Tue, 20 Oct 2020

12:45 - 13:30

A Randomised Subspace Gauss-Newton Method for Nonlinear Least-Squares

Zhen Shao
((Oxford University))
Abstract

We propose a subspace Gauss-Newton method for nonlinear least squares problems that builds a sketch of the Jacobian on each iteration. We provide global rates of convergence for regularization and trust-region variants, both in expectation and as a tail bound, for diverse choices of the sketching matrix that are suitable for dense and sparse problems. We also have encouraging computational results on machine learning problems.

Tue, 17 Nov 2020

14:15 - 15:15
Virtual

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Susan Sierra
(Edinburgh University)
Abstract

Let W be the Witt algebra of vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W.  We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir.

We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W).  We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores.  We then lift this classification to Sym(Vir), and use it to show that if λ0, then (zλ) is a maximal Poisson ideal of Sym(Vir).

Mon, 16 Nov 2020

16:00 - 17:00

Introduction to sieve theory and a variation on the prime k-tuples conjecture

Ollie McGrath
Abstract

Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the prime k-tuples conjecture which involves sums of two squares. No knowledge of sieves is required!

Mon, 26 Oct 2020

16:00 - 17:00
Virtual

From curves to arithmetic geometry: Parshin's trick

Jay Swar
Abstract

In 1983, Faltings proved Mordell's conjecture on the finiteness of K-points on curves of genus >1 defined over a number field K by proving the finiteness of isomorphism classes of isogenous abelian varieties over K. The "first" major step from Mordell's conjecture to what Faltings did came 15 years earlier when Parshin showed that a certain conjecture of Shafarevich would imply Mordell's conjecture. In this talk, I'll focus on motivating and sketching Parshin's argument in an accessible manner and provide some heuristics on how to get from Faltings' finiteness statement to the Shafarevich conjecture.

Wed, 09 Sep 2020

16:00 - 17:00

An elementary proof of RH for curves over finite fields

Jared Duker Lichtman
Abstract

The Riemann hypothesis (RH) is one of the great open problems in mathematics. It arose from the study of prime numbers in an analytic context, and—as often occurs in mathematics—developed analogies in an algebraic setting, leading to the influential Weil conjectures. RH for curves over finite fields was proven in the 1940’s by Weil using algebraic-geometric methods. In this talk, we discuss an alternate proof of this result by Stepanov (and Bombieri), using only elementary properties of polynomials. Over the decades, the proof has been whittled down to a 5 page gem! Time permitting, we also indicate connections to exponential sums and the original RH.
 

Mon, 19 Oct 2020

16:00 - 17:00

Khovanskii's Theorem and Effective Results on Sumset Structure

Michael Curran
Abstract

A remarkable theorem due to Khovanskii asserts that for any finite subset A of an abelian group, the cardinality of the h-fold sumset hA grows like a polynomial for all sufficiently large h. However, neither the polynomial nor what sufficiently large means are understood in general. We obtain an effective version of Khovanskii's theorem for any AZ whose convex hull is a simplex; previously such results were only available for d=1. Our approach also gives information about the structure of hA, answering a recent question posed by Granville and Shakan. The work is joint with Leo Goldmakher at Williams College.

Mon, 12 Oct 2020

16:00 - 17:00
Virtual

Classical and elliptic polylogarithms

Nil Matthes
(Oxford)
Abstract

The Dirichlet class number formula gives an expression for the residue at s=1 of the Dedekind zeta function of a number field K in terms of certain quantities associated to K. Among those is the regulator of K, a certain determinant involving logarithms of units in K. In the 1980s, Don Zagier gave a conjectural expression for the values at integers s 2 in terms of "higher regulators", with polylogarithms in place of logarithms. The goal of this talk is to give an algebraic-geometric interpretation of these polylogarithms. Time permitting, we will also discuss a similar picture for Hasse--Weil L-functions of elliptic curves.
 

Tue, 20 Oct 2020

14:15 - 15:15
Virtual

Subspace arrangements and the representation theory of rational Cherednik algebras

Stephen Griffeth
(Universidad de Talca)
Abstract

I will explain how the representation theory of rational Cherednik algebras interacts with the commutative algebra of certain subspace arrangements arising from the reflection arrangement of a complex reflection group. Potentially, the representation theory allows one to study both qualitative questions (e.g., is the arrangement Cohen-Macaulay or not?) and quantitative questions (e.g., what is the Hilbert series of the ideal of the arrangement, or even, what are its graded Betti numbers?), by applying the tools (such as orthogonal polynomials, Kazhdan-Lusztig characters, and Dirac cohomology) that representation theory provides. This talk is partly based on joint work with Susanna Fishel and Elizabeth Manosalva.

Mon, 02 Nov 2020

16:00 - 17:00

Stochastic Ricci flow on surfaces

JULIEN DUBEDAT
(Columbia University)
Abstract

The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an infinite-dimensional gradient flow. We introduce a natural 'Langevin' version of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

 

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