In this talk I will discuss recent joint work with Dominik Gruber where 
we find a reasonable model for random (infinite) Burnside groups, 
building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the 
quotient of a free group of rank r by the normal subgroup generated by 
all elements of the form g^n; quotients of such groups are called 
Burnside groups.  In 1902, Burnside asked whether any such groups could 
be infinite, but it wasn't until the 1960s that Novikov and Adian showed 
that indeed this was the case for all large enough odd n, with later 
important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic 
groups in the 1980s and 90s, he observed that random quotients of free 
groups have interesting properties: depending on exactly how one chooses 
the number and length of relations one can typically gets hyperbolic 
groups, and these groups are infinite as long as not too many relations 
are chosen, and exhibit other interesting behaviour.  But one could 
equally well consider what happens if one takes random quotients of 
other free objects, such as free Burnside groups, and that is what we 
will discuss.
 

We present a construction which associates to a KdV equation the lamplighter group. 
In order to establish this relation we use automata and random walks on ultra discrete limits. 
It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy 
invariants of closed manifolds.

In this talk I will articulate and contextualize the following sequence of results.

The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.
Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.  
In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. 

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.  

In 1982, Gromov introduced bounded cohomology to give estimates on the minimal volume of manifolds. Since then, bounded cohomology has become an independent and active research field. In this talk I will give an introduction to bounded cohomology, state many open problems and relate it to other fields. 

I will discuss the basics of normal surface theory, and how they were used to give an algorithm for deciding whether a given diagram represents the unknot. This version is primarily based on Haken's work, with simplifications from Schubert and Jaco-Oertel.
 

The celebrated Lang-Vojta Conjecture predicts degeneracy of S-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier’s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at S-units. This is a joint work with A. Turchet.

In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.

In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.

I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture.