Scientific

"Quantitative weak mixing" is the term used to bound the dimensions of spectral measures of a measure-preserving system. This type of study has gained popularity over the last decade, led by a series of results of Bufetov and Solomyak for a large class of flows which include general one-dimensional tiling spaces as well as translation flows on flat surfaces, as well as results on quantitative weak mixing by Forni. In this talk I will present results which extend the results for flows to higher rank parabolic actions, focusing on quantitative results for a broad class of tilings in any dimension. The talk won't assume familiarity with almost anything, so I will define all objects in consideration.

Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). The key insight is to relate the counting problems to the equidistribution of circles under geodesic flow. In this talk I will discuss how to deduce circle equidistribution and counting problem asymptotics from mixing. The talk will involve many pictures and focus on the case of hyperbolic surfaces, however, the arguments presented will be general and their application to counting on general affine symmetric spaces will be explained at the end of the talk.

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. I will report on a joint work with Lingfei Yi, where we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems. This confirms a central conjecture of the geometric Langlands program for hypergeometrics. The key tool we use is the notion of rigid automorphic data due to Zhiwei Yun. This talk is based on the preprint arXiv:2006.10870.

One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.

In this talk I will report on joint work in progress with A. Craw and T. Schedler on the birational geometry of quiver varieties. We give an explicit local description of the birational transformations that occur under variation of GIT for quiver varieties. The main consequence of this local picture is that one can show that all Q-factorial terminalizations of quiver varieties (excluding the (2,2) case) can be obtained by VGIT. I will try to explain what our results mean in two concrete classes of examples. Namely, for framed affine Dynkin quivers (corresponding to wreath product quotient singularities) and star shape quivers (corresponding to hyperpolygon spaces).