18 - 19.10.2022
Venue: Lipschitz Hall, Math Center, Endenicher Allee 60, 53115 Bonn
Organizers: Franca Hoffmann, Tim Laux
The Global Math Network program aims to encourage BIGS PhD students to spend an extended research stay abroad. BIGS provides funding for research stays with a duration of 4 weeks up to 4 months either in one of the partner institutions or another mathematical institute worldwide.
Program:
| Tuesday, October, 18th 14:00 |
François Charles (l'École Normale Supérieure, Paris): Geometry of numbers in finite and infinite dimension I will discuss the appearance of Euclidean lattices and their associated Gaussian measures in various questions of diophantine and arithmetic geometry. I will explain how some aspects of the classical study of Euclidean lattices can be extended to infinite dimension, and how such infinite-dimensional objects come up naturally in various arithmetic problems and provide one more facet of the analogy between number fields and algebraic curves. Joint work with Jean-Benoît Bost. |
| Tuesday, October, 18th 14:30 |
Weinan E (Peking University & Princeton University): Towards a Mathematical Theory of Machine Learning Given a machine learning model, what are the class of functions that can be approximated by this particular model efficiently, in the sense that the convergence rate for the approximation, estimation and optimization errors does not deteriorate as dimensionality goes up? We address this question for three classes of machine learning models: The random feature model, two-layer neural networks and the residual neural network model. During the process, we will also summarize the current status of the theoretical foundation of deep learning, and discuss some of the key open questions. |
| Wednesday, October, 19th 15:30 |
Koji Fujiwara (Kyoto University): The rates of growth of hyperbolic groups The growth rates of a finitely generated group is a fundamental object in the study of groups. It is also very much related to geometry. For example, Milnor proved in the 60s that if a compact manifold has negative sectional curvature, then its fundamental group has exponential growth. I will discuss the structure of the set of growth rates of a hyperbolic group, which is a finitely generated group of "negative curvature". |
| Wednesday, October, 19th 15:45 | Sylvia Serfaty (Courant Institute, New York): Systems of points with Coulomb interactions Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the ''mean-field'' derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature. |
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