Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Geometry & Representation - Chris Brav (Higher School of Economics, Moscow)

Monday, November 11th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Chris Brav (Higher School of Economics, Moscow)

Title: Cartan's magic without the formulas.

Abstract:  The Cartan calculus concerns vector fields on a smooth variety X acting on differential forms via Lie derivative and contraction, with Cartan's magic formula expressing the relation between the two actions: Lie derivative is the graded commutator of the de Rham differential with contraction. On a smooth variety, the magic formula can be checked in local coordinates, while for singular schemes (and more general prestacks) it is necessary to work with the tangent complex, where it is no longer feasible to give explicit local formulas. Interpreting the magic formula as giving Griffiths transversality for the Gauss-Manin connection of the universal infinitesimal deformation of X, we are able to construct a formula-free, chain level Cartan calculus using the tangent complex of a singular scheme, and to establish the compatibility of this calculus with the noncommutative calculus of Hochschild cochains acting on Hochschild chains. This is joint work with Nick Rozenblyum.

Department Colloquium - Ari Arapostathis (UT Austin)

Ari Arapostathis (UT Austin)

Friday, November 8th, 2019

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Ari Arapostathis (UT Austin)

Title: Lower bounds on the rate of convergence for heavy-tailed driven SDEs motivated by large scale stochastic networks.

Abstract: We show that heavy-tailed Levy noise can have a dramatic effect on the rate of convergence to the invariant distribution in total variation. This rate deteriorates from the usual exponential to strictly polynomial under the presence of heavy-tailed noise. To establish this, we present a method to compute a lower bound on the rate of convergence. We should keep in mind that standard Foster-Lyapunov theory furnishes only an upper bound on this rate. To motivate the study of such systems, we describe how L\'evy driven stochastic differential equations arise in the study of stochastic queueing networks. This happens when the arrival process is heavy-tailed, or the system suffers asymptotically negligible service interruptions. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity, and we show that these conditions are sharp. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence to the stationary distribution in total variation is polynomial, and we provide a sharp quantitative characterization of this rate via matching upper and lower bounds. We conclude by presenting analogous results on convergence in the Wasserstein distance.

This talk is based on joint work with Hassan Hmedi, Guodong Pang and Nikola Sandric.

Ari Arapostathis is a professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and holds the Texas Atomic Energy Research Foundation Centennial Fellowship in Electrical Engineering. He received his BS from MIT and his PhD from U.C. Berkeley, in 1982. He is a Fellow of the IEEE, and was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. His research has been supported by several grants from the National Science Foundation, the Air-Force Office of Scientific Research, the Army Research Office, the Office of Naval Research, DARPA, the Texas Advanced Research/Technology Program, Samsung, and the Lockheed-Martin Corporation.

Number Theory Seminar - Anup Dixit (Queen's University)

Thursday, November 7th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: On the distribution of certain sequences and a prime number theorem.

Abstract: The fourth problem of Landau conjectures that there are infinitely many primes of the form $n^2+1$. Inspired by this, we consider sequences of the form $\{[n^{\alpha}] + 1\}$, where $[x]$ denotes the greatest integer less than or equal to $x$. In this talk, we will discuss how often the elements of such a sequence lie in a given arithmetic progression for $\alpha<2$ and also establish an analogue of prime number theorem for $\alpha<1$. This is joint work with Prof. M. Ram Murty.

Statistics & Biostatistics - Yanglei Song (Queen's University)

Wednesday, November 6th, 2019

Time: 11:30-12:30 Place: Jeffery Hall 225

Speaker: Yanglei Song (Queen's University)

Abstract: I will start with a discussion on the recent development in the normal approximation (a.k.a. central limit theorem) and bootstrap for the sum of high dimensional random vectors, empirical processes and U-processes. Statistical applications will also be provided. Then I will talk about a piece of ongoing work in this line, with Xiaohui Chen and Kengo Kato. Its abstract is as follows: This paper studies the non-asymptotic inference for the supremum of an incomplete, non-degenerate U-process. The process is indexed by a function class of order r, whose complexity possibly increases with the sample size n. For each function, its corresponding U-statistic involves the average of O(n^r) numbers, which is prohibitively demanding even for moderate r. Thus we study its incomplete version, where each subsample of size r is included in the average with a very small probability. We first approximate the supremum of such incomplete U-process by that of an appropriate Gaussian process in the Kolmogorov distance and then propose valid bootstrap methods to address the practical issue of unknown covariance function. Finally, we discuss its application in testing the qualitative features, such as convexity, of nonparametric functions.

Geometry & Representation - Anne Dranowski (University of Toronto)

Monday, November 4th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Anne Dranowski (University of Toronto)

Title: Generalized orbital varieties and MV modules.

Abstract:  Let O be the conjugacy class of a nilpotent matrix, and let C be its closure. By work of Joseph and Spaltenstein, the irreducible components of the subvariety of uppertriangular matrices in C, (aka orbital varieties,) can be labeled by standard Young tableaux. We explain how this labeling generalizes to the intersection of C and and a Slodowy slice, S. This question is motivated by the fact (due to Mirkovic-Vybornov) that such intersections are related to the Mirkovic-Vilonen (MV) construction of a cohomological crystal basis of GL(m). By D., the Mirkovic-Vybornov isomorphism maps the generalized orbital varieties to the MV cycles such that the crystal structure on tableaux matches the crystal structure on MV cycles. Our labeling enables us to determine equations of MV cycles and therefore compare the MV basis to another basis in bijection with tableaux - Lusztig's dual semicanonical basis - under the magnifying glass of the Duistermaat-Heckman measure.

Dynamics, Geometry, & Groups - Giulio Tiozzo (Queen's University)

Friday, November 1st, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Giulio Tiozzo (Queen's University)

Title: Entropy and drift for Gibbs measures on geometrically finite manifolds.

Abstract: The boundary of a simply connected, negatively curved manifold carries two natural types of measures: on one hand, Gibbs measures such as the Patterson-Sullivan measure and the SRB measure. On the other hand, harmonic measures arising from random walks. We prove that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, drift and critical exponent, extending the previous formulas of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold (or more generally, a CAT(-1) space) is geometrically finite but not convex cocompact, harmonic measures are singular with respect to Gibbs measures.