Department of Mathematics and Statistics

Tuesday, October 9th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: On the Euler-Kronecker constants.

Abstract: In 2006, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$, which is a generalization of the Euler-Mascheroni constant. This constant surprisingly arises in several seemingly unrelated aspects of analytic number theory. Ihara studied this constant systematically and produced bounds on $gamma_K$ under GRH. In this talk, we prove unconditional bounds for $\gamma_K$ in some cases and discuss its connection to the Brauer-Siegel theorem.

Friday, October 5th, 2018

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

Speaker: Aaron Smith (Queen's University)

Title: Mixing and the Glass Transition.

Abstract: Supercooled liquid forms when a liquid is cooled below its usual freezing temperature without entering a crystalline solid phase. As supercooled liquids continue to get colder, they exhibit something called the glass transition: they remain disordered, but start to otherwise behave much like solids. This glass transition is important for many materials, including rubbers and colloids, but is not theoretically well-understood. In this talk I will introduce a simple model for the glass transition that is easy to understand but difficult to study. I will then introduce two related families of models, introduced by physicists, that seem to give similar "glassy" behaviour. Finally, I will present some heuristics and recent results on the relaxation and mixing behavior of these two models. My results in this talk are from joint and ongoing work with Paul Chleboun, Alessandra Faggionato, Fabio Martinelli, Natesh Pillai and Cristina Toninelli.

Aaron Smith works in the areas of applied probability, with a focus on Markov chain Monte Carlo and related methods from computational statistics or statistical physics. He also has interest in in data mining and machine learning. He obtained his Ph.D. at Stanford in Mathematics, and was an undergraduate student at Queen's University. He held short-term appointments at Federal government of Canada, Brown University (applied math), and Harvard (statistics). He is currently an assistant professor in the Department of Mathematics and Statistics at the University of Ottawa.

Thursday, October 4th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Pei-Lun Tseng (Queen's University)

Title: A-Valued Theory

Abstract:  In order to study operator-valued probability, we need some basic knowledge of C*-algebras. In this talk, I am trying to review what is a C*-algebra. and some properties of C*-algebras. The GNS construction will be covered, which is known as the follows: every C*-algebras can be represented as a algebra of bounded linear operators on a Hilbert space. This construction gives us good starting point to study matrices over a C*-algebra. If time permits, I will introduce right Hilbert A-modules.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, October 2nd, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: On the values of the Epstein zeta function.

Abstract: Given a positive definite binary quadratic form, $Q(X,Y)$, the Epstein zeta function attached to $Q$ is given by $Z_Q(s) = \sum_{m,n} Q(m,n)^{-s}$, where the sum is over all tuples $(m,n)$ in $\mathbb{Z} \times \mathbb{Z}$, excluding $(0,0)$. This series converges absolutely for $Re(s)>1$. In this talk, we will present a result by J. R. Smart that 'evaluates' $Z_Q(k)$ for positive integers $k > 1$.

Friday, September 28th, 2018

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

Speaker: Jon Chaika (University of Utah)

Title: Horocycle orbits in strata of translation surfaces.

Abstract: Ergodic theory is concerned with describing the long term behavior of orbits as time evolves. Ratner, Margulis, Dani and many others, showed that the horocycle flow have strong measure theoretic and topological rigidity properties that allow a good understanding of every such orbit. Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, showed that the action of $SL(2,\mathbb{R})$, and its upper triangular subgroup, on strata of translation surfaces have similar rigidity properties. We will describe how some of these results fail for the horocycle flow on strata of translation surfaces. In particular, 1) There exist horocycle orbit closures with fractional Hausdorff dimension; 2) There exist points which do not equidistribute under the horocycle flow with respect to any measure; 3) There exist points which equidistribute under the horocycle flow with respect to a measure, but they are not in the topological support of that measure. No familiarity with these objects will be assumed and the talk will begin with motivating the subject of dynamics and ergodic. This is joint work with John Smillie and Barak Weiss.

Jon Chaika works in the field of Dynamical systems. He did his undergraduate at the University of Iowa, obtained his Ph.D. from Rice, then went to the University of Chicago before coming to the University of Utah.