Abstracts

Week of February 26, 2023

February 27, 2023
Group Actions, Geometry and Dynamics 3-manifolds built out of 1-dimensional actions (Joint with Geometry/Topology seminar) 4:00pm -
LOM206

In a recent joint work with KyeongRo Kim and Hongtaek Jung, we show that a group of circle homeomorphisms is a 3-manifold group if it preserves a veering pair of invariant laminations. The proof has two parts - the topological part and dynamical part. We will try to explain both aspects. In some sense this is a partial converse to Thurston’s universal circle theorem. 
Geometry, Symmetry and Physics t-structures on the equivariant derived category of the Steinberg variety 4:30pm -
LOM 214

The Steinberg variety and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in zero and positive characteristics. Based on arXiv:2302.05782 and work in progress.
February 28, 2023
Geometry & Topology Anosov Flows on 3-manifolds 4:15pm -
LOM 206

Anosov flows are rich examples of dynamical systems, they include the geodesic flows on unit tangent bundles of hyperbolic surfaces, and many other examples. This talk is about how dynamics, geometry and topology interact in dimension 3 via some longstanding open questions: Which 3-manifolds support Anosov flows? Which 3-manifolds support many topologically distinct Anosov flows? What invariants can be used to distinguish them? I will describe some of the state of the art, and recent work with Thomas Barthelmé, Steven Frankel, and Sergio Fenley that provides new topological invariants towards this classification problem.
March 1, 2023
Colloquium Structure theorems for actions of homeomorphism groups. 4:10pm -
The groups Homeo(M) and Diff(M) of homeomorphisms or diffeomorphisms of a manifold M have many striking parallels with finite dimensional Lie groups.  In this talk, I’ll describe some of these, and explain joint work with Lei Chen, that gives a structure theorem for actions of homeomorphism and diffeomorphism groups on other spaces, analogous to some classical results for actions of locally compact Lie groups.  As applications, we answer many concrete questions towards classifying all actions of Diff(M) on other manifolds (many of which are nontrivial, for instance Diff(M) acts naturally on the unit tangent bundle of M…) and resolve several threads in a research program initiated by Ghys.  I’ll aim to give both a broad overview and several basic applications in the talk.
March 2, 2023
Group Actions, Geometry and Dynamics Joint equidistribution of maximal flat cylinders and holonomies in Anosov homogeneous spaces 4:00pm -
LOM 206

 Margulis–Mohammadi–Oh (‘14) described the asymptotic joint equdistribution of the closed geodesics and holonomies of a geometrically finite rank one locally symmetric manifold as their lengths grow to infinity.
 We obtain an Anosov analogue of this result. More precisely,  in the quotient of a higher rank semisimple Lie group by an  Anosov subgroup, non-trivial closed orbits under a maximal real-split torus give rise to maximal flat cylinders. We prove an analogous joint equidistribution result of the maximal flat cylinders and holonomies with circumferences growing to infinity. 

This is joint work with Elijah Fromm.
Analysis Recent results in elliptic homogenization. 4:00pm -

I will discuss some recent results and open problems in elliptic homogenization. This will include my work with Armstrong and Kuusi on large scale regularity for solutions of periodic operators
March 3, 2023
Friday Morning Seminar On syzygy categories of dimer tree algebras 9:30am -
LOM 215

A dimer tree algebra A is the Jacobian algebra of a quiver Q (without loops and 2-cycles) with a canonical potential such that (i) every arrow lies in a chordless oriented cycle; and (ii) the dual graph is a tree. The category of non-projective syzygies over A is equivalent to the stable category of (maximal) Cohen Macaulay modules as well as to the singularity category of A. It is a triangulated 3-Calabi-Yau category.
We give a geometric model for the syzygy category for dimer tree algebras in terms of 2-diagonals in a polygon with checkerboard pattern.
This checkerboard pattern can be extended to a dimer model on the disk.
Geometric Analysis and Application Intermediate curvature and a generalization of Geroch's conjecture 2:00pm -
LOM 215

In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to positive Ricci curvature for m =  1, and positive scalar curvature for m = n-1) on closed orientable manifolds with topology $N^n = M^{n-m} x \mathbb{T}^m$ for $n \leq 7$. Our proof uses a slicing constructed by minimization of weighted areas, the associated stability inequality, and estimates on the gradients of the weights and the second fundamental form of the slices. This is joint work with Simon Brendle and Sven Hirsch.