Veranstaltungen

Hofer-Wysocki-Zehnder proved that every strictly convex energy hypersurface in R^4 possesses a disk-like global surface of section. They asked whether a systole, i.e. a periodic orbit of least action, must span such a disk-like global surface of section. In my talk, I will give an affirmative answer to this question. Moreover, I will discuss some implications of this result concerning normalized symplectic capacities. This is based on joint work in progress with Abbondandolo and Kang.

We will study the high-dimensional statistical problem of multi-spiked tensor PCA, where the goal is to infer a finite number of unknown, orthogonal signal vectors (or spikes) from noisy observations of a p-tensor. I will present our recent results on the sample complexity required for stochastic gradient descent to efficiently recover the signal vectors from natural initializations. In particular, we will show that it is possible to recover a permutation of all spikes provided a number of sample scaling as N^{p-2}, aligning with the computational threshold identified in the rank-one tensor PCA problem. The recovery process is governed by a sequential elimination phenomenon. As one correlation exceeds an explicit critical threshold, all correlations that share a row or column index become sufficiently small to be negligible, allowing the subsequent correlation to grow and become macroscopic. The order in which correlations become macroscopic is determined by their initial values and the associated signal-to-noise ratios. Based on recent joint work with Gérard Ben Arous (NYU) and Cédric Gerbelot (NYU).

Let u(x,t) be the temperature distribution of a d-dimensional convex domain at time t with given initial temperature u(x,0) and insulating boundary. The hot-spots conjecture of Rauch asserts that for large times, the maximum of the function x -> u(x,t) is taken near the boundary of the domain. Equivalently, the conjecture asserts that the first nontrivial Neumann Laplace eigenfunction of a convex domain takes its maximum (and minimum) in the boundary. A general philosophy in convex analysis is that dimension free statements about convex sets imply an analogous, dimension-free statement about log-concave measures. In this talk I will construct the log-concave analogue to the hot spots conjecture, and construct a counterexample for it in high dimensions.

The coherent state transform, under various names, appears in many fields of mathematics and physics. It is associated with representations of a group. In this talk we are concerned with the problem of minimizing the entropy of the coherent state transform and we explain how complex analysis can be used to achieve this in certain settings. We discuss various open questions

In this joint work with G. Forni and M. Radziwill we study the orbits of horocycle flows when sampled at semi-primes (integers with at most two prime factors). We show that if the lattice is arithmetic then such orbits equidistribute towards Haar measure for every non-periodic point. During the talk I will outline a new criterion for semi-primes, and describe main ideas of the proof.

One can use Bloch's higher Chow groups to compute the usual Chow groups of moduli spaces. This involves first computing the necessary higher Chow groups, and then computing the connecting homomorphism of the localization exact sequence. I will explain general techniques for performing these computations and give examples for the integral Chow groups of moduli spaces of genus 1 curves.

The Boundary Element Method (BEM) is a discretization technique commonly employed for the accurate and rapid numerical solution of constant-coefficient second-order partial differential equations (PDEs) in the complement of an obstacle, e.g., electromagnetic scattering problems. The BEM exploits the fundamental solution of the PDE to reduce the number of unknowns compared to the Finite Element Method for the same level of error. However, it is a non-local method and thus leads to full linear systems. For this reason, the BEM linear systems are often solved iteratively, and good preconditioners often turn out to be a key ingredient to ensure a fast resolution. In this talk, we present the motivation and the practical and mathematical challenges for extending the BEM to geometric settings involving non-manifold boundaries. We will first summarize the recent advances in the mathematical formalization of this problem. We will then present a particular preconditioning technique for "multi-screen" obstacles based on substructuring (domain decomposition). This work is in collaboration with Xavier Claeys and Ralf Hiptmair.

It was shown by Hans Föllmer that the Ito formula may be derived as an analytical change of variable formula for smooth functions of irregular trajectories with finite quadratic variation, without making use of any probabilistic methods. We show how this idea may be pushed farther to construct an Ito calculus for causal functionals of irregular paths with finite (and non-zero) p-th order variation for any p>1. In particular we will discuss pathwise integration and change of variable formulae for functionals of irregular trajectories, as well as pathwise analogues of martingales and Doob-Meyer decompositions. These results have natural applications to pathwise optimal control and model-free formulations in mathematical finance. References: R Cont, R Jin (2024) Fractional Ito calculus, Transactions of the American Mathematical Society, Ser. B 11, 727-761. H Chiu, R Cont (2022) Causal Functional Calculus. Transactions of the London Mathematical Society, Volume 9, No. 1 December 2022, 237-269. R Cont, N Perkowski (2019) Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Transactions of the American Mathematical Society (Series B), Volume 6, 161-186.

Local curves are an important class of threefolds which are indicative of the general behaviour of curve counting invariants. We give a closed formula for all descendent stable pair invariants of (absolute) local curves and use these to prove structure results conjectured by Pandharipande and Pixton. We then use our results to give a new approach to the Bethe ansatz story in physics/representation theory.