L4

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term $\mathrm{Tr}(\Phi^4)$. By methods from quantum field theory we show that also the quartic case is exactly solvable. All cumulants can be expressed as composition of elementary functions with the inverse of another elementary function. For infinite matrices the inversion gives rise to hyperlogarithms and zeta values as familiar from quantum field theory. For finite matrices the elementary functions are rational and should be viewed as branched covers of Riemann surfaces, in striking analogy with the topological recursion of the Kontsevich model. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable.
 

Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).
 

Exceptional holonomy manifolds come with certain geometric data that include a Ricci flat metric. Finding examples is therefore very difficult. The task can be made more tractable by imposing symmetry.  The focus of this talk is the case of torus symmetry. For a particular rank of the torus, one gets a natural parameterisation of the orbit space in terms of so-called multi-moment maps. I will discuss aspects of the local and global geometry of these 'toric' exceptional holonomy manifolds. The talk is based on joint work with Andrew Swann.

In this talk we present p-order methods for unconstrained minimization of convex functions that are p-times differentiable with Hölder continuous p-th derivatives. We establish worst-case complexity bounds for methods with and without acceleration. Some of these methods are "universal", that is, they do not require prior knowledge of the constants that define the smoothness level of the objective function. A lower complexity bound for this problem class is also obtained. This is a joint work with Yurii Nesterov (Université Catholique de Louvain).
 

I will explain the basics of 2-representation theory and will explain an approach to classifying 'simple' 2-representations of the Hecke 2-category (aka Soergel bimodules) for finite Coxeter types.

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.
 

Whereas the spectral properties of random matrices has been the subject of numerous studies and is well understood, the statistical properties of the corresponding eigenvectors has only been investigated in the last few years. We will review several recent results and emphasize their importance for cleaning empirical covariance matrices, a subject of great importance for financial applications.