Spectra of surfaces and MCG actions on random covers
Abstract
The Ivanov conjecture is equivalent to the statement that every covering map of surfaces has the so-called Putman-Wieland property. I will discuss my recent work with Vlad Marković, where we prove it for asymptotically all coverings as the degree grows. I will give some overview of our main tool: spectral geometry, which is related to objects like the heat kernel of a hyperbolic surface, or Cheeger connectivity constant.
Coadmissible modules over Fréchet-Stein algebras
Abstract
Let K be a non-archimedean field of mixed characteristic (0,p), and let L be a finite extension of
the p-adic numbers contained in K. The speaker is interested in the continuous representations of a
given L-analytic group G in locally convex (usually infinite dimensional) topological vector spaces over K.
This is, up to technicalities, equivalent to studying certain topological modules over the locally
analytic distribution algebra D(G,K) of G. But doing algebra with topological objects is hard!
In this talk, we present an excellent remedy, found by Schneider and Teitelbaum in the early 2000s.
16:00
Traces of random matrices over F_q, and short character sums
Abstract
16:00
A new approach to modularity
Abstract
In the 1960's Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.
16:00
The dispersion method and beyond: from primes to exceptional Maass forms
Abstract
16:00
The Metaplectic Representation is Faithful
Abstract
Iwasawa algebras are completed group rings that arise in number theory, so there is interest in understanding their prime ideals. For some special Iwasawa algebras, it is conjectured that every non-zero such ideal has finite codimension and in order to show this it is enough to establish the faithfulness of the modules arising from the completion of highest weight modules. In this talk we will look at methods for doing this and apply them to the specific case of the metaplectic representation for the symplectic group.
16:00
On entropy of arithmetic functions
Abstract
In this seminar, I will talk about a notion of entropy of arithmetic functions and some properties of this entropy. This notion was introduced to study Sarnak's Moebius Disjointness Conjecture.
16:00
Quantitative bounds for a weighted version of Chowla's conjecture
Abstract
The Liouville function $\lambda(n)$ is defined to be $+1$ if $n$ is a product of an even number of primes, and $-1$ otherwise. The statistical behaviour of $\lambda$ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of $+1$'s and $-1$'s. For example, the two-point Chowla conjecture predicts that the average of $\lambda(n)\lambda(n+1)$ over $n < x$ tends to zero as $x$ goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.
16:00
A friendly introduction to Shimura curves
Abstract
Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties. Their less famous cousins, Shimura curves, are also very interesting, and very concrete.
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.