Scientific

Quantitative weak mixing for random substitution tilings

Speaker: 
Rodrigo Treviño
Date: 
Thu, Jul 2, 2020
Location: 
Zoom
Conference: 
Pacific Dynamics Seminar
West Coast Dynamics Seminar
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Abstract: 

"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.

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Unique ergodicity of horocycle flows on compact quotients of SL(2,R) following Coudène

Speaker: 
Jon Chaika
Date: 
Tue, Jun 30, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

Furstenberg proved that the horocycle flow on any compact quotient of SL(2,R) is uniquely ergodic. This has been generalized by many people. I will present a proof due to Yves Coudène, which I find elegant and can prove some of the generalizations of Furstenberg's theorem too.

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The counting formula of Eskin and McMullen

Speaker: 
Paul Apisa
Date: 
Tue, Jun 9, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

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.

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Specification and the measure of maximal entropy

Speaker: 
Vaughn Climenhaga
Date: 
Tue, Jun 23, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

There are various proofs that a transitive uniformly hyperbolic dynamical system has a unique measure of maximal entropy. I will outline a proof due to Bowen that uses the specification and expansivity properties, focusing on the example of shift spaces. If time permits, I will describe how Bowen's proof works for equilibrium states associated to nonzero potential functions.

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Geometric Langlands for hypergeometric sheaves

Speaker: 
Masoud Kamgarpour
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

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.

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On generalized hyperpolygons

Speaker: 
Laura Schaposnik
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan (arXiv:2001.06911). After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we shall conclude the talk by mentioning some partial results on current work on the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

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Finding mirrors for Fano quiver flag zero loci

Speaker: 
Elana Kalashnikov
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

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.

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Exact Lagrangians in conical symplectic resolutions

Speaker: 
Filip Zivanovic
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

Conical symplectic resolutions are a vast family of holomorphic symplectic manifolds that appear in representation theory, algebraic and differential geometry, and also in theoretical physics. Their typical examples arise from the hyperkähler quotient construction (quiver and hypertoric varieties) but also from the representation theory of Lie algebras (resolutions of Slodowy varieties, slices in affine Grassmannians). In this talk, I will focus on their symplectic topology. In particular, we find families of non-isotopic exact Lagrangian submanifolds in them arising from different C*-actions. These Lagrangians have a very nice symplectic topology; in particular, we conjecture (work in progress) that all of their Floer-theoretic invariants are completely determined by their topology. At the end of the talk, I will discuss the special cases of Nakajima quiver varieties and resolutions of Slodowy varieties, where their count becomes feasible and interesting in its own.

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Birational geometry of quiver varieties

Speaker: 
Gwyn Bellamy
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

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).

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Euler numbers of Hilbert schemes of points on simple surface singularities

Speaker: 
Hiraku Nakajima
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

We prove the conjecture by Gyenge, Némethi and Szendrői in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points Hilbn(C2/Γ) on a simple singularity C2/Γ, where Γ is a finite subgroup of SL(2). This is based on my preprint arXiv:2001.03834.

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