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
Location: Unitobler, F-112
30 seats, 54m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS197, part 1: Numerical differential geometry
Unitobler, F-112 
 
10:00am - 12:00pm

Numerical Differential Geometry

Chair(s): Tingran Gao (THE UNIVERSITY OF CHICAGO, United States of America), Ke Ye (Chinese Academy of Sciences)

The profound theory of differential geometry have interacted with the computational and statistical communities in the past decades, yielding fruitful outcomes in a wide range of fields including manifold learning, Riemannian optimization, and geometry processing. This minisymposium encourages researchers from applied differential geometry, optimization, manifold learning, and geometry processing to share their perspectives and technical tools on problems lying in the intersection of geometry and computations.

 

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

 

Introduction to Numerical Differential Geometry

Ke Ye
Chinese Academy of Sciences

It is quite a common phenomenon that data sets have special geometric strucutres, hence it is natural and convenient to model the problem on manifolds. In this talk, we provide an overview of the emerging impact of numerical differential geometry in areas of modern mathematical data science, including but not limited to manifold learning, Bayesian optimization, and geometry processing. As an illustrative example, we will present the framework and application of Riemannian optimization, with an emphasis on differential geometry as a guiding principle in the design and analysis of optimization algorithms.

 

A Riemannian Proximal Gradient Descent Method with Optimal Convergence Rate

Wen Huang
Xiamen University

We consider solving nonconvex and nonsmooth optimization problems with Riemannian manifold constraints. Such problems have received considerable attention due to many important applications such as sparse PCA, sparse blind deconvolution, robust matrix completion. In the Euclidean setting, proximal gradient method is an excellent method for solving nonconvex nonsmooth problems. However, in the Riemannian setting, the related work is still limited. In this talk, we briefly review the existing Riemannian proximal gradient methods and give an accelerated Riemannian proximal gradient with convergence analysis. Numerical experiments are used to demonstrate the performance of the proposedmethod.

This is joint work with Ke Wei at Fudan University.

 

Semi-Riemannian Manifold Optimization

Tingran Gao
The University of Chicago

We introduce a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice. In addition, for many interesting matrix manifolds, closed-form expressions for geodesics and parallel-transports are much easier to obtain under the semi-Riemannian metric.

 
3:00pm - 5:00pmMS197, part 2: Numerical differential geometry
Unitobler, F-112 
 
3:00pm - 5:00pm

Numerical Differential Geometry

Chair(s): Tingran Gao (THE UNIVERSITY OF CHICAGO, United States of America), Ke Ye (Chinese Academy of Sciences)

The profound theory of differential geometry have interacted with the computational and statistical communities in the past decades, yielding fruitful outcomes in a wide range of fields including manifold learning, Riemannian optimization, and geometry processing. This minisymposium encourages researchers from applied differential geometry, optimization, manifold learning, and geometry processing to share their perspectives and technical tools on problems lying in the intersection of geometry and computations.

 

(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)

 

Anisotropic Diffusion Kernels to Compare Distributions

Xiuyuan Cheng
Duke University

We introduce a kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions from finitely-many multivariate samples, where the kernel is anisotropic. The kernel computes the affinity between n data points and a set of nR reference points, where nR can be drastically smaller than n. When the unknown distributions are locally low-dimensional, the proposed MMD test can be more powerful to distinguish certain alternatives, which is theoretically characterized by the spectral decomposition of the kernel. The consistency of the test is proved as long as the magnitude of the distribution departure is of a higher order than n^{-1/2}, and a finite-sample lower bound of the testing power is provided. The test is applied to flow cytometry and diffusion MRI datasets, which motivate the proposed approach to compare distributions.

 

Coupled Geometric and Topological Basis for Data-Driven Shape Reconstruction

Qixing Huang
The University of Texas at Austin

We introduce a data-driven geometry reconstruction method with provable guarantees. A key enabler for the robustness of our shape recovery, with geometric and topological fidelity, is a new coupled basis representation that combines a voxelized implicit form of the shape geometry, and a vectorized persistent diagram of the shape topology. Our method optimizes an objective function that enforces the agreement between the reconstructed shape and the input point cloud, regularizes geometry and topology of the reconstruction with data and enforces the consistency between the geometric prior and the topological prior. We show how to solve this optimization problem effectively by combing spectral initialization under the geometric representation alone and gradient-descent refinement under the coupled representation. In particular, we show that the spectral initialization does not need to be accurate, as the refinement procedure is able to improve the topology of the reconstruction. Experimental results on synthetic and real datasets justify the usefulness of our approach.

 

Intrinsic Gaussian processes on complex constrained domains

Mu Niu
Plymouth University

We propose a class of intrinsic Gaussian processes (in-GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregular-shaped spaces arising as subsets or submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate spatial domains arising as complex sub- sets of Euclidean space. in-GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the proposed approach is to utilise the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computation- ally feasible covariance kernels. This enables in-GPs to be practically applied in great generality, while existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the in-GP approach are illustrated through simulation studies and data examples.

 

Locally Linear Embedding on Manifold

Nan Wu
Duke University

Locally Linear Embedding(LLE), is a well known manifold learning algorithm published in Science by S. T. Roweis and L. K. Saul in 2000. In this talk, we provide an asymptotic analysis of the LLE algorithm under the manifold setup. We establish the kernel function associated with the LLE and show that the asymptotic behavior of the LLE depends on the regularization parameter in the algorithm. We show that on a closed manifold, asymptotically we may not obtain the Laplace--Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. The talk is based on the joint work with Hau-tieng Wu.

 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmRoom free
Unitobler, F-112 
3:00pm - 5:00pmMS178: Geometric design for fabrication
Unitobler, F-112 
 
3:00pm - 5:00pm

Geometric design for fabrication

Chair(s): Helmut Pottmann (KAUST, Saudi Arabia)

Geometric modeling in the early design phase typically consists of pure shape design with little or no consideration of material properties, functionality and fabrication. The separation of geometry from engineering and manufacturing results in a costly product development process with multiple feedback loops. This minisymposium presents recent research on computational design tools which respect material properties and constraints imposed by function and fabrication. To achieve high performance, the additional constraints are closely tied to an adapted geometric representation or even formulated in terms of 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)

 

Geometric modeling of flank CNC machining

Michael Barton
BCAM, Bilbao

Geometric modeling is very closely related to manufacturing in situations when objects modeled in the digital realm are subsequently manufactured. The leading manufacturing technology is Computer Numerically Controlled (CNC) machining and this talk will focus on the finishing stage called flank milling. At this stage of machining, high accuracy of few micrometers for objects of size of tens of centimeters is needed and therefore the path-planning algorithms have to be carefully designed to respect synergy between the geometry of the milling tool and the input object. I will discuss two recent projects that look for the best initialization of a conical milling tool and the sequential path-planning algorithm. Finally, I will discuss future research directions towards machining with custom-shaped milling tools.

 

Modeling developable surfaces through orthogonal geodesics

Michael Rabinovich
ETH Zurich

We present a discrete theory for modeling developable surfaces through quadrilateral meshes satisfying simple angle constraints, termed discrete orthogonal geodesic nets (DOGs). Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. We prove and experimentally demonstrate strong ties to smooth developable surfaces including a set of convergence theorems. We show that the constrained shape space of DOGs is locally a manifold of a fixed dimension, apart from a set of singularities, implying that generally DOGs are continuously deformable. Smooth flows can then be constructed by a smooth choice of vectors on the manifold’s tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations

 

Developability of triangle meshes

Oded Stein
Columbia University

Developable surfaces can be fabricated by smoothly bending flat pieces of material without stretching or shearing. This enables a variety of fabrication methods, such as fabrication from flat material or 5-axis CNC milling. We introduce a discrete definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines, and show the importance of both of these properties. This definition provides a starting point for algorithms in developable surface modeling - we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient-based optimization of an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We also explore applications of this energy to developable design and manufacturing.

 

Statics-aware design of freeform architecture

Johannes Wallner
TU Graz

The design of 3D structures for architecture is not only geometric but involves financial, legal and statics considerations. It would be very valuable if design tools could incorporate some of these aspects already in an early state of design, in an interactive manner. In this presentation we show examples of how statics - both as a constraint and as an optimization target - can feature in the design of wide-span lightweight structures. We discuss a discretization of the Airy stress potential and its connection to selfsupporting surfaces and weight optimization.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS150, part 1: Fitness landscapes and epistasis
Unitobler, F-112 
 
10:00am - 12:00pm

Fitness landscapes and epistasis

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

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

 

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

 

Introduction to fitness landscapes and epistasis

Lisa Lamberti
ETHZ, Switzerland

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

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

 

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

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

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

 

A mechanistic approach to understanding multi-way interactions between mutations

Michael Harms
University of Oregon, USA

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

 

Understanding the biophysics of molecules from large functional assays

Jakub Otwinowski
MPI for Dynamics and Self-Organization, Germany

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

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

Fitness landscapes and epistasis

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

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

 

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

 

Shape theory, landscape topography and evolutionary dynamics

Joachim Krug, Malvika Srivastava
Uni Koeln, Germany

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

 

Graphs, polytopes, and unpredictable evolution

Kristina Crona
American University, Washington, USA

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

 

Computational complexity as an ultimate constraint on evolution

Artem Kaznatcheev
University of Oxford, UK

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

 

Tropical Principal Component Analysis and its Applications to Phylogenomics

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

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

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS128, part 1: 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)

 

Multilinear systems, determinantal resultants and the multiparameter eigenvalue problem

Matias Bender, Jean-Charles Faugère, Angelos Mantzaflaris, Elias Tsiagaridas
Inria, France

Multivariate resultant matrices characterize the roots of a polynomial system and reduce their computation to an eigenvalue problem. However, determinantal formulas (i.e. without extraneous factors) for the resultant do not exist for arbitrary systems. They have been constructed mostly for unmixed systems, that is, systems of polynomial equations with a common Newton polytope of special structure. In this talk we derive determinantal formulations for the multivariate resultant of structured systems with distinct supports per equation. We focus on mixed multilinear polynomial systems, that is multilinear systems with different supports per equation. These systems have applications to the Multiparameter Eigenvalue Problem (MEP).

 

Algorithmic aspects of the rational interpolation problem

Carlos D'Andrea
University of Barcelona

The Rational Interpolation Problem is an extension of the classical Polynomial Interpolation one, and there are several approaches to it: Euclidean algorithm, Linear Algebra with structured matrices, barycentric coordinates, orthogonal polynomials, computation of syzygies,.... We will review some of these methods along with some recent bounds in the complexity of their computation.

 

Computing Gröbner basis for sparse polynomial systems

Matias Bender, Jean-Charles Faugère, Elias Tsigaridas
Inria, France

Gröbner bases are one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example, several problems in computer-aided design, robotics, vision, biology, kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms. An approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gröbner bases over this algebra. Prior to our work, the algorithms that follow this approach benefited from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. In this talk, I will present the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, I will use it to compute Gröbner basis in the standard algebra and solve sparse polynomials systems over the torus. The complexity of the algorithm depends on the Newton polytopes and it is similar to the complexity of the solving techniques involving the sparse resultant. Hence, this algorithm closes a 25-years gap between the strategies to solve systems using resultants and Gröbner bases. Additionally, for particular families of sparse systems, I will use the multigraded Castelnuovo-Mumford regularity to improve the complexity bounds.

 

Real solving polynomial systems with interval method

Zafeirakis Zafeirakopoulos1, Mahmut Levent Doğan2
1Gebze Technical University, 2ODTÜ

Given an ideal I in a polynomial ring K[x_1, x_2, ... ,x_n], let its i-th elimination ideal be the ideal I_i= I cap K[x_1, x_2, ..., x_{n-i}]. The standard method for computing elimination ideals is by computing the Groebner basis G for I with respect to an elimination order and due to the elimination property of Groebner bases, the Groebner basis of I_i is G cap K[x_1, x_2, ..., x_{n-i}]. In this work we are interested in computing the i-th elimination ideal, without computing G, the Groebner basis of I, first. We do that by using the resultant system. The resultant system, introduced in van der Waerden's Modern Algebra, is a polynomial system having the properties we expect from a resultant. The resultant system contains a large number of polynomials, and thus it is not computationally efficient. We present an analysis of how we can improve the computation of the resultant system. Elimination ideals is a useful tool when dealing with parametric polynomial systems. Such systems often appear in applications and it was one of our original motivations, especially applications in combinatorics and motion planning.

 
3:00pm - 5:00pmMS137, part 3: Symbolic Combinatorics
Unitobler, F-112 
 
3:00pm - 5:00pm

Symbolic Combinatorics

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

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

 

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

 

Polynomial Reduction and Super Congruences

Qing-Hu Hou
Tianjin University

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. As applications, we derived two infinite families of super-congruences.

 

Diagonals, determinants, and rigidity

Christoph Koutschan
Radon Institute for Computational and Applied Mathematics

Diagonals of rational functions occur naturally in lattice statistical mechanics and enumerative combinatorics. We find that the diagonals of certain families of rational functions can be expressed as pullbacked hypergeometric 2F1 functions. On the other hand, the enumeration of combinatorial objects is often encoded by determinants. We study several families of binomial determinants that count the number of lozenge tilings of hexagonal domains with holes. Also graphs play a prominent role in combinatorics, and we are particularly interested in the aspect of rigidity. Using a novel combinatorial algorithm for computing the number of complex realizations of a maximally rigid graph, we explore exhaustively the Laman numbers of graphs with up to 13 vertices.

 

Central Limit Theorems from the Location of Roots of Probability Generating Functions

Marcus Michelen
University of Pennsilvania

For a discrete random variable, what information can we deduce from the roots of its probability generating function? We consider a sequence of random variables X_n taking values between 0 and n, and let P_n(z) be its probability generating function. We show that if none of the complex zeros of the polynomials P_n(z) are contained in a neighborhood of 1 in the complex plane then a central limit theorem occurs, provided the variance of X_n isn't subpolynomial in n. This result is sharp a sense that will be made precise, and thus disproves a conjecture of Pemantle and improves upon various results in the literature. This immediately improves a multivariate central limit theorem of Ghosh, Liggett and Pemantle, and has ramifications for certain variables that arise in graph theory contexts. This is based on joint work with Julian Sahasrabudhe.

 

Periodic Pólya urns and an application to Young tableaux

Michael Wallner
TU Wien

Pólya urns are urns where at each unit of time a ball is drawn uniformly at random and is replaced by some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time mod p.

Our key tool are generating functions, which encode all possible urn compositions after a certain number of steps. The evolution of the urn is then translated into a system of differential equations and we prove that the moment generating functions are D-finite. From these we derive asymptotic forms of the moments. When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions.

Furthermore, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions. This is joint work with Cyril Banderier and Philippe Marchal.

 

Date: Saturday, 13/Jul/2019
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.

 
3:00pm - 5:00pmRoom free
Unitobler, F-112