In this talk, we revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi, Musil, and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell���s inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers. This is a joint work with Ali Barki, Sergey Bobkov, and Cyril Roberto.

I will explain how to describe form factors of single-trace half-BPS operators in planar N=4 super Yang Mills theory using the T-dual Wilson loop picture. After reviewing earlier results for operators in the stress-tensor multiplet, I will present the dual Wilson loop description for the so-called MHV form factors of half-BPS operators. The general proposal relates these form factors to the matrix elements of a null periodic super Wilson loop with outgoing states composed of zero-momentum scalars. I will present perturbative tests of this description at weak coupling. I will then explain how to obtain exact result at finite coupling in the collinear limit using the Wilson loop Operator Product Expansion. I will conclude with general comments and speculations about form factors of unprotected operators such as the Konishi operator.

Part II. A statistical physics analysis using random matrix theory and mean-field methods.

Mutations of quivers were introduced by Fomin and Zelevinsky in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. I will discuss a geometric interpretation of this construction: these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of relatively small volume with proper actions of Coxeter groups. If time permits, I will discuss a generalization of the construction of Barot and Marsh leading to a new invariant of bordered marked surfaces, and relation to extended affine Weyl groups. The talk is based on joint works with Anna Felikson, John Lawson and Michael Shapiro.

Given a graph G = (V,E), consider the set of all discrete-time, reversible Markov chains with equilibrium distribution uniform on V and transitions only across edges E of the graph. We establish a Cheeger-type inequality for the fastest mixing time using the vertex conductance of G. We also consider chains with almost-uniform invariant distribution. Time permitting, we also discuss a construction of a continuous-time chain with exactly-uniform invariant distribution and average jump-rate 1, and mixing time bounded by the d^2 log(n), where d is the graph diameter and n is the number of vertices.

I will discuss some of the recent developments in the Krylov complexity. In particular, I will focus on the applications of the Krylov basis techniques to the modular Hamiltonian evolution and I will discuss a new angle on entanglement entropy in QCD at high energies. Based on arXiv:2306.14732 [hep-th] and work in progress.

Part I, Introduction and theoretical basis: how time-reversing Langevin dynamics one can create images from white noise.

How often does a randomly chosen variety have a point? Answering this question depends on the family of varieties in question, how we decide to order them, and what kinds of points we are looking for. Motivated by rational points, we endeavor to explicitly describe how often a randomly chosen variety is everywhere locally soluble. When our family is described by the fibers of a suitable morphism, this likelihood is equal to the product of local probabilities at each place and in some cases may be computed exactly. In particular, in joint work with Lea Beneish we find that for almost 97% of integral binary sextic forms f(x,z), the superelliptic curve y^3 = f(x,z) is everywhere locally soluble, with the local factors described explicitly by rational functions. Time permitting, we will discuss ongoing work on determining how often a cubic hypersurface has a rational point.

Sequential Monte Carlo Samplers (SMCS) constitute a widely used class of SMC algorithms that calculate normalizing constants and simulate complex multi-modal target distributions. Typically, SMCS utilizes a process known as annealing, which propagates solutions from a tractable reference distribution to the intractable target through a continuous path of increasingly complex distributions. SMCS delivers state-of-the-art performance when adequately tuned, although this can pose a challenge for current tuning methods, yielding a random run-time and compromising the normalizing constant's unbiasedness.

In this talk, we intend to describe all the components of an SMCS algorithm and their influence on the variance of the normalizing constant. Specifically, we will demonstrate that SMCS exhibits fundamentally different behaviour in large particle and dense schedule limits. The dense schedule limit reveals the natural geometry induced by annealing, which can pinpoint optimal performance and tune the number of particles, number of annealing distributions, annealing schedules, the resampling schedule, and the path. Lastly, we propose an efficient, black-box algorithm for tuning SMCS that delivers optimal performance within a fixed, user-specified computation budget, all while preserving the unbiasedness of the normalizing constant.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed\DSEMIC differential operators with piecewise constant coefficients are thus included.