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4:00pm to 5:00pm - RH 340N - Geometry and Topology Ezra Getzler - (Northwestern) Generalizing Lie theory to higher dimensions - the De Rham theorem on simplices and cubes There is a generalization of Lie theory from Lie algebras to differential graded Lie algebras. Ordinary Lie theory involves first order ordinary differential equations. Higher Lie theory may be understood as a non-linear generalization of the de Rham theorem on simplicial complexes (in Dupont's formulation), as against graphs. In this talk, we present an alternate approach to this theory, using the more elementary de Rham theorem on cubical complexes.
Along the way, we will need an interesting relationship between cubical and simplicial complexes, which has recently become better known due to its use in Lurie's theory of straightening for infinity categories.
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4:00pm to 5:30pm - RH 340P - Logic Set Theory Derek Levinson - (UCLA) Unreachability of $\Gamma_{2n+1,m}$ We prove from ZF + AD + DC that there is no sequence of distinct $\Gamma_{1,m}$ sets of length $\aleph_{m+2}$. This is the optimal result for the pointclass $\Gamma_{1,m}$ by earlier work of Hjorth. We also get a bound on the length of sequences of $\Gamma_{2n+1,m}$ sets using the same techniques. |
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4:00pm - ISEB 1200 - Differential Geometry Chi Fai Chau - (UC, Irvine) A free boundary problem in pseudoconvex domains A domain with C^2 boundary in complex space is called pseudoconvex if it has a C^2 defining function with positive complex hessian on its boundary. Pseudoconvexity is a generalization of convexity. It can be realised as a domain with geometric condition on the boundary and its topology can be studied by Morse theory. In this talk, we will discuss the Morse index theorem for free boundary minimal disks for partial energy in strictly pseudoconvex domain and the relation between holomorphicity and stability of the free boundary minimal disk. We will also give an example to illustrate the necessity of strict pseudoconvexity in our index estimate. |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Gavin Kerrigan - (UCI) Deep Generative Models in Infinite-Dimensional Spaces Deep generative models have seen a meteoric rise in capabilities across a wide array of domains, ranging from natural language and vision to scientific applications such as precipitation forecasting and molecular generation. However, a number of important applications focus on data which is inherently infinite-dimensional, such as time-series, solutions to partial differential equations, and audio signals. This relatively under-explored class of problems poses unique theoretical and practical challenges for generative modeling. In this talk, we will explore recent developments for infinite-dimensional generative models, with a focus on diffusion-based methodologies. |
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9:00am to 10:00am - Zoom - Inverse Problems Tapio Helin - (LUT University) Next frontier of Bayesian Inverse Problems: Optimal Experimental Design |
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4:00pm to 5:00pm - RH 306 - Colloquium Edray Goins - (Pomona College) Quasi-Critical Points of Toroidal Belyi Maps A Belyi map \( \beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}) \) is a rational function with at most three critical values; we may assume these values are \( \{ 0, \, 1, \, \infty \} \). Replacing \( \mathbb{P}^1 \) with an elliptic curve \( E: \ y^2 = x^3 + A \, x + B \), there is a similar definition of a Belyi map \( \beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})\). Since \( E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R}) \) is a torus, we call \( (E, \beta) \) a Toroidal Belyi pair. There are many examples of Belyi maps \( \beta: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C}) \) associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree \( N \), the inverse image \( G = \beta^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr) \) is a set of \( N \) elements which contains the critical points of the Belyi map. In this project, we investigate when \( G \) is contained in \( E(\mathbb{C})_{\text{tors}} \).
This is joint work with Tesfa Asmara (Pomona College), Erik Imathiu-Jones (California Institute of Technology), Maria Maalouf (California State University at Long Beach), Isaac Robinson (Harvard University), and Sharon Sneha Spaulding (University of Connecticut). This was work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).
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1:00pm - MSTB 124 - Graduate Seminar Shaozhuan Li - (UCI) Counseling Center presentation |