In this talk I will discuss the Nevanlinna-Pick interpolation problem under indeterminacy conditions. In terms of rational functions, the first and second kind, we obtain the explicit formula for the Nevanlinna matrix. The solutions to the interpolation problem are described through linear fractional transformations including Nevanlinna functions. This is a direct analog of the Nevanlinna formula for the Hamburger moment problem.

I will discuss the physics of high energy, many-particle states from two complementary perspectives. First, I will present a new method for using data from conformal field theories to compute observables in more general QFTs, which can be used to numerically study many properties of many-particle states. Then I will consider an analytic approach to a particular set of these states, those near threshold, where many features become largely theory-independent.

We describe the constructions of implosion and contraction for
complex-symplectic or hyperkahler manifolds. Implosions are examples
of nonreductive quotients in geometric invariant theory. We can
interpret both constructions in terms of a generalisation of the
Moore-Tachikawa category that was introduced in physics.

The contact process is a model for the spread of an infection in a graph. Vertices can be either healthy or infected\DSEMIC infected vertices recover with rate 1 and send the infection to each neighbor with rate lambda. A key question of interest is: if we start the process with a single infected vertex, can the infection survive forever with positive probability? This typically depends on the graph and on the value of lambda\DSEMIC for instance, on integer lattices, there is a critical value of lambda at which the survival probability changes from zero to strictly positive. However, on graphs that include vertices of high degree, such as Galton-Watson trees with heavy-tailed offspring distributions, it has been observed that the infection survives with positive probability for all values of lambda, no matter how small. This is because high-degree vertices sustain the infection for a long time and send the infection to each other. In this work, we investigate this survival-for-all-lambda phenomenon for a modification of the contact process, which we introduce and call the penalized contact process. In this new process, vertex u transmits the infection to neighboring vertex v with rate lambda/max(degree(u),degree(v))^mu, where mu>0 is an additional parameter (called the penalization exponent). This is inspired by considerations from social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that the introduction of this penalty factor introduces a rich range of behavior for the phase diagram of the contact process on Galton-Watson trees. We also show corresponding results for the penalized contact process on finite graphs obtained from the configuration model.

Joint work with J��lia Komj��thy and Zsolt Bartha.

In this talk, I will discuss correlation functions in 6d (2, 0) theories of two 1/2-BPS operators inserted away from a 1/2-BPS surface defect. In the large central charge limit the leading connected contribution corresponds to sums of tree-level Witten diagram in AdS7�����S4 in the presence of an AdS3 defect. I will show that these correlators can be uniquely determined by imposing only superconformal symmetry and consistency conditions, eschewing the details of the complicated effective Lagrangian. I will present the explicit result of all such two-point functions, which exhibits remarkable hidden simplicity.

In these lectures we will discuss various aspects of conformal field theories in Lorentzian signature. First, we will study the general properties of Lorentzian correlation functions, including their global conformal structure and the relation to Euclidean correlators. We will then consider the Regge limit of correlation functions and how this limit requires the introduction of complex spin. We will define complex spin using the Lorentzian inversion formula, and interpret it in terms of non-local light-ray operators. Finally, we will discuss applications of light-ray operators to even shape observables.

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.

In recent years we've greatly expanded our understanding of entanglement in many-body quantum systems; both how it behaves in ground states, and how it grows out-of-equilibrium. While entanglement is very difficult to measure in experiments, it has nevertheless driven progress in 1) the classification of quantum phases of matter and 2) strategies for efficiently simulating many-body systems on classical and quantum computers. I will review some recent progress in these directions. Along the way I will summarise some older results on how entanglement grows in many-body systems, briefly highlight some connections to holography, and present a conjecture about the asymptotic computational difficulty of calculating transport in many-body systems.

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.