of normalized condition number that can be found, for example, in the book "Complexity and real computation" by Blum, Cucker, Shub and Smale. We find such a sequence of polynomials. Actually, the condition number of our polynomials is bounded by the square root of N, which we prove is a lower bound for the normalized condition number, meaning that our sequence has, up to some constant, optimal condition number.

 
10:00am - 12:00pmMS176: Algebraic geometry for kinematics and dynamics in robotics
Unitobler, F-113 
 
10:00am - 12:00pm

Algebraic geometry for kinematics and dynamics in robotics

Chair(s): Noémie Jaquier (Idiap Research Institute, Switzerland), Sylvain Calinon (Idiap Research Institute)

A fundamental problem in robotics is to characterize the kinematics of the robotic mechanism, i.e. to infer the relationship between the joint configuration and the position of the end-effector of the robot, typically the gripper. Motions of robotics mechanisms, essentially composed by rigid links connected by joints, are often characterized using the group of rigid body motions SE(3). Exploiting Lie algebra properties, kinematics problems can be formulated as systems of polynomial equations that can be solved using algebraic geometry tools. Algebraic geometry can further be used to study the dynamics properties of robotics mechanisms, i.e. the effect of forces and torques on the robot motions.

The goal of this minisymposium is to show the practical interest of algebraic geometry to analyze and control kinematic and dynamic motions of robotic systems in various applications such as solving inverse kinematic and dynamic problems, tracking manipulability ellipsoids or analyzing robots workspace. Furthermore, this minisymposium aims at bringing together mathematicians and roboticists to discuss further challenges in robotics involving application and development of algebraic geometry tools.

 

(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)

 

Some Applications of Classical Algebraic Geometry in Robotics

Jon Selig
London South Bank University

The purpose of this talk is to give a brief overview of some problems in Robotics and how they can be viewed in terms of classical algebraic geometry. The group of proper rigid-body displacements plays a fundamental role in Robotics and a lot of Mechanical Engineering. Although this is not an algebraic group, using dual quaternions, it can be modelled as an open set in a six-dimensional non-singular quadric. Many of the linear subspaces of this quadric have important interpretations in terms of Robotics. In particular, lines through the identity element correspond to either rotational or translational one-parameter subgroups. In turn these correspond to mechanical joints in a robot or mechanism. Some other linear subspaces will be discussed. Series composition of joints then give rise to Segre varieties in the quadric and intersections of these varieties solve some important enumerative problems in Robotics. For some serial chains the displacements of the final link are the solution to a purely geometrical problem. For example, the displacements that maintain the contact between a point and a fixed plane. In these cases the solution lies in the intersection of the quadric representing the group of all rigid-body displacements and another non-singular quadric. This can be used to study parallel robots. Leading us to consider other realisations of the group. The standard 4-by-4 representation of the group gives a variety of degree 8 in P12. The rotation group SO(3) here is mapped to a Veronese variety. To look at the Gough-Stewart platform, a particular type of parallel robot, it is useful to look at an old idea, pentaspherical coodinates. This is equivalent to a realisation of the group of rigid-body displacements as a subgroup of the conformal group of R3. Finally some other application to robot motion and dynamics will be briefly discussed.

 

A modular approach for kinematic and dynamic modeling of complex robotic systems using algebraic geometry

Shivesh Kumar1, Andreas Müller2
1DFKI Bremen, 2Johannes Kepler University

Parallel mechanisms are increasingly being used as a modular subsystem units in the design of modern robotic systems for their superior stiffness and payload to weight ration. This leads to series-parallel hybrid robots which combine the advantages of both serial and parallel topologies but also inherit their kinematic complexity. One of the main challenges in modeling and simulation of these complex robotic systems is the existence of kinematic loops. Standard approaches in multi-body kinematics and dynamics adopt numerical resolution of loop closure constraints which leads to accuracy and inefficiency problems. These approaches give you a limited understanding of the geometry of the system. Recently, approaches from computational algebraic geometry have enabled a global description of the kinematic behavior of these complex systems. In this talk, we present a modular and analytical approach towards exploiting these algebraic methods for kinematics and dynamics modeling. This approach forms the basis of a software workbench called Hybrid Robot Dynamics (HyRoDyn). Further, we demonstrate its application in multi-body simulation and control of a complex series-parallel humanoid.

 

Kinematics Analysis of Serial Manipulators via Computational Algebraic Geometry

Zijia Li
Johannes Kepler University

Kinematic singularities of a redundant serial manipulator with 7 rotational joints are analyzed and their effects on the possible self-motion are studied. We obtain the numerical kinematic singularities through algebraic varieties and demonstrate this on the kinematically redundant serial manipulator KUKA LBR iiwa. The algebraic equations for determining the variety are derived by taking the determinant of the 6-by-6 submatrix of the Jacobian matrix of the forward kinematics. By the primary decomposition, the singularities can be classified. Further analysis of the kinematic singularities including the inverse kinematics of the redundant manipulator provides us with valuable insights. Firstly, there are kinematic singularities where the inverse kinematics has no effect on the self-motion and cannot be used to avoid obstacles. Secondly, there are kinematic singularities, which lead to a single closed-loop connection with the serial redundant manipulator, so that a kinematotropic mechanism is achieved. Then we show the result of kinematic singularities of several industry robots which are obtained similarly. A special inverse kinematics analysis of a (2n+1)R serial manipulator is also presented in the end.

 

Robot manipulability tracking and transfer

Noémie Jaquier, Sylvain Calinon
Idiap Research Institute

Body posture influences human and robots performance in manipulation tasks, as appropriate poses facilitate motion or force exertion along different axes. In robotics, manipulability ellipsoids are used to analyze, control and design the robot dexterity as a function of the articulatory joints configuration. These ellipsoids can be designed according to different task requirements, such as tracking a desired position or applying a specific force.

In the first part of this talk, we present a manipulability tracking formulation inspired by the classical inverse kinematics problem in robotics. Our formulation uses the Jacobian of the map from the joint space to the manipulability space. This relationship demands to consider that manipulability ellipsoids lie on the manifold of symmetric positive definite matrices, which is here tackled by exploiting tensor-based representations and Riemannian geometry.

In the second part of the talk, we show how this tracking formulation can be combined with learning from demonstration techniques to transfer manipulability ellipsoids between robots. The presented approaches are illustrated with various robotic systems, including robotic hands, humanoids and dual-arm manipulators.

 
10:00am - 12:00pmMS196: Algebro-geometric methods for social network modelling
Unitobler, F-121 
 
10:00am - 12:00pm

Algebro-geometric methods for social network modelling

Chair(s): Kayvan Sadeghi (University College London, United Kingdom)

Algebraic and geometric methods have recently been proposed for statical random (social) network models. These methods could be described in three categories:

1) Understanding the geometry of the network models, especially the exponential random graph models (ERGMs) in order to understand the (mis)behaviour of such models in the asymptotic settings, commonly known as degeneracy of such models, which occurs commonly. In addition, many ERGMs are in fact curved exponential families, and understanding the geometry of the parameter space is of great importance.

2) Finding the model polytope of network models, i.e. the polytope of all sufficient statistics for every network of fixed size n in order to determine the existence of the MLE for such models and also to demonstrate which parameters are actually estimable.

3) Understanding the Markov bases of random network models specified by a multi-homogeneous ideal. This is directly relevant to the goodness-of-fit testing problems for network models as well as simulating from these models.
In this minisymposium some of the experts of the field of random network analysis demonstrate the latest developments on the algebro-geometric methods as described above.

The minisymposium would consist of four speakers, some of whom have already agreed to reset their papers. A tentative list is as follows (I will ad them to the list when the attendance is finalized by the authors):

 

(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)

 

Goodness-of-fit testing for log-linear network models

Despina Stasi
Illinois Institute of Technology

We define and study degree and block-based ERGMs called log-linear ERGMs. These models admit a correspondence to contingency table models which gives us access to categorical data analysis tools. We use these tools in combination with sampling tools stemming from discrete mathematics and algebraic statistics to produce a non-asymptotic goodness-of-fit test of network data to these models.

 

Cores, shell indices and the degeneracy of a graph limit

Johannes Rauh
Max-Plack Institute

The k-core of a graph is the maximal subgraph in which every node has degree at least k, the shell index of a node is the largest k such that the k-core contains the node, and the degeneracy of a graph is the largest shell index of any node. After a suitable normalization, these three concepts generalize to limits of dense graphs (also called graphons). In particular, the degeneracy is continuous with respect to the cut metric.

 

On Exchangeability in Network Models

Kayvan Sadeghi
University College London, United Kingdom

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size. We also show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts. We then characterize all possible conditional independence structures for finitely exchangeable random graphs.

 
10:00am - 12:00pmMS185, part 3: 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)

 

Subcovers and codes on a class of trace-defining curves

Guilherme Tizziotti
Federal University of Uberlandia

In this work, we construct some class of explicit subcovers of the curve Xn,r defined over Fq^n by affine equation yq^(n-1)+...+yq+y=xq^(n-r)+1-xq^n+q^(n-r). These subcovers are defined over Fq^n by affine equation gs(y)=xq^n+q^(n-r)-xq^(n-r)+1, where gs(y) is a q-polynomial of degree qs. The Weierstrass semigroup H(P), where P is the only point at infinity on such subcovers, is determined for 1 ≤ s ≤ 2r-n+1 and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained.

 

On Weierstrass semigroup at $m$ points on curves of the form $f(y)=g(x)$

Alonso Sepúlveda Castellanos
Federal University of Uberlandia

In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain m points on curves X with plane model of the form f(y)=g(x) over Fq, where f(T),g(T) in Fq[T]. Our results were obtained using the concept of discrepancy, for given points P and Q on X. This concept was introduced by Duursma and Park, and allows us to make a different and more general approach than that used to certain specific curves studied earlier.

 

Pure gaps on curves with many rational places

Ariane Masuda
NYC College of Technology

We consider the algebraic curve defined by ym=f(x) where m≥2 and f(x) is a rational function over Fq. We extend the concept of pure gap to c-gap and obtain a criterion to decide when an s-tuple is a c-gap at s rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places. We present the parameters of codes constructed using our families of pure gaps. This is joint work with Bartoli, Montanucci, and Quoos.

 

Non projective Frobenius algebras and linear codes

Javier Lobillo Borrero
Universidad de Granada

We extend the notion of a Frobenius algebra, dropping the projectivity condition, to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one’s options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the McWilliam identities that may be obtained in this context.

 
10:00am - 12:00pmMS145, part 4: 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)

 

Post-quantum signature schemes and more from supersingular isogenies

Ward Beullens
KU Leuven

To be completed.

 

Algorithmic aspects of cryptographic invariant maps from isogenies

Florian Hess
University of Oldenburg

We discuss some algorithmic aspects of candidate cryptographic invariant maps from isogenies, in particular those presented by Boneh, Glass, Krashen, Lauter, Sharif, Silverberg, Tibouchi and Zhandry in their paper on multiparty non-interactive key exchange.

 

Verifiable Delay Functions from Isogenies and Pairings

Luca De Feo
Ecole Polytechnique

We present a (non-post-quantum) framework for proving statements on isogeny walks in supersingular graphs. The framework can be seen as a combination of the BLS signature scheme with the supersingular isogeny graphs popularized by the key exchange protocols SIDH and CSIDH.

An instatiation of the framework for signature and interactive identification was already suggested in a 2010 patent owned by Microsoft; however the most interesting new application we obtain is a Verifiable Delay Function, whereby an isogeny walk of "great" length between two elliptic curves is made public, and the framework produces a succinct and easily verifiable proof of isogeny evaluation (similar to a proof of work).

This is joint work with S. Masson, C. Petit and A. Sanso.

 

Cryptographic goals beyond key exchange and signatures

Jeff Burdges
GNUnet

We shall discuss some cryptographic problems beyond key exchange and signatures for which practical post-quantum protocols would be much appreciated. These come in two flavours depending upon motivation, protocols desired for a more ethical applications that protect metadata, and protocols used in modern consensus algorithms.

 
1:30pm - 2:30pmIP10: Kathryn Hess Bellwald: Topological adventures in neuroscience
vonRoll, Fabrikstr. 6, 001 
 
1:30pm - 2:30pm

Topological adventures in neuroscience

Kathryn Hess Bellwald

EPFL, Switzerland

Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast. Topology has, for example, been successfully applied to objective classification of neuron morphologies and to automatic detection of network dynamics. In this talk I will focus on the algebraic topology of brain structure and function, describing results obtained by members of my lab in collaboration with the Blue Brain Project on digitally reconstructed microcircuits of neurons in the rat cortex. In particular, I will describe our on-going work on the topology of synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use.

 
1:30pm - 2:30pmIP10-streamed from 001: Kathryn Hess Bellwald: Topological adventures in neuroscience
vonRoll, Fabrikstr. 6, 004 
2:30pm - 3:00pmCoffee break
Unitobler, F wing, floors 0 and -1 
3:00pm - 5:00pmMS131, part 2: Computations in algebraic geometry
Unitobler, F005 
 
3:00pm - 5:00pm

Computations in algebraic geometry

Chair(s): Diane Maclagan (University of Warwick), Gregory G. Smith (Queen's University)

This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications.

 

(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 semigroup and cone of effective divisor classes on a hypersurface in a toric variety

Michael Stillman
Cornell University

The computation of the semigroup (or even the cone) of all effective divisor classes on a Calabi–Yau hypersurface of a toric variety is an important open problem, whose solution would have a number of applications in theoretical physics. This is a difficult computation, with no known effective algorithms. We present computational tools and algorithms for investigating this and related problems, and describe some results in this direction.

 

On subring counting and simultaneous monomialization

Anne Frübhis-Krüger
University of Hanover

The task of determining the order zeta function for certain number rings (which is just a sophisticated form of counting subrings) gives rise to a particular kind of p-adic integrals. The domain of integration of these stubbornly withstands standard techniques, including even an out-of-the-box Hironaka-style resolution of singularities. However, choosing centers of blow-ups using the structural properties of the problem, a simultaneous monomialization of the conditions can be achieved, making the problem again accessible to usual methods. This talk is based on joint work with Josh Maglione, Bernd Schober, and Christopher Voll.

 

Fröberg-Macaulay conjectures for algebras

Mats Boij
Royal Institute of Technology (KTH)

In a joint work with Aldo Conca, we look at what should correspond to Macaulay’s Theorem and Fröberg’s Conjecture for the Hilbert function of subalgebras of standard graded polynomial rings. Upper bounds correspond to generic forms and lower bounds correspond to strongly stable monomial ideals.

 

Singular value decomposition for complexes

Frank-Olaf Schreyer
Saarland University

In this talk, the concept of singular value decomposition of complexes will introduced and applied to the computation of syzygies.

 
3:00pm - 5:00pmRoom free
Unitobler, F006 
3:00pm - 5:00pmMS171, part 2: Grassmann and flag manifolds in data analysis
Unitobler, F007 
 
3:00pm - 5:00pm

Grassmann and flag manifolds in data analysis

Chair(s): Chris Peterson (Colorado State University, United States of America), Michael Kirby (Colorado State University), Javier Alvarez-Vizoso (Max-Planck Institute for Solar System Research in Göttingen)

A number of applications in large scale geometric data analysis can be expressed in terms of an optimization problem on a Grassmann or flag manifold.The solution of the optimization problem helps one to understand structure underlying a data set for the purposes such as classification, feature selection, and anomaly detection.

For example, given a collection of points on a Grassmann manifold, one could imagine finding a Schubert variety of best fit corresponds to minimizing some function on the flag variety parameterizing the given class of Schubert varieties.

A number of different algorithms that exist for points in a linear space have analogues for points in a Grassmann or flag manifold such as clustering, endmember detection, self organized mappings, etc

The purpose of this minisymposium is to bring together researchers who share a common interest in algorithms and techniques involving Grassmann and Flag varieties applied to problems in 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)

 

A dual subgradient approach to computing an optimal rank Grassmannian circumcenter

Tim Marrinan
Université de Mons

This talk concerns the circumcenter of a collection of linear subspaces. When the subspaces are k-dimensional subspaces of n-dimensional Euclidean space, this can be cast as an infinity-norm minimization problem on a Grassmann manifold, Gr(k,n). For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate circumcenter on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. In this talk we propose and solve an optimization problem parametrized by the rank of the circumcenter. The solution can be computed approximately using a dual subgradient algorithm. By scaling the objective and penalizing the information lost by the rank-k circumcenter, we jointly recover an optimal dimension, k*, and a central subspace on Gr(k*,n) that best represents the correlated subspace of the data.

 

Low Rank Representations of Matrices using Nuclear Norm Heuristics

Silvia Dinica
Romanian Senate

The connection between the entries of an Euclidean distance matrix and the nuclear norm of the matrix in the positive semidefinite cone given by the one to one correspondence between the two cones. In the case when the Euclidean distance matrix is the distance matrix for a complete k-partite graph, the nuclear norm of the associated positive semidefinite matrix can be evaluated in terms of the second elementary symmetric polynomial evaluated at the partition.

For k-partite graphs the maximum value of the nuclear norm of the associated positive semidefinite matrix is attained in the situation when we have equal number of vertices in each set of the partition. This result can be used to determinea lower bound on the chromatic number of the graph.

 

Grassmann Tangent-Bundle Means

Justin Marks
Gonzaga University

Applications of geometric data analysis often involve producing collections of subspaces, such as illumination spaces for digital imagery. For a given collection of subspaces, a natural task is to find the mean of the collection. A robust suite of algorithms has been developed to generate mean representatives for a collection of subspaces of xed dimension, or equivalently, a collection of points on a particular Grassmann manifold. These representatives include the flag mean, the normal mean, and the Karcher mean. In this talk, we catalogue the types of means and present comparative heuristics for the suite of mean representatives. We respond to, and at times, challenge, the conclusions of a recent paper outlining various means built via tangent-bundle maps on the Grassmann manifold.

 
3:00pm - 5:00pmRoom free
Unitobler, F011 
3:00pm - 5:00pmRoom free
Unitobler, F012 
3:00pm - 5:00pmMS167, part 4: 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)

 

Massively parallel methods with applications in tropical geometry

Dominik Bendle1, Kathrin Bringmann2, Arne Buchholz3, Janko Boehm1, Christoph Goldner4, Hannah Markwig4, Mirko Rahn5, Yue Ren6, Benjamin Schroeter7
1Technische Universität Kaiserslautern, 2Universität Köln, 3Universität des Saarlandes, 4Eberhard Karls Universität Tübingen, 5Fraunhofer ITWM, 6Max Planck Institute for Mathematics in the Sciences, Germany, 7Binghamton University

In this talk, I will discuss the use of massively parallel methods in the context of tropical geometry. I will first address the technical framework, which combines the computer algebra system Singular with the workflow management system GPI-Space. I will then focus on the computation of tropicalizations, and algorithms to determine generating series for Gromov-Witten invariants via Feynman integrals.

 

Tropical Grassmannians Gr_p(3,8) and the Dressian Dr(3,8)

Dominik Bendle1, Janko Boehm1, Yue Ren2, Benjamin Schroeter3
1Technische Universität Kaiserslautern, 2Max Planck Institute for Mathematics in the Sciences, Germany, 3Binghamton University

The pointwise valuation of an algebraic variety is a polyhedral complex, the tropical variety, which carries information about the algebraic set. A class of prominent examples are tropical Grassmannians Gr_p(d,n) over fields of characteristic p which are set theoretically included in tropical prevarieties, called Dressians.
These fans have close connections to many mathematical areas, e.g., matroid theory, mathematical biology, cluster algebras and physics. More precisely, the Dressian Dr(d,n) parametrizes d-dimensional tropical linear spaces in n-dimensional space also known as valuated matroids, while the Grassmannian Gr_p(d,n) contains those that are realizable. Moreover, these moduli spaces agree for d=2, and additionally they parametrize both all phylogenetic trees, and tropical curves of genus zero.

I will introduce these fans with their natural fan structure, inherit from Gröbner bases and regular subdivisions. Moreover, I report about new theoretical results additional to Janko Böhm's presentation of massiv parallelized computations. My focus will be on the relationship between Gr_0(3,8) and Dr(3,8).

 

Computing unit groups of curves

Justin Chen1, Sameera Vemulapalli2, Leon Zhang1
1UC Berkeley, 2Princeton University

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus.

Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.

 

A numerical algorithm for tropical membership

Taylor Brysiewicz
Texas A&M University

In 2012, Hauenstein and Sottile proposed a numerical oracle for the Newton polytopes of a hypersurface. Drawing from ideas of Hept and Theobald, we describe how this algorithm may be used to numerically verify membership in tropical hypersurfaces.

 
3:00pm - 5:00pmMS153, part 2: Symmetry in algorithmic questions of real algebraic geometry
Unitobler, F021 
 
3:00pm - 5:00pm

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)

 

Orbit closures in the Zariski spectrum of the infinite polynomial ring

Mario Kummer
TU Berlin

We study orbit closures of zero sets of prime ideals under the action of the infinite symmetric group. This leads, for instance, to a characterization of invariant prime ideals of the polynomial ring with n sets of infinitely many variables in terms of families of subvarieties of affine n-space. We also discuss semi-algebraic aspects over the real numbers. This is joint work with Cordian Riener.

 

Sum-of-squares hierarchy for symmetric formulations.

Adam Kurpisz
ETH Zurich

We preset a method for proving Sum-of-Squares (SoS) lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semi-definiteness to the analysis of "well-behaved" univariate polynomial inequalities.

We illustrate the technique with few applications. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that frac{n+2d-1}{2} levels of the SoS hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. As a special case we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. Additionally, we show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.

Finally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.

 

Symmetry Preserving Interpolation

Erick Rodriguez Bazan
INRIA

I will talk about multivariate interpolation in the presence of symmetry. Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. In my presentation, I will show how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases. This allows to construct bases of invariant interpolation spaces in blocks, capturing the inherent redundancy in the computations. I will also show that the so constructed symmetry adapted interpolation bases alleviate the computational cost of any interpolation problem and automatically preserve any equivariance of this interpolation problem might have.

 

Separating invariants of finite groups

Fabian Reimers
TU Munich

Let X be an affine variety with an action of an algebraic group G (over an algebraically closed field K). A subset (e.g. a subalgebra) of the invariant ring K[X]^G is called separating if it has the same capability of separating the orbits as the whole invariant ring. In this talk we focus on finite groups and show how the existence of a separating set of small size, or a separating algebra which is a complete intersection, is related to the property of G being a reflection (or bireflection) group. Theorems of Serre, Dufresne, Kac-Watanabe and Gordeev about linear representations are extended to this setting of G-varieties.

 
3:00pm - 5:00pmMS129, part 2: Sparsity in polynomial systems and applications
Unitobler, F022 
 
3:00pm - 5:00pm

Sparsity in polynomial systems and applications

Chair(s): Timo de Wolff (Technische Universität Berlin, Germany), Mareike Dressler (University of California, San Diego, CA, USA)

In this session we bring together researchers working in different areas involving sparsity in applications and sparse polynomial systems. The principle of sparsity is to represent a structure by functions, e.g., polynomials, with as few variables or terms as possible. It is ubiquitous in various areas and problems, where algebra and geometry play a key role. Recently, it has been succesfully applied to problems such as sparse interpolation, polynomial optimization, sparse elimination, fewnomial theory, or tensor decomposition.

This minisymposium provides an opportunity to learn about a selection of these recent developments and explore new potential applications of sparsity.

 

(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)

 

Filling a much-needed gap in the literature

Bruce Reznick
University of Illinois Urbana-Champaign, IL, USA

Many of the early examples in the study of psd forms which are not sos (such as the Motzkin form) arise from monomial substitution into the arithmetic-geometric inequality. Thirty years ago, the speaker gave a necessary and sufficient condition for such a form to be sos or not ({it Math. Ann.,} 283 (1989), 431-464 (MR 90i.11043)). He also announced that certain results would appear as part of the "in preparation" paper "Midpoint polytopes and the map $x_i mapsto x_i^k$". That paper never appeared, and this talk is an attempt to reconstruct the missing material.

 

Computing elimination ideals of likelihood equations

Xiaoxian Tang1, Timo de Wolff2, Rukai Zhao1
1Texas A&M University, TX, USA, 2Technische Universität Berlin, Germany

We develop a probabilistic algorithm for computing elimination ideals of likelihood equations. We show experimentally that it is far more efficient than directly computing Groebner bases or the known interpolation method for medium to large size models. Furthermore, we deduce discriminants of the elimination ideals, which play a central role in real root classification. In particular, we compute the discriminants of the 3 by 3 matrix model and one Jukes-Cantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively).

 

Nonnegative polynomials and circuit polynomials

Jie Wang
Peking University, China

The concept of sums of nonnegative circuit polynomials (SONC) was introduced as a new certificate of nonnegativity of polynomials, which was proved to be efficient in many cases. It is natural to ask which types of nonnegative polynomials admit SONC decompositions and how big the gap between the PSD cone and the SONC cone is. In this talk, we will consider these questions. Moreover, we clarify an important fact that every SONC polynomial decomposes into a sum of nonnegative circuit polynomials with the same support, which reveals the advantage of SONC decompositions for certifying nonnegativity of sparse polynomials compared with the classical SOS decompositions.

 

An Experimental Classification of Maximal Mediated Sets

Oguzhan Yürük, Timo de Wolff, Olivia Röhrig
Technische Universität Berlin, Germany

Maximal mediated sets (MMS), first introduced by Bruce Reznick, arise as a natural structure in the study of nonnegative polynomials supported on circuits. Due to Reznick's, de Wolff's, and Iliman's results, given a nonnegative polynomial $f$ supported on a circuit $C$ with vertex set $Delta$, $f$ is a sum of squares if and only if the non-vertex element of $C$ is in the MMS of $Delta$. In this project, we classify MMS experimentally. As a main theoretical result, we show that an MMS is determined by its underlying lattice. This is joint work with Olivia Röhrig and Timo de Wolff.

 
3:00pm - 5:00pmMS127, part 3: The algebra and geometry of tensors 2: structured tensors
Unitobler, F023 
 
3:00pm - 5:00pm

The algebra and geometry of tensors 2: structured tensors

Chair(s): Elena Angelini (Università degli studi di Siena), Enrico Carlini (Politecnico di Torino), Alessandro Oneto (Barcelona Graduate School of Mathematics)

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. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven.

 

(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)

 

Varieties of tensor decompositions and multi secants to curves and surfaces

Kristian Ranestad
University of Oslo

For a point p and a variety X in a projective space, the X -rank of p is the minimal n such that p lies on an n-secant (n-1)-space to X. I consider the variety V(p,X) of such n-secants, will recall the cases when X is a rational or elliptic curve and explain some surface cases. A very nice result of Iliev and Manivel for tensors C^3 x C^3 have applications on these issues.

 

Varieties of Hankel matrices and their secant varieties

Hirotachi Abo
University of Idaho

The maximal minors of the generic k x n Hankel matrix (also known as the catalacticant matrix) defines a rational map from the projective n-space to the Grassmann variety of (k-1)-planes in the projective n-space. The image of the projective n-space under the rational map is called the Hankel variety of (k-1)-planes, which is birationally equivalent to the projective n-space if n is sufficiently large compared with k. This talk concerns higher secant varieties of the Hankel variety of (k-1)-planes. The main focus of the talk is set on the defectivity of the Hankel variety of lines.

 

Tensor decomposition, sparse representation and moment varieties

Bernard Mourrain
INRIA

Tensor decomposition problems appear in many areas such as Signal Processing, Quantum Information Theory, Algebraic Statistics, Biology, Complexity Analysis, etc as a way to recover hidden structures from data. The decomposition is a representation of the tensor as a weighted sum of a minimal number of indecomposable terms. This problem can be seen as a sparse recovery problem from sequences of moments. We will develop this analogy and present an algebraic approach to address the decomposition problem, via duality and Hankel operators. We will analyze the varieties of moments associated to low rank decompositions, investigate their defining equations and some of their properties that can be exploited in the decomposition problem. Links with the Hilbert scheme of points will be presented. Examples exploiting these properties will illustrate the approach.

 

The Distance Function from the Variety of Rank One Partially-Symmetric Tensors

Luca Sodomaco
Università di Firenze

Let X be a Segre-Veronese product of projective spaces and denote with X* its dual variety. In this talk, we outline the main properties of the ``Euclidean Distance polynomial'' (ED polynomial) of X*, as a remarkable example of a more general theory on ED polynomials developed in a recent work with Ottaviani. Given a tensor T, the roots of the ED polynomial of X* at T correspond to the singular values of T. Moreover, we describe the variety of tensors that fail to have the expected number of singular vector tuples, counted with multiplicity. This variety is, in general, a non-reduced hypersurface and its equation is, up to scalars, the leading coefficient of the ED polynomial of X*.

 
3:00pm - 5:00pmMS199, part 2: Applications of topology in neuroscience
Unitobler, F-105 
 
3:00pm - 5:00pm

Applications of topology in neuroscience

Chair(s): Kathryn Hess Bellwald (Laboratory for topology and neuroscience, EPFL, Switzerland), Ran Levi (University of Aberdeen, UK)

Research at the interface of topology and neuroscience is growing rapidly and has produced many remarkable results in the past five years. In this minisymposium, speakers will present a wide and exciting array of current applications of topology in neuroscience, including classification and synthesis of neuron morphologies, analysis of synaptic plasticity, and diagnosis of traumatic brain injuries.

 

(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)

 

Simplicial convolutional neural networks for in-painting of cochains

Gard Spreemann
Laboratory for topology and neuroscience, EPFL, Switzerland

We use the simplicial Laplacian to define convolutional neural networks over simplicial complexes in a way that naturally generalizes classical CNNs. This provides us with the tools to build networks that perform in-painting of simplicial cochains while respecting the underlying topological structure. This is joint work with Stefania Ebli and Michaël Defferrard.

 

Using topological data analysis to classify certain stimuli in the Blue Brain reconstruction

Jason Smith
University of Abedeen , UK

The Blue Brain Project's digital reconstruction of a rat's neocortical column allows us to study the effect of certain stimuli on the brain. The insertion of a stimulus into the model causes information to propagate through the column, creating activity patterns that are not well understood. Using techniques from applied topology and combinatorics we attempt to characterise the firing patterns of different stimuli. Using this characterisation we then apply the methods to an unknown sequence of stimuli of the same type and attempt a classification of those stimuli.

 

Topology and neuroscience

Daniela Egas Santander
Laboratory for topology and neuroscience, EPFL, Switzerland

I will present some of the applications of topology and topological data analysis to neuroscience through an exploration of the collaboration between the applied topology group at EPFL and the Blue Brain Project. In particular, I will describe how we are using topology to further understand learning or simulations of voltage sensitive dye experiments.

 

Application of topological data analysis to the detection of mild cognitive impairment

Alice Patania
Indiana University

Identifying subjects with cognitive deficits as early as possible is critical in pursuing treatments for Alzheimer’s Disease. However, in the mildly symptomatic stages, pathological brain atrophy can be subtle and overpowered in signal by aging. Applying persistent homology, we are able to build coarse descriptors of the overall cortical thickness of each subject and isolate atrophy features that are indicative of MCI. These 0- and 1-persistence features can be used to build integrated persistent homological kernels which retain the meaningful homological information of brain atrophy. Using a support vector machine approach, we show how building a coarse descriptor of the cortical topology improves discriminative power of whole brain atrophy biomarkers at the MCI stage and homological features prove useful in identifying individuals with early stages of cognitive impairment.

 
3:00pm - 5:00pmMS164, part 4: Algebra, geometry, and combinatorics of subspace packings
Unitobler, F-106 
 
3:00pm - 5:00pm

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 fourth session is "Symplectic and real algebraic geometry in frame 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)

 

Symplectic Geometry and Frame Theory

Clayton Shonkwiler
Colorado State University

Geometric tools are increasingly important in the study of (finite) frames, which are simply redundant bases that are useful in signal processing and other applications where robustness to noise and erasures are important. Symplectic geometry was developed as the right general setting for Hamiltonian mechanics and in practice is often quite closely related to complex geometry: for example, smooth projective varieties are always symplectic manifolds. This is a promising, though mostly unexplored, collection of tools to be applied to the theory of frames in complex vector spaces.

The slogan is that any property which can be characterized as a level set of a moment map is likely to be amenable to symplectic techniques. In particular, unit-norm tight frames – which are particularly useful for applications – arise as the level set of a natural Hamiltonian group action on the set of complex matrices of a given size.

In this talk I will describe how symplectic tools can be used to generalize the frame homotopy theorem of Cahill–Mixon–Strawn and to give new insight into the Paulsen problem.

 

Symplectic Geometry, Optimization and Applications to Frame Theory

Tom Needham
Ohio State University

In recent work with Clayton Shonkwiler, we show that any space of complex frames (considered up to global rotations) with a prescribed frame operator can naturally be endowed with an extra geometric structure called a symplectic form. The goal of this talk is to explain how classical results from symplectic geometry can be used to provide theoretical guarantees for the convergence of optimization algorithms arising in frame theory. More specifically, spaces of frames with prescribed frame operator admit torus actions which are compatible with the symplectic structure (the torus actions are Hamiltonian). A result of Duistermaat says that gradient flows of certain functionals associated to Hamiltonian actions have no spurious local minima. We will discuss applications of this framework to the Paulsen problem from frame theory.

 

The optimal packing of eight points in the real projective plane

Hans Parshall
Ohio State University

How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in order to maximize the minimum angle between pairs of lines? Conway, Hardin and Sloane (1996) produced numerical line packings for $n leq 55$ that they conjectured to be optimal in this sense, but until now only the cases $n leq 7$ have been solved. We will discuss the resolution, joint with Dustin Mixon, of the case $n = 8$. Drawing inspiration from recent work on the Tammes problem, we proceed by enumerating potential contact graphs for an optimal configuration and eliminating those that violate various combinatorial and geometric constraints. The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and we convert this numerical packing to an exact packing through cylindrical algebraic decomposition. We will further describe some potential improvements to our approach that could yield more exact optimal packings.

 

Spherical configurations with few angles

William J. Martin
Worcester Polytechnic Institute

Let X be a spherical code in d-dimensional space. The degree of X is the number of inner products <u,v> that occur as u and v range over pairs of distinct elements from X. We are interested in spherical codes of small degree that arise from, or give rise to, association schemes. We will discuss equiangular lines, real mutually unbiased bases, work of Kodalen on "linked simplices" and joint work with Kodalen on "orthogonal projective doubles". A set of k full-dimensional simplices on the unit sphere is said to be "linked" if only two possible angles occur between vectors in distinct simplices. Given a graph G with vertex set V, a set L of lines through the origin in d-dimensional space is an "orthogonal projective double" of G if there is a bijection V --> L that maps adjacent pairs of vertices to orthogonal pairs of lines and non-adjacent pairs to lines forming some fixed angle between zero and 90 degrees. There is one aspect of this study involving elementary algebraic geometry. The ideal of X is the set of polynomials in d variables that vanish on each point in X and our goal is to determine, for each of the families mentioned above, a generating set for this ideal consisting of polynomials all having lowest possible total degree. This talk is based in part on joint work with my student Brian Kodalen and is supported by the US National Science Foundation.

 
3:00pm - 5:00pmMS136, part 3: 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)

 

Inversion of polynomial systems and polar maps

Remi Bignalet Cazalet
Université de Bourgogne

Given F=V(f) a reduced projective hypersurface defined by a homogeneous polynomial f in several variables, the gradient of f defines a rational map P_f between two projective spaces of the same dimension, called polar map of F. In general, it is a problem to describe all the homaloidal hypersurfaces, that is the hypersurfaces F=V(f) such that P_f is birational (i.e. P_f is an isomorphism between two Zariski opens) which aims to distinguish specific singular locus of the projective hypersurfaces. The classification of reduced homaloidal complex curves (i.e. when the base field is the complex field) was established by I.V.Dolgachev. It is formed by the smooth conics, the unions of three general lines and the unions of a smooth conics with one of its tangent. When the base field has characteristic p>2, the three curves in Dolgachev's classification are still homaloidal and a problem becomes to establish if they are the only ones. In this talk, I will explain how this question can be related to an analysis of the syzygies of the jacobian ideal of the hypersurfaces and I will show an explicit example of a homaloidal curve of degree 5 if p=3. This can be viewed algebraically as a study of the difference between the Rees and the symmetric algebra of the jacobian ideal or, equivalently, as a study of the variations of the Milnor number of the curves with respect to reduction modulo p.

 

Singularities and radical initial ideals

Alexandru Constantinescu
Freie Universität Berlin

What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Emanuela De Negri, Matteo Varbaro and myself conjecture that the answer is: Only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply in particular, that only curves of genus zero have such a degeneration. We proved this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. Consequences for rational and F-rational singularities of algebras with straightening laws will also be discussed.

 

Syzygies and gluing for semigroup rings

Philippe Gimenez
Universidad de Valladolid

Two numerical semigroups can be glued to obtain another numerical semigroup in higher embedding dimension. This concept was originally introduced to classify numerical semigroups that are complete intersections and it was later generalized to arbitrary numerical semigroups and semigroups in higher dimension. In this talk, we will construct the syzygies of the semigroup ring k[C] of a semigroup C obtained by gluing two semigroups A and B in terms of the syzygies of k[A] and k[B]. This will provide formulas for several invariants like Betti numbers, projective dimension and Hilbert series. We will use our construction to show that gluing two semigroups in higher dimension is not as easy as in the numerical case.

This is a joint work with Hema Srinivasan (Missouri University, USA).

 

Specialization of rational maps

Yairon Cid Ruiz
Universitat de Barcelona

We consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian integral domain. We will show specific applications to certain classes of rational maps. This is joint work with Aron Simis.

 
3:00pm - 5:00pmRoom free
Unitobler, F-111 
3:00pm - 5:00pmRoom free
Unitobler, F-112 
3:00pm - 5:00pmRoom free
Unitobler, F-113 
3:00pm - 5:00pmMS139, part 3: 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)

 

From random forests to regulatory rules: extracting interactions in high-dimensional genomic data

Karl Kumbier
University of California, Berkeley

Individual genomic assays measure elements that interact in vivo as components of larger molecular machines. Understanding the connections between such high-order interactions and complex biological processes from gene regulation to organ development presents a substantial statistical challenge. Namely, to identify high-quality interaction candidates from combinatorial search spaces in genome-scale data. Building on Random Forests (RFs), Random Intersection Trees (RITs), and through extensive, biologically inspired simulations, we developed the iterative Random Forest algorithm (iRF). iRF trains a feature-weighted ensemble of decision trees to detect stable, high-order interactions with same order of computational cost as RF. We define a functional relationship between interacting features and responses that decomposes RF predictions into a collection of interpretable rules, which can be used to evaluate interactions in terms of their stability and predictive accuracy. We demonstrate the utility of iRF for high-order interaction discovery in several genomics problems, where iRF recovers well-known interactions and posits novel, high-order interactions associated with gene regulation. By refining the process of interaction recovery, our approach has the potential to guide mechanistic inquiry into systems whose scale and complexity is beyond human comprehension.

 

Probabilistic tensors and opportunistic Boolean matrix multiplication

Petteri Kaski
Aalto University

We introduce probabilistic extensions of classical deterministic measures of algebraic complexity of a tensor, such as the rank and the border rank. We show that these probabilistic extensions satisfy various natural and algorithmically serendipitous properties, such as submultiplicativity under taking of Kronecker products. Furthermore, the probabilistic extensions enable strictly lower rank over their deterministic counterparts for specific tensors of interest, starting from the tensor <2,2,2> that represents 2-by-2 matrix multiplication. By submultiplicativity, this leads immediately to novel randomized algorithm designs, such as algorithms for Boolean matrix multiplication as well as detecting and estimating the number of triangles and other subgraphs in graphs. Joint work with Matti Karppa (Aalto University).

Reference: https://doi.org/10.1137/1.9781611975482.31

 

Discrete Models with Total Positivity

Dane Wilburne
York University

We consider the case of a discrete graphical loglinear model whose underlying distribution is assumed to be multivariate totally positive of order 2. In particular, we study the implications of total positivity on interactions between the random variables, the marginal polytope associated to the model, and model selection through maximum likelihood estimation. We also compare these results to recent work in Gaussian setting. This is joint work with Helene Massam.

 
3:00pm - 5:00pmMS134, part 7: 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)

 

An Asymmetric MacWilliams Identitity for Quantum Stabilizer Codes

Tefjol Pllaha
Aalto University

It was discovered in 2007 that a quantum channel is asymmetric with respect to errors. Namely, the bit-flip errors are more likely than the phase-flip errors. This motivates the study of asymmetric weight enumerators. We restrict ourselves to quantum stabilizer codes over Frobenius rings, for which we use character theory to prove asymmetric versions of the MacWilliams Identity.

 

Code-based crypto for small servers

Tanja Lange
Eindhoven University of Technology

Deployment of high-confidence code-based cryptography is hampered by the large keys associated with Goppa codes.This talk shows how to make use of the structure of encryption in code-based cryptography and how to combine this with tree hashing for confirming the integrity of the public key to use code-based cryptography for tiny, stateless network servers.

 

Reproducible Codes and Cryptographic Applications

Edoardo Persichetti
Florida Atlantic University

In this talk I will present a work in progress on structured linear block codes. The investigation starts from well-known examples and generalizes them to a wide class of codes that we call reproducible codes. These codes have the property that they can be entirely generated from a small number of signature vectors, and consequently admit matrices that can be described in a very compact way. I will show some cryptographic applications of this class of codes and explain why the general framework introduced may pave the way for future developments of code-based cryptography.

 

Hyperelliptic point-counting in genus 3 and higher, the RM case

Simon Abelard
University of Waterloo

The problem of counting points on hyperelliptic curves defined over finite fields has been studied for decades by number theorists and cryptographers. This work studies the case of large characteristic, using methods inspired by Schoof and Pila's algorithms. The cornerstone of this approach is to carefully model the torsion by polynomial systems and solve them using appropriate methods (resultants, geometric resolution, Groebner bases). In practice, the exponential dependency in the genus makes it hard to use these point-counting algorithms in genus larger than 2. Restricting to curves with explicit real multiplication, however, we can drastically reduce the size of our polynomial systems, even in arbitrary genus. In genus 3, the subsequent complexity gain allowed us to achieve a record computation over a 64-bit prime field. Part of this is joint work with P. Gaudry and P.-J. Spaenlehauer.

 
3:00pm - 5:00pmMS162, part 2: Applications of finite fields theory
Unitobler, F-123 
 
3:00pm - 5:00pm

Applications of finite fields theory

Chair(s): Antoine Joux (University of Sorbonne), Giacomo Micheli (EPFL), Violetta Weger (University of Zurich, Switzerland)

The theory of finite fields is one of the most important meeting points of Algebraic Geometry, Computer Science, and Number Theory. One of the most important challenges in the area is to develope the theory of finite fields in connection with useful applications, in particular in secure communication, coding theory, and pseudorandom number generation. In this minysimposium we plan to bring together experts from many different areas of the mathematics of communication who share the common interest towards the theory of finite fields. Our main purpose is to provide an overview of some of the cutting-edge research in the field, and to lay the fundations for new collaborations among researchers interested in applications of the theory of finite fields.
In the cryptographic setting, we focus on new post-quantum cryptographic schemes (Marco Baldi, Antoine Joux) and cryptanalysis (Gohar Kyureghyan, Yann Rotella). For pseudorandomness we propose construction of new pseudorandom generators (Federico Amadio Guidi, Laszlo Merai) and construction of polynomials over finite fields with given properties which are interesting for applications (Andrea Ferraguti).

 

(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)

 

Public key encryption and key exchange from LDPC codes: LEDAcrypt

Paolo Santini
Marche Polytechnic University

The pioneering work of McEliece in 1978 paved the way for code-based cryptography, which is still today a promising research area for the development of cryptographic primitives characterized by high efficiency and, most importantly, quantum resistance. Among several variants of the McEliece cryptosystem employing families of codes other than the original family of Goppa codes, those based on low-density parity-check (LDPC) codes have been shown able to achieve compact public keys and high algorithmic efficiency. This talk will recall the basic concepts of LDPC code-based cryptography, and then describe two primitives for asymmetric cryptography based on LDPC codes that are candidates to the NIST post-quantum cryptography standardization initiative: LEDAkem and LEDApkc.

 

Cryptological properties of mappings of finite fields

Gohar Kyureghyan
University of Rostock

Mappings used in some of cryptological primitives must be highly nonlinear, since linear ones are easy to predict. In this talk, we present several notions for optimal nonlinearity. We discuss connections between the different concepts and review known constructions and major open challenges in this research area.

 

Pseudorandom walks on elliptic curves

Laszlo Merai
RICAM

We give an overview of pseudorandom number generators (PRNGs) based on elliptic curves over finite fields. Many PRNGs are defined via a recursion law Pn = ψ(Pn-1) for some initial point P0 in E and a rational map (morphism) ψ:E → E of the curve E. An example for such PRNGs is the so-called power generator, where ψ is a scalar multiplication: ψ: P → eP for some integer e ≥ 2. We consider in detail the case when ψ is an arbitrary endomorphism of the curve.

We present bounds on the discrepancy and linear complexity of the obtained sequences.

 

Fractional Jumps and pseudorandom number generation

Federico Amadio Guidi
University of Oxford

In this talk we discuss a new construction of full orbit sequences in affine spaces over finite fields via Fractional Jumps of transitive projective automorphism, that is joint work with S. Lindqvist and G. Micheli. In dimension 1, our construction covers entirely the case of Inversive Congruential Generator (ICG) sequences. We explain how the sequences produced using Fractional Jumps enjoy the same discrepancy bounds as ICG sequences, but are less expensive to compute, thus representing a good source for pseudorandom number generation.