L6

Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. I'll discuss some recent refinements of Gallagher's theorem, one of which is joint work with Niclas Technau. A key new ingredient is the correspondence between Bohr sets and generalised arithmetic progressions. It is hoped that these are the first steps towards a metric theory of multiplicative diophantine approximation on manifolds. 

Seminal work of Hida tells us that if a modular eigenform is ordinary at p then we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it says that we can find q-expansions with coefficients in p-adic analytic function of the weight variable k that when evaluated at positive integers give the q-expansion of classical eigenforms. His construction of these families uses mainly the geometry of the modular curve and its ordinary locus.
In a joint work with Marc-Hubert Nicole, we obtained similar results for Drinfeld modular forms over function fields. After an extensive introduction to Drinfeld modules, their moduli spaces, and Drinfeld modular forms, we shall explain how to construct Hida families for ordinary Drinfeld modular forms.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. 
In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)
 

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree \Delta is rapidly mixing for k \ge \Delta+2. In 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper k-colorings for k > (11/6)\Delta, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the (11/6)\Delta barrier for general graphs by showing rapid mixing for k > (11/6 - \eta)\Delta for some positive constant \eta. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda’s 2007 survey paper on Glauber dynamics for colorings. 

This is joint work with Michelle Delcourt and Luke Postle.

Multiple zeta values are generalizations of the special values of Riemann's zeta function at positive integers. They satisfy a large number of algebraic relations some of which were already known to Euler. More recently, the interpretation of multiple zeta values as periods of mixed Tate motives has led to important new results. However, this interpretation seems insufficient to explain the occurrence of several phenomena related to modular forms.

The aim of this talk is to describe an analog of multiple zeta values for complex elliptic curves introduced by Enriquez. We will see that these define holomorphic functions on the upper half-plane which degenerate to multiple zeta values at cusps. If time permits, we will explain how some of the rather mysterious modular phenomena pertaining to multiple zeta values can be interpreted directly via the algebraic structure of their elliptic analogs.

I will describe an extremely easy construction with formal group laws, and a 
slightly more subtle argument to show that it can be done in a coordinate-free
way with formal groups.  I will then describe connections with a range of other
phenomena in stable homotopy theory, although I still have many more 
questions than answers about these.  In particular, this should illuminate the
relationship between the Lambda algebra and the Dyer-Lashof algebra at the
prime 2, and possibly suggest better ways to think about related things at 
odd primes.  The Morava K-theory of symmetric groups is well-understood
if we quotient out by transfers, but somewhat mysterious if we do not pass
to that quotient; there are some suggestions that dilation will again be a key
ingredient in resolving this.  The ring $MU_*(\Omega^2S^3)$ is another
object for which we have quite a lot of information but it seems likely that 
important ideas are missing; dilation may also be relevant here.
 

The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th power of the augmentation ideal}\}$.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.
In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some quotient $\delta_n/\gamma_n$.
Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.
 

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

NSOP1 is a class of first order theories containing simple theories, which contains many natural examples that somehow slip-out of the simple context.

As in simple theories, NSOP1 theories admit a natural notion of independence dubbed Kim-independence, which generalizes non-forking in simple theories and satisfies many of its properties.

In this talk I will explain all these notions, and in particular talk about recent progress (joint with Nick Ramsey) in the study of Kim-independence, showing transitivity and several consequences.