Research Seminar, SJSU

Research Seminar, SJSU

Please join us the first and third Mondays of the month at 3:00pm in MH 223 for the Research Seminar. This schedule will be updated as the semester proceeds.

Schedule

September 19, 2016

Jose Rodriguez (University of Chicago)

Title: Numerical computation of Galois groups

Abstract:

The Galois/monodromy group of a family of equations (or of a geometric problem) is a subtle invariant that encodes the structure of the solutions. In this talk, we will use numerical algebraic geometry to compute Galois groups. Our algorithm computes a witness set for the critical points of our family of equations. With this witness set, we use homotopy continuation to construct a generating set for the Galois group. Examples from optimization will be stated (maximum likelihood estimation and formation shape control). Joint work with Jonathan Hauenstein (University of Notre Dame) and Frank Sottile (Texas A&M). Reference: http://arxiv.org/abs/1605.07806

October 3, 2016

Elizabeth Gross (SJSU)

Title: Geometry of Exponential Graph Models

Abstract:

Exponential random graph models (ERGMs) form one of the most flexible class of statistical models for network data. ERGMs are defined by a set of network statistics, e.g. the number of edges, the number of 2-paths, the number of triangles, etc, and thus give rise to interesting graph theoretical questions. In this talk we will focus on ERGMs where the triangle count is a defining statistic; these models are useful for modeling networks with a transitivity effect such as social networks. After introducing ERGMs, we will show how goodness-of-fit testing can be performed by understanding the integer points on an algebraic variety, which will lead us into a discussion on ideals and varieties, and more broadly, computational algebraic geometry.

October 17, 2016

Anton Leykin (Georgia Tech)

Title: Solving polynomial systems via homotopy continuation and monodromy

Abstract:

We develop an algorithm to find all solutions of a generic system in a family of polynomial systems with parametric coefficients using numerical homotopy continuation and the action of the monodromy group. We argue that the expected number of homotopy paths that this algorithm needs to follow is roughly linear in the number of solutions. Our software implementation demonstrates the practicality of the method: one of the examples shows how to find all equilibria of a chemical reaction network by solving a large system of equations in less than one second. (Joint work with Timothy Duff, Cvetelina Hill, Anders Jensen, Kisun Lee, and Jeff Sommars.)

November 7, 2016

Anna Seigal (UC Berkeley)

Title: Real Rank Two Geometry

Abstract:

A tensor is real rank two if it can be written as a sum of two real outer products of vectors. Similarly, the real rank two locus of an algebraic variety is the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set and its boundary. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. We also examine the real rank two locus of a curve in three-dimensional space. This talk is based on joint work with Bernd Sturmfels.

Novemeber 21, 2016

Anastasia Chavez (UC Berkeley)

Title: The Dehn-Sommville relations and the Catalan Matroid

Abstract:

The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is a simplicial polytope the Dehn--Sommerville relations condense the f-vector into the g-vector, which has length equal to the ceiling of (d+1)/2. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which ceiling[ (d+1)/2]-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid. This is joint work with Nicole Yamzon.

December 5, 2016

Christopher O'Neill (UC Davis)

Title: Shifting numerical monoids

Abstract:

A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.

Past Semesters