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…

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…

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.

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…

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…

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…

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.…

We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented.  One goal is to study the invariance properties of families of ideals…

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…

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…

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.…

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…

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…

 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…

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…

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…

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.…