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3:00pm to 4:00pm - Zoom - Analysis Arunima Bhattacharya - ( University of Washington, Seattle) Hessian Estimates for the Lagrangian mean curvature equation In this talk, we will discuss a priori interior estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian phase. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data on a uniformly convex, bounded domain.
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4:00pm - Zoom - Differential Geometry Guoyi Xu - (Tsinghua University) The construction of the splitting maps For a geodesic ball with non-negative Ricci curvature and almost |
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9:00am to 10:00am - Zoom - Inverse Problems Gitta Kutyniok - (Ludwig-Maximilians-Universität München) Graph Convolutional Neural Networks: The Mystery of Generalization |
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10:00am to 11:00am - - Mathematical Physics Maximilian Pechmann - (University of Tennessee, Knoxville) Bose-Einstein condensation in one-dimensional noninteracting Bose gases in the presence of soft Poissonian obstacles We study Bose--Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on $\mathbb R$ with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With arbitrarily high probability when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the $r$th mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed strength an arbitrarily high probability for type-I g-BEC is also obtained by allowing sufficiently many one-particle states to be macroscopically occupied. |
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3:00pm to 4:00pm - Zoom: https://uci.zoom.us/j/95528784206 - Number Theory Jacob Mayle - (University of Illinois, Chicago) Local-Global Phenomena for Elliptic Curves
A local-global principle is a result that allows us to deduce global information about an object from local information. A well-known example is the Hasse-Minkowski theorem, which asserts that a quadratic form represents a number if and only if it does so everywhere locally. In this talk, we'll discuss certain local-global principles in arithmetic geometry, highlighting two that are related to elliptic curves, one for torsion and one for isogenies. In contrast to the Hasse-Minkowski theorem, we'll see that these two results exhibit considerable rigidity in the sense that a failure of either of their corresponding everywhere local conditions must be rather significant. |