MenuEvents for 11/22/2024 from all calendars
Mathematical Biology Seminar
Time: 10:00AM - 11:00AM
Location: ZOOM
Speaker: Declan Stacy, University of Virginia
Title: Stochastic Extinction: An Average Lyapunov Function Approach
Abstract: In the analysis of stochastic models of ecology, epidemics, and turbulence it is a central problem to determine the stability of an invariant subset M_0 of M for an M-valued Markov process X_t. For example, ecologists wish to determine whether a subset of species will coexist or go extinct. Using average Lyapunov functions we can show that (under some mild technical assumptions) both the stability of M_0 and the rate of convergence of X_t to M_0 (in the stable case) can be determined entirely by a type of Lyapunov exponent which only depends on the steady state behavior(s) of X_t on M_0. This is an improvement over previous results which require constructing a Lyapunov function and analyzing its rate of change on the entirety of M. In the talk we will omit the proofs of the main results and instead focus on examples of applying the theory to various stochastic differential equations and jump processes used to model ecosystems and epidemics.
Algebra and Combinatorics Seminar
Time: 3:00PM - 3:50PM
Location: BLOC 302
Speaker: Chun-Hung Liu, TAMU
Title: Disjoint paths problem with group-expressable constraints
Abstract: We study an extension of the k-Disjoint Paths Problem where, in addition to finding k disjoint paths joining k given pairs of vertices in a graph, we ask that those paths satisfy certain constraints expressable by abelian groups. We give an O(n^8) time algorithm to solve this problem under the assumption that the constraint can be expressed as avoiding a bounded number of group elements; moreover, our O(n^8) algorithm allows any bounded number of such constraints to be combined. Group-expressable constraints include, but not limited to: (1) paths of length r modulo m for any fixed r and m, (2) paths passing through any bounded number of prescribed sets of edges and/or vertices, and (3) paths that are long detours (paths of length at least r more than the distance between their ends for fixed r). The k=1 case with the modularity constraint solves problems of Arkin, Papadimitriou and Yannakakis from 1991. Our work also implies a polynomial time algorithm for testing the existence of a subgraph isomorphic to a subdivision of a fixed graph, where each path of the subdivision between branch vertices satisfies any combination of a bounded number of group-expressable constraints. In addition, our work implies similar results addressing edge-disjointness. It is joint work with Youngho Yoo.
Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: M. Forbes, UIUC
Title: Approximating Transcendence Degree in Medium Characteristic
Abstract: A set of multivariate polynomials F=(f_1,...,f_m) are algebraically independent if there is no non-zero polynomial P such that P(f_1,...,f_m)=0. The transcendence degree of the set F is the maximum size of any algebraically independent subset. We consider the problem of computing the transcendence degree, when the polynomials F are given succinctly by algebraic circuits. This problem is motivated by applications to the polynomial identity testing problem, as well as the task of computing the dimension of algebraic varieties. It is currently known that this problem is efficiently solvable over fields of characteristic zero, and also when the field is of exponentially large characteristic. We show that the transcendence degree can be efficiently approximated when the characteristic is polynomially large.
Free Probability and Operators
Time: 4:00PM - 5:00PM
Location: BLOC 306
Speaker: Junchen Zhao, Texas A&M University
Title: Free products and rescalings involving non-separable abelian von Neumann algebras
Abstract: For a non-separable self-symmetric abelian von Neumann algebra A, we study rescalings of the free product of n copies of A with LF_r to define a new mutually non-isomorphic continuous family of non-separable interpolated free products that has a rescaling formula and a free product addition formula. Explicit computations will be given to demonstrate well-definedness of this family, their free product, and free products with finite-dimensional or hyperfinite von Neumann algebras. Using this family, we can also show the Murray-von Neumann fundamental group of the infinite free product of A is all of (0, \infty). This is joint work with Ken Dykema.
