Veranstaltungen

The goal of this series of talks is to present a proof, at a higher level of detail than usual, of the following theorem. Let H be any smooth function on R^4 and let Y be any compact and regular level set. Then Y admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which in addition have dense union in Y. This improves on a 2018 result by Fish-Hofer, which showed the existence of one proper invariant subset. Related results were proved jointly with Dan Cristofaro-Gardiner for area-preserving surface diffeomorphisms and 3D Reeb flows. I will mention them if there is time.

The study of turbulent phenomena is related to the mathematical analysis of weak solutions to the incompressible Euler equations with non-trivial dissipation measure. In 1949 Lars Onsager predicted the maximal regularity compatible with a non-trivial dissipation measure for Euler solutions. In this talk, we discuss some refined properties of the dissipation measure in the critical and supercritical regimes, according to the Onsager conjecture. In particular, we focus on the improved regularity of the dissipation in negative Besov spaces for non-conservative Euler solutions, laying down a connection between accumulation of the dissipation on lower dimensional sets and intermittency for incompressible Euler.

Toeplitz matrices form a rich and ubiquitous class of possibly non-normal matrices. Their asymptotic spectral analysis in high dimension is well-understood, as illustrated by the strong Szegö limit theorem for Toeplitz determinants. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this talk, we explore the spectrum of a banded Toeplitz matrix perturbed by a random matrix in the asymptotic of high dimension. We show that the outlier eigenvalues are driven by a low-dimensional random analytic matrix field alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. Along the way, we present new variations around the strong Szegö limit theorem. The talk is based on a joint work with Mireille Capitaine and François Chapon.

More information: https://www.cambridge.org/core/books/risk-revealed/6C0AEEEA9CF39A538F87BE8F435F8821call_made

In 1992, Ghys introduced a remarkable class of flows called quasi-Fuchsian flows. Namely, for a pair of metrics g_1 and g_2 of constant curvature -1 on a closed surface M, corresponding to points in Teichmueller space [g_1] and [g_2], respectively, he constructed an Anosov flow \phi_{[g_1], [g_2]} on the bundle of positive half-lines over M, whose weak stable and unstable foliations are smoothly conjugated to that of the geodesic flows of g_1 and g_2, respectively. In fact, in 1993 Ghys also showed that any Anosov flow on a 3-manifold with smooth weak stable/unstable bundles is smoothly conjugate to a quasi-Fuchsian flow or a suspension of a diffeomorphism of the 2-torus. In this talk, I will give an alternative 'PDE theoretic' description of quasi-Fuchsian flows as certain thermostat flows on the unit tangent bundle of the Blaschke metric uniquely determined by a conformal class on M and a holomorphic quadratic differential, satisfying `coupled vortex equations'. Joint work with Gabriel Paternain.

In analogy with the classical situation of a general algebraic curve, Hurwitz-Brill-Noether addresses the question which linear systems appear on a general k-gonal curve of genus g. In recent years, due to impressive work of Pflueger, H. Larson, Jensen, Ranganathan and others an answer to this question has been put forward, using a mix of tropical and degeneration methods. I will discuss a radically new approach to this problem using Bridgeland stability conditions and present new applications, for instance concerning  the construction of Hurwitz-Brill-Noether generic curves defined over number fields. Joint with S. Feyzbakhsh and A. Rojas.

Incompressible magnetohydrodynamics (MHD) and the incompressible Godunov-Peshkov-Romenski (GPR) model share a similar structural framework, with key properties such as energy conservation, incompressibility, and involutions. In this talk, we demonstrate how to preserve these essential properties at the fully discrete level using compatible finite element methods, combined with a tailored time integration scheme. Furthermore, we explore both linear and nonlinear stabilization strategies necessary for convection-dominated regimes, examining their interplay with structure preservation. In particular, we show that such stabilizations affect only energy conservation. This research was conducted in collaboration with M. Dumbser from the University of Trento.

This talk aims to present several mathematical optimization problems/algorithms for AI such as the LLM training, tunning and inferencing. In particular, we describe how classic optimization models/theories can be applied to accelerate and improve the Training/Tunning/Inferencing algorithms that are popularly used in LLMs. On the other hand, we show breakthroughs in classical Optimization (LP and SDP) Solvers aided by AI-related techniques such as first-order and ADMM methods, the low-rank SDP theories, and the GPU Implementations. Bio: Yinyu Ye is currently the K.T. Li Professor of Engineering at Department of Management Science and Engineering and Institute of Computational and Mathematical Engineering, Stanford University; and visiting chair professor of Shanghai Jiao Tong University. His current research topics include Continuous and Discrete Optimization, Data Science and Applications, Algorithm Design and Analyses, Algorithmic Game/Market Equilibrium, Operations Research and Management Science etc.; and he was one of the pioneers on Interior-Point Methods, Conic Linear Programming, Distributionally Robust Optimization, Online Linear Programming and Learning, Algorithm Analyses for Reinforcement Learning & Markov Decision Process and nonconvex optimization, and etc. He and his students have received numerous scientific awards, himself including the 2006 INFORMS Farkas Prize (Inaugural Recipient) for fundamental contributions to optimization, the 2009 John von Neumann Theory Prize for fundamental sustained contributions to theory in Operations Research and the Management Sciences, the inaugural 2012 ISMP Tseng Lectureship Prize for outstanding contribution to continuous optimization (every three years), the 2014 SIAM Optimization Prize awarded (every three years).

The action of Hecke operators on cuspidal modular forms of weight 2 can be realized as the action of an algebra of algebraic cycles on the square of a smooth projective curve, with composition being defined in terms of intersection theory. Having in mind applications to the arithmetic of the associated Galois representations, one would like to generalize this description to the case of higher weight. The goal of the talk is to introduce this circle of ideas and to explain the construction of a so-called motivic Hecke algebra, providing such a generalization. This actually works in the much broader context of automorphic forms living in the cohomology of any PEL Shimura variety.

Quantifying uncertainty in multivariate regression is crucial across many real-world applications. However, existing approaches for constructing prediction regions often struggle to capture complex dependencies, lack formal coverage guarantees, or incur high computational costs. Conformal prediction addresses these challenges by providing a robust, distribution-free framework with finite-sample coverage guarantees. In this study, we offer a unified comparison of multi-output conformal techniques, highlighting their properties and interrelationships. Leveraging these insights, we propose two families of conformity scores that achieve asymptotic conditional coverage: one can be paired with any generative model, while the other reduces computational overhead by utilizing invertible generative models. We then present a large-scale empirical analysis on 32 tabular datasets, comparing all methods under a consistent code base to ensure fairness and reproducibility.

The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. In this talk, I will introduce recent advances toward solving this conjecture and discuss relevant results about the bounds of genus of curves in projective threefolds.