L6

A group is coherent if all its finitely generated subgroups are finitely presented. Aside from some easy cases, it appears that coherence is a phenomenon that occurs only among groups of cohomological dimension 2. In this talk, we will give many examples of coherent and incoherent groups, discuss techniques to prove a group is coherent, and mention some open problems in the area.

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

Consider a connected graph and choose a subset of its vertices. From this simple setup, Iyama and Wemyss define a collection of real hyperplanes known as an intersection arrangement, going on to classify all tilings of the affine plane that arise in this way. These "local" generalisations of Coxeter combinatorics also admit a nice wall-crossing structure via Dynkin involutions and longest Weyl elements. In this talk I give an analogous classification in the hyperbolic setting using the data of an "overextended" ADE diagram with three distinguished vertices. I then discuss ongoing work applying intersection arrangements to parametrise notions of stability conditions for preprojective algebras.

Suppose a finite group satisfies the following property: If you take two random elements, then with probability bigger than 5/8 they commute. Then this group is commutative. 

Starting from this well-known result, it is natural to ask: Do similar results hold for other laws (p-groups, nilpotent groups...)? Are there analogous results for infinite groups? Are there phenomena specific to the infinite setup? 

We will survey known and new results in this area. New results are joint with Gideon Amir, Maria Gerasimova and Gady Kozma.

Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and concerns work joint with Louis-Pierre Arguin and Asher Roberts.

Smooth generic representations of $GL_n$ over a $p$-adic field $F$, i.e. representations admitting a nondegenerate Whittaker model, are an important class of representations, for example in the setting of Rankin-Selberg integrals. However, in recent years there has been an increased interest in non-generic representations and their degenerate Whittaker models. By the theory of Bernstein-Zelevinsky derivatives we can associate to each smooth irreducible representation of $GL_n(F)$ an integer partition of $n$, which encodes the "degeneracy" of the representation. By using these "highest derivative partitions" we can define a stratification of the category of smooth complex representations and prove the surprising fact that all of the strata categories are equivalent to module categories over commutative rings. This is joint work with David Helm.

Motivated by a recent experiment on a superconducting quantum
information processor, I will discuss the Floquet quantum Ising model in
the presence of integrability- and symmetry-breaking random fields. The
talk will focus on the relation between boundary spin correlations,
spectral pairings, and effects of the random fields. If time permits, I
will also touch upon self-similarity in the dynamic phase diagram of
Fibonacci-driven quantum Ising models.
 

I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.