I will revisit the question of what is the holographic dual of N=6 supersymmetric Chern-Simons theories.

Over the last few years, it has become clear that working in sectors of large global charge leads to significant simplifications when studying strongly coupled CFTs. It allows us in particular to calculate the CFT data as an expansion in inverse powers of the large charge.
In this talk, I will introduce the large-charge expansion via the simple example of the O(2) model and will then apply it to a number of other systems which display a richer structure, such as non-Abelian global symmetry groups.
 

I will discuss joint work with Eero Saksman (Helsinki) describing the statistical behavior of the Riemann zeta function on the critical line in terms of complex Gaussian multiplicative chaos. Time permitting, I will also discuss connections to random matrix theory as well as some recent joint work with Saksman and Adam Harper (Warwick) relating powers of the absolute value of the zeta function to real multiplicative chaos.

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

In the last thirty years there was a lot of progress in understanding the asymmetric simple exclusion process (ASEP). Much less is currently known about the multi-species extension of ASEP. In the talk I will discuss the connection of such an extension to random walks on Hecke algebras and its probabilistic applications. 

I will talk about how to get large deviations estimates for randomly rotated matrix models using the asymptotics of spherical (aka orbital, aka HCIZ) integrals. Compared to the talk I gave last week in integrable probability conference I will concentrate on random  matrices rather than symmetric functions.

Quiver varieties are one of the main objects of study in Geometric Representation Theory. Defined by Nakajima in 1994, there has been a lot of research on them, but there is still a lot to be yet discovered, especially about their geometry. In this seminar, I will talk about their first use in Geometric Representation Theory as providing geometric representations of symmetric Kac-Moody Lie algebras.

Email @email to get a link to the Jitsi meeting room (it is included in the weekly announcements).

Non-uniformly elliptic functionals are variational integrals like
\[
(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,
\]
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$
\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\  \partial_{z}^{2} F(x,z)} $$
may blow up as $|z|\to \infty$. 
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$
w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x
$$
or
$$
w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.
$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).

I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.