Events for 10/26/2018 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Dylan Allegretti, University of Sheffield

Title: Categorified canonical bases and framed BPS states

Abstract: In a famous paper from 2006, Fock and Goncharov introduced a moduli space of framed PGL(2,C)-local systems on a surface with boundary. This moduli space has the structure of a cluster variety, and the algebra of regular functions on this cluster variety has a canonical vector space basis. In this talk, I will describe a family of graded vector spaces which categorify Fock and Goncharov's canonical basis. In certain cases, these vector spaces arise as the cohomology of moduli spaces of stable quiver representations as predicted by the physics of BPS states in N=2 field theories.

Working Seminar in Groups, Dynamics, and Operator Algebras

Time: 2:00PM - 2:50PM

Location: BLOC 506A

Speaker: Xin Ma, Texas A&M University

Title: Topological full groups of one-sided shifts of finite type V

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Anastasia Chavez, University of California, Davis

Title: Dual Equivalence Graphs and CAT(0) Combinatorics

Abstract: In this talk we will explore the combinatorial structure of dual equivalence graphs G_lambda. The vertices are Standard Young tableaux of fixed shape lambda that allows us to further understand the combinatorial structure of G_lambda, and the edges are given by dual Knuth equivalences. The graph G_lambda is the 1-skeleton of a cubical complex C_lambda. One can ask whether the cubical complex is CAT(0); this is a desirable metric property that allows us to describe the combinatorial structure of G_lambda very explicitly. We will discuss the CAT(0) characterization of Ardila--Owen--Sullivant. It is constructive and provides an algorithm for determining when a cubical complex is CAT(0). Using their characterization, we prove that C_lambda is CAT(0) if and only if lambda is a hook. This is joint work with John Guo.

Seminar on Banach and Metric Space Geometry

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Jari Taskinen, University of Helsinki

Title: Schauder bases and the decay rate of the heat equation

Abstract: Joint work with José Bonet (Valencia) and Wolfgang Lusky (Paderborn) We consider the classical Cauchy problem for the linear heat equation with integrable initial data f = f(x) in the Euclidean space R^N \ni x. As well known, the conventional solution formula implies that for generic f, the sup-norm (w.r.t. x) of the solution u(x,t) decays at the speed rate t^{-N/2} for large times t. Faster decay rates are possible for special initial data. Our aim is to present a new approach to this phenomenon and also to show that initial data leading to faster decay rates is in a certain sense not so rare. Accordingly, given a weight function w = w(x) growing rapidly at the infinity, we construct Schauder bases (e_n)_{n=1}^\infty in the Banach space L_w^p (R^N) , 1 < p < \infty or p=1, with the following property: given an arbitrary natural number m, one can find M such that for any initial data f belonging to the closed linear span of (e_n)_{n=M}^\infty , the solution of the Cauchy problem for the heat equation decays at least at the rate t^{-m} in the sup-norm. In particular, the subspace of initial data leading to "fast" decay is finite codimensional. Moreover, the mentioned basis can be constructed as a "perturbation" of any given basis. The proof is based on a construction of bases which annihilate an infinite sequence of bounded linear functionals. We also discuss the background of the problem and possible generalizations.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Dylan Allegretti , University of Sheffield

Title: The monodromy of meromorphic projective structures

Abstract: A projective structure on an oriented surface S is an atlas of charts mapping open subsets of S into the Riemann sphere. There is a natural map from the space of projective structures to the PGL(2,C) character variety of S which sends a projective structure to its monodromy representation. In this talk, I will describe a meromorphic analog of this construction. I will introduce a moduli space parametrizing projective structures with poles at a discrete set of points. I will explain how, in this setting, the object parametrizing monodromy data is a type of cluster variety. This is joint work with Tom Bridgeland.