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4:00pm - RH 340N - Geometry and Topology Christopher Kuo - (USC) Symplectic Geometry and Sheaves Sheaves have long been classical tools for studying the topology of manifolds. Symplectic geometry, which encodes topological information about a manifold via its cotangent bundle, has revealed a profound connection to sheaf theory through the microlocal framework developed by Kashiwara and Schapira. Remarkably, many important symplectic invariants can now be computed using sheaves. In this talk, I will survey several well-known applications of sheaf theory in symplectic geometry and also consider the reverse perspective: how symplectic geometry provides constructions and insights that deepen our understanding of sheaf theory. This latter viewpoint is central to obtaining a global version of the microlocal Riemann-Hilbert correspondence in joint work with Côté, Nadler, and Shende. |
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4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Siting Liu - (UC Riverside) Wasserstein Proximal Operators Describe Score-Based Generative Models and Resolve Memorization We focus on the fundamental mathematical structure of score-based generative models (SGMs). We formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled PDEs: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Next, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. The WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Our mathematically informed kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications. This is a joint work with Benjamin J. Zhang, Markos A. Katsoulakis, Wuchen Li and Stanley J. Osher. |
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4:00pm to 5:30pm - RH 440 R - Logic Set Theory Julian Talmor Eshkol - (UC Irvine) Stationary Partition Relations (3) The partition relation A→(P)^μ_λ, despite its brevity, is remarkably expressive. This fundamental combinatorial principle asserts that every λ-coloring of μ-sized subsets of A is constant on a subset in the class P. By adjusting the parameters A, μ, λ, and P, one can express a wide variety of large cardinal principles, including weak compactness, Ramsey-ness, and even supercompactness. In this talk, we focus on the case where A is an uncountable cardinal κ and P is the class of stationary subsets of κ. By work of Baumgartner 1977, it turns out that this corresponds to certain ineffability properties of κ. We also describe how this fits into a different hierarchy of ineffability properties described in current joint work with Matthew Foreman and Menachem Magidor. |
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1:00pm to 2:00pm - RH 440R - Dynamical Systems Davi Obata - (Brigham Young University) Absolute continuity of stationary measures In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired by Tsujii's “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan. |
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4:00pm - RH306 - Differential Geometry Charles Cifarelli - (SUNY Stonybrook) Explicit complete Calabi-Yau metrics and Kähler-Ricci solitons Since Calabi's original paper, the Calabi Ansatz has been central for constructions in Kähler geometry. Calabi himself used it to construct complete Ricci-flat metrics on the total space of the canonical bundle of a Kähler-Einstein Fano manifold (B, \omega_B), generalizing some well-known examples coming from physics. Over the years, work of Koiso, Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, Chi Li, and others have shown that the Calabi Ansatz can be used to produce complete Kähler-Ricci solitons, important singularity models for the Kähler-Ricci flow, on certain line bundles over (B, \omega_B). In this talk, I'll explain a generalization of these results to the total space of some higher-rank direct-sum vector bundles over (B, \omega_B). In our case the Calabi Ansatz is not suitable, and we instead use the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. The construction produces new examples of complete shrinkers, steadies, and Calabi-Yau metrics, as well as recovering some known ones. |