We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 
 

Abstract: A quasimap from a curve to a GIT quotient is a map to the stack quotient that is generically stable. The geometry of Laumon spaces (an open subset of quasimaps from P^1 to the flag variety) is closely related to the representation theory of gl_n. It has been shown that one can construct an action of gl_n on the cohomology of Laumon spaces via geometric correspondences, and this cohomology can be identified with dual Verma modules of gl_n under this action. The full moduli space of quasimaps provides a natural compactification of Laumon spaces. I will explain how to construct an action of gl_n on the equivariant cohomology of these moduli spaces and explore its relation to tilting modules in Category O.

Given a finitely supported probability measure on a Kleinian group, Kaimanovich showed that the Poisson boundary of the associated random walk is the boundary of the hyperbolic space equipped with the hitting measure. It has been conjectured that the hitting measure is singular to conformal measures of the Kleinian group. In this talk, we mainly focus on how Mostow’s rigidity can be generalized to show the expected singularity when the Kleinian group is not convex cocompact, which is my joint work with Andrew Zimmer. We also discuss other applications of this general machinery besides the singularity conjecture.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 

Abstract: This talk will discuss regularity results for harmonic maps into a certain kind of metric space, called Euclidean buildings.  This will include a discussion of what these spaces are, the behavior of harmonic maps into these spaces, and the connection to superrigidity results for lattices.  We will discuss previously known regularity results before presenting a recent theorem of Breiner, Mese, and myself which generalizes these considerably. 

We are interested in surface subgroups of the coaffine subgroup of GL(4,R) via their actions on 3-dimensional projective space.  Many such surface subgroups are (projective)-Anosov and preserve a properly convex subset of projective space, on which they act with compact 3-manifold quotient with surface boundary.  In this talk, we explain how to identify the boundary of the locus of Anosov representations in terms of the shape of the unit ball for the “stable norm” on homology with respect to the relevant asymmetric metric on the surface.  We use this description and certain dual best Lipschitz maps to describe the structure of the limit sets for these degenerate surface groups.  There is, in particular, an oriented geodesic lamination of maximal stretch whose endpoints are all crushed to the same point in the limit set.  This is joint work with Marit Bobb.

Random band matrices have entries concentrated in a narrow band of width W around the main diagonal, modeling systems with spatially localized interactions.

We consider one-dimensional random band matrices with bandwidth W >> N^½, general variance profile, and arbitrary entry distributions. We establish complete isotropic delocalization, quantum unique ergodicity (eigenstate thermalization), and Wigner-Dyson universality in the bulk of the spectrum. The key technical input is a family of local laws capturing the spatial decay of resolvent entries, established using a combination of Ornstein-Uhlenbeck dynamics and Green function comparison (the Zigzag strategy). Based on joint work with László Erdős.

I will define conical symplectic singularities and give some examples. Then I’ll discuss their quantizations.

Abstract:  We define a family of Turaev-Viro type invariants of hyperbolic 3-manifolds with totally geodesic boundary from the 6j-symbols of the modular double of U_q(sl(2; R)), and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the “1-loop term” the adjoint twisted Reidemeister torsion of the double of the manifolds. This is a joint work with Tianyue Liu, Shuang Ming, Xin Sun and Baojun Wu.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 

Abstract: Quantitative unique continuation states that eigenfunctions of the Laplacian on smooth closed manifolds cannot vanish faster than exponentially in its eigenvalue in any open set. It is interpreted physically that the probability of a quantum particle to appear in the classically forbidden region (total energy < potential) is at least exponentially small, also known as quantum tunnelling. In this informal talk, we will discuss the strategy to prove it on both closed and open manifolds with specified end structures (like cones or cylinders) and discuss open problems.

There are many topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), complementary component types of a curve, mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we focus on the so called inf invariant (shortest length metric) associated to a filling curve, its relationship with the geometric self-intersection number, and its relation to the optimal metric that is tailored to produce the minimum length. While clearly the geometric self-intersection number is a type invariant, we address whether the inf invariant can distinguish between curves that have the same self-intersection. This is joint work with Ara Basmajian.

This is the last lecture in the series. I will discuss the restriction functor for Harish-Chandra modules.

Abstract: A periodic pencil of flat connections on a smooth algebraic variety X is a linear family of flat connections ∇(s1, …, sn) = d − Xr i=1 Xn j=1 sjBijdxi , where {xi} are local coordinates on X and Bij : X → MatN are matrixvalued regular functions. A pencil is periodic if it is generically invariant under the shifts sj 7→ sj + 1 up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic p, the p-curvature operators {Ci , 1 ≤ i ≤ r} of a periodic pencil ∇ are isospectral to the commuting endomorphisms C ∗ i := Pn j=1(sj − s p j )B (1) ij , where B (1) ij is the Frobenius twist of Bij . This allows us to compute the eigenvalues of the p-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.

Abstract:  Questions regarding the complexity of the simplex method in linear programming  turn out to be related in special cases to questions about partially ordered sets.  Exploring this connection led us to study lattices arising as 1-skeleta of simple polytopes, obtaining the homotopy type of the intervals in these posets as well as a geometric way of constructing lattice-theoretic joins.  In recent joint work with Christian Gaetz, we generalized this to the setting of directionally simple polytopes, motivated by questions about Bruhat interval polytopes.  This allowed us to  generalize a result of Athanasiadis, Edelman and Reiner that the reduced expressions for any fixed permutation are highly connected under braid moves and commutation moves, obtaining an analogous statement for the BCFW bridge decompositions of reduced plabic graphs (a family of graphs arising in the study of the totally nonnegative real part of the Grassmannian); we also determined the homotopy type for each interval in posets arising as 1-skeleta of Bruhat interval polytopes.   This talk will not assume background in this area and will mention several open questions.  

Abstract: In this talk, we will talk about the proof of that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. As a corollary, it implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. This addresses a question posed by Evans-Spruck and De Giorgi. The talk is based on the joint work with Tang-Kai Lee.

Essential embedded surfaces in an irreducible 3-manifold correspond to non-trivial splittings of its fundamental group. In joint work with Khánh Lê, we give some conditions on the fundamental group of a Haken hyperbolic 3-manifold which guarantee that any other 3-manifold group with the same set of finite quotients must have a non-trivial splitting. Using one of these conditions, we show that every finite regular cover of an aspherical integer homology three sphere with positive first Betti number will have first Betti number at least four and this is optimal. We will also discuss examples of Haken 3-manifolds to which the theorems in this work apply.