MenuAlgebra and Combinatorics Seminar
Spring 2017
| Date: | January 20, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Jens Forsgård, Julia Plavnik, and Eric Rowell, Texas A&M University |
| Title: | Organizational Meeting |
| Date: | February 3, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Li Ying, Texas A&M University |
| Title: | Stability of the Heisenberg Product on Symmetric Functions. |
| Abstract: | The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and Kronecker product. I will give the definition of this product and describe some properties of it. One well known thing about the Kronecker product of Schur functions is the stability phenomenon discovered by Murnaghan in 1938. I will give an analogous result for the Heisenberg product of Schur functions. |
| Date: | February 10, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Xiaoxian Tang, Texas A&M University |
| Title: | Computing bounds for equiangular lines in Euclidean spaces |
| Abstract: | Determining the maximum number of equiangular lines in a r-dimensional Euclidean vector space is an open problem in combinatorics, frame theory, graph theory, linear algebra and many related areas. So far the exact maximum number is only known for a few small dimensions. In this talk, we compute the upper bound of number of equiangular lines by combing the classical pillar decomposition and the semi-definite programming (SDP) method. Our computational results show an explicit bound, which is strictly less than the well-known Gerzon's bound for the dimensions between 44 and 400. Particularly, when the angles is arccos(1/5) or arccos(1/7), we dramatically improve the known SDP bounds. |
| Date: | February 17, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Timo de Wolff, Texas A&M University |
| Title: | A Positivstellensatz for Sums of Nonnegative Circuit Polynomials |
| Abstract: | Deciding nonnegativity of real polynomials is a fundamental problem in real algebraic geometry and polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. The standard certificates for nonnegativity are sums of squares (SOS). In practice, SOS based semidefinite programming (SDP) is the standard method to solve polynomial optimization problems. In 2014, Iliman and I introduced an entirely new nonnegativity certificate based on sums of nonnegative circuit polynomials (SONC), which are independent of sums of squares. We successfully applied SONCs to global nonnegativity problems. In Summer 2016, Dressler, Iliman, and I proved a Positivstellensatz for SONCs, which provides a converging hierarchy of lower bounds for constrained polynomial optimization problems. These bounds can be computed efficiently via relative entropy programming. In this first of two talks on the topic I will give a brief overview about SONCs and Positivstellensätze in general and then introduce and prove our Positivstellensatz. The second, corresponding talk will occur directly afterwards in the geometry seminar. |
| Date: | February 24, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Luis David Garcia-Puente, Sam Houston State University |
| Title: | Counting arithmetical structures |
| Abstract: | Let G be a finite, simple, connected graph. An arithmetical structure on G is a pair of positive integer vectors d,r such that (diag(d)−A)r=0, where A is the adjacency matrix of G. Arithmetical graphs were introduced in the context of arithmetical geometry by Lorenzini in 1989 to model intersections of curves. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n−1, n−1), and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles. |
| Date: | March 3, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Yuyu Zhu, Texas A&M University |
| Title: | A Fast Algoritm for Complex Feasibility |
| Abstract: | Complex feasibility is the problem of deciding if a system of polynomials with integer coefficients has a complex solution. Koiran proved that under the assumption of generalized Riemann Hypothesis, this problem is in the polynomial hierarchy. We will talk about a fast algorithm to determine the satisfiability of the system based on this result. We will also look for weaker assumptions aided with results and conjectures on prime density. |
| Date: | March 24, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Xingwei Wang , Nankai University and Texas A&M University |
| Title: | Infinite log-monotonicity and finite difference of combinatorial sequences |
| Abstract: | Log-concavity and log-convexity of combinatorial sequences is related with unimodality of combinatorial sequences, real rootedness of polynomials and asymptotically normal distribution of combinatorial sequence. In this talk, we will introduce the notion of infinitely log-monotonic sequences originated from complete monotonic functions and show that some log-behaviors of combinatorial sequences can be deduced from this property. We will show many combinatorial sequence satisfying this property. Especially, we will discuss the finite difference of the logarithms of the partition function and the overpartition function. |
| Date: | March 31, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Elizabeth Gross , San Jose State University |
| Title: | Combinatorial and algebraic problems in systems biology |
| Abstract: | Systems biology focuses on modeling complex biological systems, such as metabolic and cell signaling networks. These biological networks are modeled with polynomial dynamical systems that can be described with directed graphs. Analyzing these systems at steady-state results in polynomial ideals with significant combinatorial structure. Using the Wnt shuttle model as an example, we will discuss some of the combinatorial and algebraic techniques available for parameter estimation and model selection. We will then look at the algebraic problems that arise when constructing new network models from smaller models through gluing operations on the corresponding directed graphs. This talk draws on joint work with Heather Harrington, Zvi Rosen, and Bernd Sturmfels, as well as joint work with Heather Harrington, Nicolette Meshkat, and Anne Shiu. |
| Date: | April 7, 2017 |
| Time: | 1:30pm |
| Location: | BLOC 506A |
| Speaker: | Matthew Titsworth, Collin College |
| Title: | Classifying fusion categories using invariant theory |
| Abstract: | Fusion categories are ubiquitous in mathematics and mathematical physics, arising in places like the representation theory of quantum groups, and it is a natural impulse to want to classify them. They can be constructed from solutions to polynomial equations which come with the actions of a gauge group G and G', its extension by automorphisms of the underlying Grothendieck ring. Monoidal equivalence of two solutions is then determined by whether or not the solutions lie in the same orbit of G' acting on the associated algebraic set X. In this talk, we consider the classification problem from the perspective of geometric invariant theory. We show that the quotients X/G and X/G' always exist and so for points in different orbits there is always a G (or G') invariant polynomial whose evaluation distinguishes them. As an application, we use this to classify fusion categories with the same fusion rules as representations of SO(2p+1)_2 with p a natural number. We construct a family of G-nvariants which uniquely distinguish equivalence classes and obtain an explicit count for fixed p. Finally, we conjecture the form of a single G'-invariant which distinguishes all equivalence classes for fixed p. |
| Date: | April 7, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Title: | Algebra and Combinatorics Course Discussion |
| Date: | April 7, 2017 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Alperen Ergür , North Carolina State University |
| Title: | Approximations to the Cone of Nonnegative Polynomials - (Joint Algebra and Combinatorics - Banach Spaces - Linear Analysis seminar) |
| Abstract: | In this talk, we discuss polyhedral approximations and approximation limits to the cone of nonnegative polynomials. Our approximation results relies on the recent work on spectral sparsification and inapproximability results are based on simple properties of Gaussian measure. Time permits, we will also discuss some open problems related to sums of squares approximation to the cone of nonnegative polynomials. |
| Date: | April 14, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Mitchell Phillipson, St. Edward's University |
| Title: | Counting Foldings in RNA |
| Abstract: | Everyone is familiar with DNA, the double stranded helix that makes us who we are. However, you may be less familiar with RNA. RNA is the single stranded cousin of DNA, responsible for many actions within cells. Because RNA is single stranded, it tends to fold onto itself; these foldings can potentially change the function of the RNA. Given an RNA sequence, it is possible that the sequence can fold in a variety of ways. An interesting combinatorial question is, how many ways can an RNA sequence fold onto itself? Further, are there RNA sequences that have a unique folding? If so can we classify them? In this talk we’ll work to answer these questions. |
| Date: | April 21, 2017 |
| Time: | 3:00pm |
| Location: | |
| Speaker: | TAGS, Rice University |
| Title: | Texas Algebraic Geometry Symposium |
| Date: | April 28, 2017 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Paul Bruillard, Pacific Northwest National Laboratory |
| Title: | Transposition Reconstruction |
| Abstract: | The Reconstruction Conjecture was formulated in 1941 and asserts that a graph on three or more vertices can be determined up to isomorphism by its vertex-deleted subgraphs. This conjecture is considered one of the foremost unsolved problems in graph theory and has been described as a graphical disease due to its contagiousness. In this talk we will review the vertex reconstruction problem and prior work. We will also review the related problem of edge reconstruction, i.e., whether or not a graph can be determined by its edge-deletion subgraphs. Finally, we will introduce the Transposition Reconstruction problem. This related problem appears to contain more information and thus may be easier than either vertex or edge reconstruction. However, many of the conventional tools are not applicable in transposition reconstruction. We will discuss progress on the transposition reconstruction problem and conclude with open questions. |
