Weekly Bulletin

tba

To an algebraic variety X one associates its group of birational transformations Bir(X). In the last decades, these groups and their rich algebraic, geometric, and dynamic structures have attracted a great deal of interest. In this talk, I will explain how the group structure of Bir(X) characterizes in many important cases the variety X. In particular, we will obtain new rationality criteria.

<div dir="auto" data-olk-copy-source="MessageBody">I will introduce the main objects from spectral theory of</div> <div dir="auto">infinite dimensional operators: the domain, the resolvent and the</div> <div dir="auto">spectrum. I will then describe an analytic family of non normal operator</div> <div dir="auto">\(D(\alpha)\) (for \(\alpha \in \mathbb{C}\)) such that \(\alpha\mapsto\mathrm{Spec}(D(\alpha))\) is discontinuous. The points of discontinuity are called</div> <div dir="auto">"magic angles" and I will show that there are infinitely many of them.</div> <div dir="auto">This is joint work with Simon Becker and Maciej Zworski.</div> <div dir="auto"> </div> <div dir="auto">The existence of these angles were predicted by Bistritzer–MacDonald in</div> <div dir="auto">2011 and were observed experimentally in 2018. They correspond to</div> <div dir="auto">specific "twisting angles" between two sheets of graphene for which we</div> <div dir="auto">observe supraconductivity.</div>

<p>We'll recall the spectral theory of the Laplacian on functions on a hyperbolic surface and show that as the genus goes to infinity, one can't obtain a uniform spectral gap above 1/4 - the bottom of the \(L^2\) spectrum of the hyperbolic plane. We'll show that this obstruction goes away when one instead considers the spectrum of the Dirac operator on spinors on a hyperbolic surface with a choice of spin structure. We'll sketch a construction of these surfaces with such a uniform spectral gap and note that these surfaces can be taken to be arithmetic.</p> <p><br>Based on joint work with Anshul Adve.</p>

We investigate the time-harmonic Maxwell equations in complex geometries, with particular emphasis on configurations that model polarization filters or Faraday cages. The domains under consideration contain perfectly conducting inclusions that are periodically distributed along a surface. We analyze the asymptotic regime in which both the size of the inclusions and the distance between them tend to zero, and we derive the resulting effective interface conditions. Depending on the geometric properties of the inclusions, the limiting system exhibits markedly different behaviors: perfect transmission, perfect reflection or polarization effects

We study bond percolation on the d-dimensional hypercube Q^d with edge retention probability p=c/d. It is well known that when c>1 is fixed, a unique giant component emerges. In this regime, we resolve conjectures of Bollobás, Kohayakawa, and Łuczak (1994) and of Benjamini and Mossel (2003), showing that the typical diameter of the giant component is Θ(d), and that the mixing time of the lazy random walk on it is Θ(d^2). In the talk, we will introduce the notion of mixing time and its connection to expansion properties of subsets of the giant. We will then discuss some of the key obstacles in obtaining this result, and in particular why classical sprinkling techniques are insufficient for this problem. Finally, we will explain how our new approach - based on analysing the effect of small perturbations and establishing stability under thinning - overcomes these obstacles. This method also yields tight large-deviation estimates for the size of the giant. Based on joint work with Michael Anastos, Lyuben Lichev, and Maksim Zhukovskii.

<p>In 1858, Helmholtz conjectured that coaxial vortex rings in 3D inviscid fluids can exhibit the "leapfrogging" dynamics, where rings repeatedly pass through one another. While well-documented experimentally, this phenomenon involves singular interactions that are difficult to capture analytically. In this talk, we present a rigorous justification of these dynamics for the 3D incompressible Euler equations. We consider a general class of initial data corresponding to N sharply concentrated coaxial vortex rings and prove that the leapfrogging motion persists on a logarithmic time scale. The core of the analysis requires new estimates to propagate the concentration of vorticity despite the singular mutual interactions required for the phenomenon to occur. </p> <p>Joint work with M. Donati, C. Lacave, and E. Miot.</p>

This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples--typically generated via a generative model or through random transformations of the original data--prior to model fitting. For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples. On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.

Let K be a finite extension of Qp and let o_K be its ring of integers. The theory of Lubin-Tate formal groups allows us to construct many integer-valued polynomials on o_K. To what extent do these polynomials generate all integer-valued polynomials on o_K? I will explain some results of de Shalit and Iceland, as well as of Ardakov, Sprang and myself on this question. I will also explain the motivation (p-adic Fourier theory) for our results.