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Wen-Shin Lee, University of Antwerp, “Sparse interpolation, Padé approximation, signal processing, and tensor decomposition”

SAS 4201

A mathematical model is called sparse if it is a combination of only a few non-zero terms. The aim of sparse interpolation is to determine both the support of the sparse linear combination and the coefficients in the representation, from a small or minimal amount of data samples. This talk centers around multi-exponential models: A…

Rainer Sinn, Georgia Tech, “Pythagoras numbers of real projective varieties”

The Pythagoras number of field F, studied in the theory of quadratic forms, is the smallest k such that every sum of squares in F is a sum of k squares. We will reinterpret this definition for coordinate rings of real projective varieties and discuss ways to give bounds on this invariant. A central concept…

Erdal Imamoglu, NC State, Algorithms for Solving Linear Differential Equations with Rational Function Coefficients

SAS 4201

We present two algorithms for computing hypergeometric solutions of a second order linear differential equation with rational function coefficients. Our first algorithm uses quotients of formal solutions, modular reduction, Hensel lifting, and rational reconstruction. Our second algorithm first tries to simplify the input differential equation using integral bases and then uses quotients of formal solutions.

Wen-shin Lee, University of Antwerp, Belgium, Identification problem in exponential analysis

The Blahut/Ben-Or/Tiwari sparse interpolation algorithm from computer algebra is closely related to Prony’s method for exponential analysis. In this talk, we focus on some challenges in exponential analysis. Estimating the spectral information of an exponential sum plays an important role in many signal processing applications. In particular, we mention the direction of arrival (DOA) problem in smart antenna technology…

Gleb Pogudin, Courant Institute of Mathematical Sciences, Algorithms for Checking Global Identifiability

SAS 4201

The following situation arises in modeling: one has a system of differential equations with parameters and wants to determine the values of these parameters measuring unknown functions (assuming that perfect noise-free measurements are possible). Usually, some of the unknown functions are impossible or very hard to measure, so only a subset of them is available…

Yang Qi, University of Chicago, On approximations and decompositions of a general tensor

SAS 4201

Tensors are closely related to secant varieties. In fact, the affine cone of the $r$th secant variety of the Segre variety is the set of tensors whose border rank is less than or equal to $r$. Similarly, we have a geometric interpretation of symmetric tensors. By studying the geometry of these secant varieties, we can…

Deane Yang, New York University, Introduction to Convex Geometry and Brunn-Minkowski Theory

SAS 4201

Convex geometry is the study of convex bodies in Euclidean space. Despite the apparent simplicity of such objects, they are a source of many deep mathematical discoveries and mysteries. This talk will present a survey of Brunn-Minkowski theory, which is the study of affine geometric invariants and inequalities satisfied by convex bodies. Unlike differential geometry,…

Harm Derksen, University of Michigan, Matrix Invariants and Complexity

SAS 4201

We consider the action of the group SL_n x SL_n on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. Visu Makam and the speaker recently proved that invariants of degree at most mn^4 generate the invariant ring. This result has interesting applications in algebraic complexity theory and is related to the notion of non-commutative rank.…

Jeaman Ahn, Kongju National University, Multivariate Hermite Interpolation via Explicit Groebner Basis

Multivariate Hermite interpolation problem asks to find a "small" polynomial that has given values of several partial derivatives at given points. It has numerous applications in science and engineering. Thus, naturally, it has been intensively studied, resulting in various beautiful ideas and techniques.  One approach is as follows. (1) Chooses a basis of the vector space of interpolating polynomials.…

John Perry, University of Southern Mississippi, The dynamic approach to Gröbner basis computation

SAS 4201

Most algorithms to compute a Gröbner basis are “static”, inasmuch as they require as input both a set of polynomials and a term ordering, and preserve the term ordering throughout the computation.  This talk presents ongoing work on “dynamic” Buchberger algorithms. First described by Sturmfels and Caboara, dynamic algorithms require only a set of polynomials…

Victor Magron, LAAS-CNRS, France, The quest of efficiency and certification in polynomial optimization

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In 2001, Lasserre introduced a nowadays famous hierarchy of relaxations, called the moment-sums of squares hierarchy, allowing one to obtain a converging sequence of lower bounds for the minimum of a polynomial over a compact semialgebraic set. Each lower bound is computed by solving a semidefinite program (SDP). There are two common drawbacks related to…

Alban Quadrat, Sorbonne University, Paris, France, An introduction to the Quillen-Suslin theorem: algorithms and applications

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In 1955, Serre conjectured that every row vector with entries in a commutative polynomial ring R=k over a field k, admitting a right inverse over R, could be completed into a square matrix whose determinant is 1. That conjecture was independently proved by Quillen and Suslin in 1976 and is nowadays called the Quillen-Suslin theorem.…

Jonathan Hauenstein, University of Notre Dame, Energy landscapes and algebraic geometry

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Broadly speaking, an energy landscape is the graph of a loss function over a parameter space.  Some examples include the potential energy landscape of a chemical reaction, measuring the fitness of a mechanism to perform many tasks, and a loss function arising from a training set in machine learning.  This talk will discuss some successes…

Gregor Kemper, Technical University of Munich, Distance geometry, algebra and drones

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Is it always possible to reconstruct a point configuration in the plane from the unlabeled set of mutual distances between the points? This and other questions translate to invariant theory and ultimately into problems about polynomial ideals. As it turns out, the following question is related: Can a drone detect the walls of a room…

Symbolic Computation: Parker Edwards, University of Notre Dame, Feature Sizes and Bottlenecks for Algebraic Manifolds

SAS 4201

 In computational topology and geometry, theoretical guarantees for algorithms often take the following form: Start with a finite sample of points from a subspace of . If the sample is "dense enough" with respect to the subspace, then the algorithm outputs a quantity of interest for the subspace, for example its Betti numbers. The quantities associated…

Symbolic Computation Seminar: Josué Tonelli Cueto, The University of Texas at San Antonio, Computing numerically the homology of semialgebraic sets

SAS 4201

Computing the homology of semialgebraic sets is a central problem in computational real algebraic geometry. However, as of today, all symbolic algorithms for this problem require still time that is doubly exponential with respect to the number of variables. The latter is so although the size of the Betti numbers is known to be singly…

Symbolic Computation Seminar: Máté Telek , University of Copenhagen, Reaction networks and a generalization of Descartes’ rule of signs to hypersurfaces

SAS 4201

The classical Descartes’ rule of signs provides an easily computable upper bound for the number of positive real roots of a univariate polynomial with real coefficients. Descartes' rule of signs is of special importance in applications where positive solutions to polynomial systems are the object of study. This is the case in reaction network theory…

Symbolic Computation Seminar: Sriram Gopalakrishnan, Sorbonne Université, The arithmetic complexity of computing Grobner bases of determinantal systems

SAS 4201

Determinantal systems are systems of polynomial equations which encode a rank deficiency of a given matrix with polynomial entries over the solution set to other polynomial equations. Such systems arise in a number of areas of computational mathematics such as polynomial optimization, real algebraic and enumerative geometry and engineering sciences such as robotics and biology.…