Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory - Siddhi Pathak (Queen's University)

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 $k > 1$.

Geometry & Representation - Ben Webster (Waterloo)

Monday, October 1st, 2018

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

Speaker: Ben Webster (University of Waterloo/Perimeter Institute)

Title: Representation theory of symplectic singularities

Abstract:  There are a lot of non-commutative algebras out there in the world, so if you want to study some of them, you have to have a theory about which are especially important. One class I find particularly interesting are non-commutative algebras which "almost" commutative and thus can be studied with algebraic geometry, giving a rough dictionary between certain non-commutative algebras and certain interesting spaces. This leads us to a new perspective on some well-known algebras, like universal enveloping algebras, and also to new ones we hadn't previously considered. The representations of the resulting algebras have a lot of interesting structure, and have applications both in combinatorics and in the construction of knot invariants.

Department Colloquium - Jon Chaika (University of Utah)

Jon Chaika, University of Utah

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.

Probability Seminar - Jamie Mingo (Queen's University)

Thursday, September 27th, 2018

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

Speaker: Jamie Mingo (Queen's University)

Title: Additive Convolution and Subordination

Abstract:  Last week I reviewed the basic facts of scalar free independence. This week I will continue with a new convolution of probability measures called the free additive convolution. We will use a tool from complex analysis called subordination to get our convolution. It will then lead to a discussion of conditional expectation which will be the basis of operator valued freeness.

Free Probability and Random Matrices Seminar Webpage:

Control Theory - Christoph Kawan (University of Passau)

Thursday, September 27th, 2018

Time: 9:00-10:30 a.m Place: Jeffery Hall 319

Speaker: Christoph Kawan (University of Passau)

Title: Robust estimation under information constraints for deterministic non-linear systems

Abstract: A fundamental problem in information-based control is to estimate the state of a dynamical system using information sent through a rate-limited channel. In this talk, we explain the concept of restoration entropy, introduced by Matveev and Pogromsky, which characterizes the smallest channel capacity above which a robust coding and estimation policy with arbitrarily small estimation error can be implemented. In particular, we provide a characterization of restoration entropy that involves no asymptotic quantities and leads to nearly optimal policies.

CYMS Seminar - Noriko Yui (Queen's University)

Thursday, September 27th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Noriko Yui (Queen's University)

Title: Four-dimensional Galois representations arising from certain Calabi--Yau threefolds

Abstract: We consider the (irreducible) four-dimensional Galois representations arising from certain Calabi--Yau threefolds over ${\bf{Q}}$ with all the Hodge numbers of the third cohomology groups equal to $1$. There are many examples of (families) of such Calabi--Yau threefolds. The modularity/automorphy of such Calabi--Yau threefolds will be the main topic of discussion. There are two venues to be considered. In one venue, we ought to count the number of rational points over finite fields of these Calabi--Yau threefolds to concoct their L-series. In the other venue, we ought to construct some modular varieties, in this case, conjecturally, Siegel modular forms of weight $3$ and genus $2$ on some paramodular subgroups of $Sp(4,{\bf{Z}})$, and then compute their L-series. Such modular forms may be constructed using Borcherds forms. The ultimate aim is to establish a Langlands correspondence between the two L-series, thereby establishing the modularity/automorphy of such Calabi--Yau threefolds.

This is a joint work with Yifan Yang (National Taiwan University).

Number Theory - Neha Prabhu (Queen's University)

Tuesday, September 25th, 2018

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

Speaker: Neha Prabhu (Queen's University)

Title: The error term in the Sato-Tate theorem of Birch.

Abstract: In 1968, Birch proved a vertical analogue of the Sato-Tate conjecture for elliptic curves showing the asymptotic behaviour of $a_E(p)$, a quantity associated to an elliptic curve $E$ mod p. An error term for this result was obtained by Banks and Shparlinski in 2009 using the results of Katz and Neiderreiter. In this talk, we shall see that this error term can also be obtained in an elementary fashion using ideas in Birch's paper. This is joint work with Ram Murty.

Geometry & Representation - Emine Yildrim (Queen's University)

Monday, September 24th, 2018

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

Speaker: Emine Yildrim (Queen's University)

Title: The bounded derived category for cominuscule posets

Abstract:  Cominuscule posets come from root posets and have connections to Lie theory and Schubert calculus. We are interested in whether the bounded derived category of the incidence algebra of a cominuscule poset is fractionally Calabi-Yau. In other words, we ask if some non-zero power of the Serre functor is a shift functor. We answer this question on the level of the Grothendieck groups. On the Grothendieck group this functor becomes an endomorphism called the Coxeter transformation. We show that Coxeter transformation has finite order for two of the three infinite families of cominuscule posets, and for the exceptional cases. Our motivation comes from a conjecture by Chapoton which states that the bounded derived category of incidence algebra of root posets is fractionally Calabi-Yau. Our result can be thought of as a parabolic analogue of Chapoton's conjecture.

Department Colloquium - Boris Levit (Queen's University)

Boris Levit, Queen's University

Friday, September 21st, 2018

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

Speaker: Boris Levit (Queen's University)

Title: Optimal Cardinal Interpolation in Approximation Theory, Nonparametric Regression, and Optimal Design

Abstract: For the Hardy classes of functions analytic in the strip around real axis of a size $2\beta$, an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery. It will be shown that this method, based on the Jacobi elliptic functions, is also optimal according to the criteria of Nonparametric Regression and Optimal Design. In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from $0$. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant's bias and variance mutually cancel each other. In the limiting case $\beta \rightarrow \infty $, the optimal interpolant converges to the well known Nyquist-Shannon cardinal sampling series.

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