Week 21.01.2024 – 27.01.2024

For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Quantum chaotic systems display correlation between eigenvalues as described by the random matrix theory (RMT). I will present three results on the universal aspects of many-body quantum chaos that go beyond the standard RMT paradigm. Firstly, I will present an exact scaling form of the spectral form factor (SFF) in a generic many-body quantum chaotic system, deriving the so-called "bump-ramp-plateau" behaviour. Secondly, I will introduce and provide an analytical solution of a generalization of SFF for non-Hermitian matrices, called Dissipative SFF, which displays a "ramp-plateau" behaviour with a quadratic ramp. Thirdly, I will provide evidences that non-Hermitian Ginibre ensemble behaviour surprisingly emerge in generic many-body quantum chaotic systems, due to the presence of many-body interaction.

It is well-known that a random walk on a (connected, finite, non-bipartite) graph converges to its invariant distribution. For some graphs it has been observed that this convergence happens rather abruptly. This phenomenon is called cutoff, and it has been established widely, but in general there is little understanding for what causes it.

In this work we consider a model with a small amount of randomness. Given some deterministic graphs, we apply a random perturbation on them and based on the strength of the perturbation, we establish a phase transition in whether the resulting graphs have cutoff.

The first part of the talk will be an overview about mixing times and cutoff. In the second part we will focus on our model and explain the ideas behind the results.

Based on joint work with Jonathan Hermon, Andela Sarkovic and Perla Sousi.