Events for 04/09/2021 from all calendars

Noncommutative Geometry Seminar

Time: 1:00PM - 2:00PM

Location: Zoom 951 5490 42

Speaker: Jean-Eric Pin, Université Paris Denis Diderot et CNRS

Title: A noncommutative extension of Mahler’s interpolation theorem

Abstract: I will report on a result recently obtained with Christophe Reutenauer. Let p be a prime number. Mahler’s theorem on interpolation series is a celebrated result of p-adic analysis. In its simplest form, it states that a function from N to Z is uniformly continuous for the p-adic metric d_p if and only if it can be uniformly approximated by polynomial functions. We prove a noncommutative generalization of this result for functions from a free monoid A* to a free group F(B) (or more generally to a residually p-finite group), where d_p is replaced by the pro-p metric. One of the challenges is to find a suitable definition of polynomial functions in this noncommutative setting.

URL: Event link

Mathematical Physics and Harmonic Analysis Seminar

Time: 2:00PM - 2:50PM

Location: Zoom

Speaker: Jared Wunsch, Northwestern University

Title: Semiclassical analysis and the convergence of the finite element method

Abstract: An important problem in numerical analysis is the solution of the Helmholtz equation in exterior domains, in variable media; this models the scattering of time-harmonic waves. The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity. I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter. The essential new tools come from semiclassical microlocal analysis. No knowledge of either FEM or semiclassical analysis will be assumed in the talk, however.

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Anton Dochtermann, Texas State University

Title: Betti numbers of random edge ideals

Abstract: We study asymptotic homological properties of random quadratic monomial ideals in a polynomial ring R = k[x_1, . . . , x_n], utilizing methods from the Erd\"os-R\'enyi model of random graphs. Here we consider a graph on n vertices and exclude an edge (corresponding to a quadratic generator of the ideal I) with probability p, and consider algebraic properties as n tends to infinity. Our main results involve fixing the edge parameter p = p(n) so that asymptotically almost surely the Krull dimension of R/I is fixed. Under these conditions we establish various properties regarding the Betti table of R/I, including sharp bounds on regularity and projective dimension and distribution of nonzero Betti numbers. These results extend work of Erman-Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Our results use collapsibility properties of random clique complexes and Garland's method regarding spectral gaps of graphs, and in particular rely on the underlying field in some cases. This is joint work with Andrew Newman.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: zoom

Speaker: M. Velasco, Universidad de los Andes, Bogotá, Colombia

Title: The geometry of SOS-multipliers on varieties

Abstract: A homogeneous polynomial F admits an SOS-multiplier certificate on a real projective variety X if there exist sums of squares g and s such that Fg=s in the homogeneous coordinate ring of X. Such an expression certifies the non-negativity of F and is thus of considerable theoretical and practical importance, giving us new algorithms for global optimization as well as many new open problems at the interface between real and complex algebraic geometry. In this talk I will present ongoing work with G. Blekherman, R. Sinn and G.G. Smith on the geometry of such non-negativity certificates. Our main results are effective bounds on the degrees of possible g's on algebraic curves and surfaces which depend on classical geometric invariants of the varieties. These bounds are, in some cases, provably optimal providing us with the strongest lower bounds on effective version of Hilbert's 17th problem.