Events for 01/31/2025 from all calendars
Algebra and Combinatorics Seminar
Time: 3:00PM - 3:50PM
Location: BLOC 302
Speaker: Youngho Yoo, TAMU
Title: Structure in group-labelled graphs and its applications
Abstract: Group-labelled graphs provide a unified framework that can encode many natural graph constraints. I will present recent work on the structure of unoriented group-labelled graphs that led to the resolution of two problems on modularity constraints, each of which had been open for over 30 years. The first, studied by Arkin, Papadimitriou, and Yannakakis in 1991, is on the existence of a polynomial time algorithm to detect cycles of length L modulo M; we settle this affirmatively for all L and M. The second, posed by Dejter and Neumann-Lara in 1987, is on an approximate packing-covering duality of cycles of length L modulo M; we settle this by characterizing the topological obstructions to this duality. Our results are proved in the general setting of group-labelled graphs and have further applications beyond modularity constraints.
Nonlinear Waves and Microlocal Analysis
Time: 3:00PM - 4:00PM
Location: BLOC 624
Speaker: Dean Baskin, Texas A&M University
Title: The Feynman propagator in a model singular setting
Abstract: I will describe a global construction of the Feynman propagator for the wave equation with an inverse square potential on Minkowski space as well as its asymptotic behavior near infinity.
Departmental Colloquia
Time: 4:00PM - 5:00PM
Location: Bloc 117
Speaker: Ionut Chifan
Title: Classification of von Neumann algebras associated with property (T) groups
Abstract: In the mid-thirties, F.\ Murray and J.\ von Neumann introduced a natural way to associate a von Neumann algebra, $L(G)$, to every countable group $G$. Understanding the structural theory of these algebras, particularly the classification of $L(G)$ in terms of $G$, quickly became a central and challenging research theme. This is because these algebras often have a very limited \emph{memory} of the underlying group. A striking illustration of this phenomenon is A.\ Connes’ celebrated 1976 result, which shows that all nontrivial amenable with infinite nontrivial conjugacy classes (icc) groups give rise to isomorphic von Neumann algebras. Thus, in this case, aside from amenability, $L(G)$ retains no additional information about the algebraic structure of $G$. In the non-amenable case, the classification remains wide open and significantly more complex. However, instances where the von Neumann algebraic structure completely retains algebraic properties of the underlying group have been discovered through Popa’s deformation/rigidity theory. In my talk, I will survey several recent advances in the classification and the structural study of von Neumann algebras arising from property (T) groups. In this context, the famous Connes Rigidity Conjecture (1982) predicts that all icc property (T) groups $G$ are completely recognizable from $L(G)$. In the first part, I will introduce the first (and currently only) known examples of groups satisfying this conjecture. These groups, known as \emph{wreath-like products}, arise naturally in the context of group-theoretic Dehn filling. Next, I will discuss a natural generalization of the Connes Rigidity Conjecture for property (T) central extensions, and introduce a class of groups that satisfy this generalization, constructed using a natural quotienting technique applied to wreath-like products. In the final part, I will explain how wreath-like product groups can also be used to advance other longstanding open problems posed by A.\ Connes, V.F.R.\ Jones and S.\ Popa, concerning
Geometry Seminar
Time: 5:00PM - 6:00PM
Location: BLOC 302
Speaker: M. Varbaro
Title: Singularities of Herzog varieties
Abstract: Let S be a polynomial ring over a field and I a homogeneous ideal. We say that I as a Herzog ideal if there exists a monomial order < on S such that in_<(I) is squarefree. A projective variety X is a Herzog variety if it admits an embedding in which it is defined by a Herzog ideal. If X is a Herzog variety with respect to a revlex order, with Constantinescu and DeNegri we proved that the smoothness of X forces S/I to be Cohen-Macaulay with negative a-invariant (hence a (F)-rational singularity). We will discuss the problem wether this happens for any Herzog variety (not necessarily w.r.t. a revlex order); this is not even clear when X is a curve. In this case, rephrasing the problem the question is: if X is a Herzog smooth projective curve, does X have genus 0? In this talk we will largely discuss this problem, giving some evidence for it and explaining why it is difficult to show it in general, giving insights on an ongoing work with Amy Huang, Jonah Tarasova and Emily Witt.