## Week of September 8, 2019

 Group Actions and Dynamics Decrease of Fourier coefficients of Furstenberg measures and renewal theory 4:15pm - LOM206 Let mu be a Borel probability measure on SL(2,R) with a finite exponential moment, such that the support of mu generates a Zariski dense subgroup in SL(2,R). We can define a unique probability measure on the circle, which is called the mu stationary measure or Furstenberg measure. We will prove, using Bourgain’s discretized sum-product estimate, that the Fourier coefficients of this measure go to zero with a polynomial speed. Starting from this result, we can obtain a spectral gap of the transfer operator, whose properties enable us to get an exponential error term in the renewal theorem in the context of random products of matrices. Geometry, Symmetry and Physics Matroidal Schur algebras and category O 4:30pm - LOM 214 The category O of Bernstein-Gelfand-Gelfand and the Schur algebra are interesting and fundamental objects.  Following motivation connected to 3-dimensional supersymmetric gauge theory, Braden-Licata-Proudfoot-Webster defined an analogue of category O associated to any hypertoric (aka toric hyperkahler) variety.  In joint work with Braden, we use the same geometry to define analogues of the Schur algebra.  In fact, we are able to define such an algebra starting from any matroid.  In work in progress with Ethan Kowalenko, we extend the construction of Braden-Licata-Proudfoot-Webster to a matroidal setting as well.  In work in progress with Jens Eberhardt, we show that these matroidal Schur algebras and matroidal category O are related to each other via a categorification.
 Algebra and Number Theory Seminar p-adic equidistribution of CM points and applications 4:15pm - LOM 205 Abstract: Consider a sequence of CM points of increasing p-adic conductor on a modular curve X. What is its limiting distribution in any of the geometric incarnations of X? Works from the 2000s give the answer for the Riemann surface X(C), and for the reduction of X modulo primes different from p. I will describe the answer in the p-adic (Berkovich) analytic setting. A weak generalisation of this result has an application to the p-adic Birch and Swinnerton-Dyer conjecture. Geometry & Topology Bounds on renormalized volume for Schottky manifolds 4:15pm - DL 431 During the talk I will introduce renormalized volume for convex co-compact hyperbolic 3-manifolds and will also describe bounds for Schottky manifolds in term of extremal lengths in the conformal surface at infinity. This will be used to partially answer a question by Maldacena about comparing renormalized volume for Schottky and Fuchsian manifolds with the same conformal boundaries.
 Undergraduate Seminar Putnam seminar 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 Weekly event for interested students to practice competition math problems in a relaxed setting. Meetings are held every Wednesday from 6:30pm to 8 in LOM 215. All are welcome to attend. Contact Pat Devlin to find out more.
 Combinatorics Seminar Approximate Spielman-Teng theorems 4:00pm - DL 431 An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ matrix $M_n$ is a statement of the following form: there exist constants $C,c > 0$ such that for all $\eta \geq 0$, $\mathbb{P}(s_n(M_n) \leq \eta) \lesssim n^{C}\eta + \exp(-n^{c})$. I will discuss a novel combinatorial approach for proving such theorems in a fairly unified manner for a variety of random matrix models.