Week 01.12.2025 – 07.12.2025

We study the high-dimensional inference of a rank-one signal corrupted by sparse noise. The noise is modelled as the adjacency matrix of a weighted undirected graph with finite average connectivity in the large size limit. Using the replica method, we analytically compute the typical value of the top eigenvalue, the top eigenvector component density, and the squared overlap between the signal vector and the top eigenvector. The solution is given in terms of recursive distributional equations for auxiliary probability density functions which are efficiently solved using a population dynamics algorithm. Specialising the noise matrix to Poissonian and Random Regular degree distributions, the critical signal strength is analytically identified at which a transition happens for the recovery of the signal via the top eigenvector, thus generalising the celebrated BBP transition to the sparse noise case. In the large-connectivity limit, known results for dense noise are recovered.

At most how many edge-disjoint Hamilton cycles does the directed random graph
contain?
It is easy to see that one cannot pack more than the minimum in-degree or the minimum out-degree of the digraph. In this talk I will discuss a recent result that shows that this trivial upper bound is in fact sharp for D(n,p), if p is slightly larger than the Hamiltonicity threshold.

Based on a joint work with Asaf Ferber.

The modern study of unitary representations of reductive groups over local fields has been greatly influenced by Arthur's conjectures on the parametrisation of the local factors of automorphic forms in the discrete spectrum. For reductive groups over real or p-adic fields, Arthur's local conjectures admit a precise formulation in the work of Adams, Barbasch, and Vogan, via the microlocal analysis of the geometric parameter space for the admissible dual of the group. In the case of p-adic groups, the geometric parameter space is given by the complex geometric setting of Kazhdan and Lusztig. An important particular case, where everything can be made precise, is the category of unipotent (in the sense of Lusztig) representations of a reductive p-adic group; these are the representations for which the Langlands parameters are unramified, in the sense of being trivial on the inertia subgroup of the Weil group. Once we fix an"infinitesimal character" for the representations, the geometry comes from the action of a complex reductive group on a complex vector space with finitely many orbits; for the Adams-Barbasch-Vogan picture, we are interested in the resulting microlocal packets and the corresponding packets of irreducible representations of the p-adic group. In the talk, I will explain the parametrisations and above constructions, give examples, and concentrate on the integral infinitesimal characters, where some surprisingly strong general conjectures about unitarisability can be formulated. 

In this talk, I will discuss some features of heavy–heavy–light–light correlators in N=4 supersymmetric Yang–Mills theory, where the light operators belong to the stress-tensor multiplet and the heavy ones correspond to giant gravitons, realised holographically as D3-branes. I will focus on the associated Integrated Correlators, for which exact expressions can be obtained despite few results are known for the correlators themselves. I will highlight several interesting properties of these integrated correlators, especially the emergence of universal structures in the strong-coupling regime. I will also discuss different integrated correlators in certain N=2 superconformal field theories, which are holographically dual to scattering of gravitons (and gluons) in the presence of D7-branes.  Surprisingly, these integrated correlators in N=2 theories are given by exactly the same strong coupling asymptotic series as those of giant gravitons, despite the fact that their weak coupling expansions are very different. 

Mengxin Xi

Title: Extrapolation of Tempered Posteriors
Abstract: Tempering is a popular tool in Bayesian computation, being used to transform a complicated posterior distribution into one that is more easily sampled. The idea is to construct a sequence (pt)0≤t≤1, with p0 typically representing the prior and p1 representing the posterior, and then to numerically approximate terms in this sequence, starting with p0 and proceeding through intermediate distributions until an approximation to p1 is obtained. Our contribution reveals that high-quality approximation of terms up to p1 is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish weak sufficient conditions under which tempered expectations are not merely smooth as a function of t, but analytic, implying that knowledge of the tempered expectation in any open t interval fully determines the posterior expectation of interest. Harnessing this result, we propose novel regression methodology for approximation of posterior expectations based on tempering and the waste-free sequential Monte Carlo method of [Dau and Chopin, 2022], illustrating its effectiveness on a number for examples.

Enrique Cacicedo Germano

Title: Calibrating credible regions for Gibbs posterior distributions with covariance matrix approximation
Abstract: Modern machine learning applications involve optimizing an evaluation metric of interest that has an objective practical meaning. The traditional Bayesian setting relies on directly modeling the data-generating process, making model misspecification a relevant concern. Alternatively, a framework based on Gibbs posterior distributions that directly link data with quantities of interest via loss functions can provide robustness for inference and uncertainty quantification. In this study, we provide a simple solution for calibrating the learning rate in Gibbs posterior distributions. The algor...

The periodic points of a discrete dynamical system control its local and global dynamical behaviour. When the system is defined over the rational numbers, one can ask about the arithmetic properties of periodic points. The central open conjecture in arithmetic dynamics asks whether there are uniform constraints on the possible periods of points for families of algebraic dynamical systems. In this talk, we will discuss this conjecture, how it generalizes the torsion conjecture—in particular, the celebrated theorems of Mazur and Merel on rational torsion of elliptic curves—and survey some recent progress on and strategies for attacking this problem.