Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Department Colloquium - Atabey Kaygun (Istanbul Tech University)

Atabey Kaygun (Istanbul Technical University)

Friday, November 1st, 2019

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

Speaker: Atabey Kaygun (Istanbul Technical University)

Title: Noncommutative Geometry for Fun and Profit.

Abstract: There are whole fields of mathematics devoted to transferring problems of geometry and topology to commutative algebra (and vice versa) and solving them. This practice produced different "dictionaries'" of terms that tell us which type of objects in the realm of geometry or topology correspond to which other types of objects in the realm of algebra. In this talk, I am going to describe such a dictionary from the perspective of a homological algebraist who forgoes commutativity "for fun and profit" going through K-theory, cyclic and Hochschild homology, Hopf algebras, and quantum groups.

Prof. Atabey Kaygun works on homological and homotopical algebra in the context of noncommutative geometry. He obtained his Ph.D. from The Ohio State University in 2005. He was a postdoctoral fellow at the University of Western Ontario, KMMF-Warsaw University, Max-Plank-Institut fur Mathematik and University of Buenos Aires before joining the faculty of Bahcesehir University in 2009. He has been an associate professor at the Istanbul Technical University since 2016. Prof. Kaygun is currently on sabbatical and visiting Queen's University.

Number Theory Seminar - Arpita Kar (Queen's University)

Thursday, October 31st, 2019

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

Speaker: Arpita Kar (Queen's University)

Title: On the distribution of certain sums of Random multiplicative functions.

Abstract: In this talk, I will discuss a result of Adam Harper regarding sums of Rademacher multiplicative functions $f(n)$, over those $n \leq x$ with $k$ distinct prime factors where $k$ is a function of $x$. In particular, we will discuss the Martingale central limit theorem due to Mcleish, and see how it establishes Harper's result.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, October 23rd, 2019

Time: 4:00-5:30 p.m Place: Jeffery Hall 319

Speaker: Mike Roth (Queen's University)

Title: Betti tables for canonical curves of genus 5 and 6.

Abstract: We will work out the Betti tables for canonically embedded curves of genus 5, and as time permits, genus 6. We will also discuss the classical theorems of Petri and Enriques-Babbage, the ‘basepoint-free pencil trick’ (due to Castelnuovo), and the modern idea of the ‘Castelnovo-Mumford regularity of a sheaf.