Department of Mathematics and Statistics

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.

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:

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).

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.

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.