Talia Fernos – Vanderbilt

In this joint work with Balasubramanya, we explore the capacity for a group acting AU-acylindrically on a finite product of delta-hyperbolic spaces to satisfy three properties introduced by Bader, Furman, and Sauer. When satisfied, these properties restrict the potential ambient group in which it can be imbedded as a lattice. In this talk, we will also discuss the classification of actions on a delta-hyperbolic space, the associated trifurcation of elliptic actions, and the relationship to normal and commensurate subgroups. We will end the talk with an open question. 

John Ratcliffe – Emeriti- Vanderbilt

In this talk, we will show how to compute the G-index of a spin closed hyperbolic 4-manifold M for a group G of symmetries of a spin structure on M. As an example, we will compute the G-index for the group G of symmetries of the fully symmetric spin structure on the Davis closed hyperbolic 4-manifold M. Our talk will involve finite groups, infinite discrete groups, and Lie groups.  The talk is based on joint work with Steven Tschantz.

Huali Zhang – Hunan University

Disconzi proposed an open Problem D, about establishing low regularity solutions for 3D relativistic Euler equations with the logarithmic enthalpy $h_0$, initial velocity $\bu_0$, and modified vorticity $\bw_0$ belonging to $H^s \times H^s \times H^{s_0} (2<s_0<s)$. Similar results have been obtained for compressible Euler equations by Wang (see also Andersson-Zhang). Compared with the non-relativistic case, the velocity varies from space-like to time-like in relativistic Euler, which brings us challenges for Problem D. In this talk, we will give a positive answer to this open Problem D for 3D relativistic Euler equations.

• Zoom link: https://vanderbilt.zoom.us/j/98625175307

Robbie Lyman – Rutgers

In geometric group theory, we love groups and graphs. Every (abstract) group has (many) Cayley graphs, each one associated with a choice of a generating set. Recently I’ve been curious about topological groups, inspired by the budding theory around and community of people inspired by mapping class groups of infinite-type surfaces and by Christian Rosendal’s breakthrough work on geometries for topological groups. I’m hoping to share out some of what I’ve learned about topological groups acting on graphs. Much of this comes from recent joint work with Beth Branman, George Domat and Hannah Hoganson.

Host: Talia Fernos

Mark de Cataldo, Stony Brook University

The P=W Conjecture in Non Abelian Hodge Theory

The classical de Rham and the Hodge Decomposition theorems deal with the singular cohomology of a projective manifold with coefficients in the non-zero complex numbers C*. Non abelian Hodge theory seeks to generalize this picture, with complex reductive groups, such as the general linear group, playing the role of the abelian C*. Instead of cohomology groups, we obtain complex algebraic varieties and their singular cohomology groups carry additional structures. The P=W Conjecture seeks to relate two of these non-classical structures. This talk is devoted to introducing the audience to this circle of ideas and related developments.

Casey Rodriguez – UNC Chapel Hill.

In this talk, we explore recent developments in the field of gradient elasticity. We begin by providing an intuitive introduction to the theory, which extends classical elasticity by incorporating higher-order spatial derivatives that capture microstructural effects. These contributions become particularly important at small spatial scales, offering a more refined description of deformation that classical models cannot account for. We then present a novel theory of three-dimensional Green-elastic bodies with gradient elastic material boundary surfaces and highlight its application to modeling brittle fracture.