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4:00pm to 5:00pm - RH 340N - Geometry and Topology Bahar Acu - (Pitzer College and Claremont Graduate University) Contact topology and geometry in high dimensions A very useful strategy in studying topological manifolds is to factor them into “smaller" pieces. An open book decomposition of an n-manifold (the open book) is a fibration that helps us study our manifold in terms of its (n-1)-dimensional fibers (the pages) and (n-2)-dimensional boundary of these submanifolds (the binding). Open books provide a natural framework for studying topological properties of certain geometric structures on smooth manifolds such as "contact structures". Thanks to open books, contact manifolds, odd dimensional smooth manifolds carrying these geometric structures, can be studied from an entirely topological viewpoint. For example, every contact 3-manifold can be presented as an open book whose pages are surfaces and binding is a knot/link. In this talk, we will talk about higher-dimensional contact manifolds and provide a setting where we study these manifolds in terms of 3D open books. We also present various results along with examples concerning geometric and topological aspects of contact and symplectic manifolds along with upcoming work concerning these special fibrations. |
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4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics Yuhua Zhu - (UCLA) A PDE-based model-free algorithm for Continuous-time Reinforcement Learning This talk addresses the problem of continuous-time reinforcement learning (RL). When the underlying dynamics remain unknown and only discrete-time observations are available, how can we effectively conduct policy evaluation and policy iteration? We first highlight that while model-free RL algorithms are straightforward to implement, they are often not a reliable approximation of the true value function. On the other hand, model-based PDE approaches are more accurate, but the inverse problem is not easy to solve. To bridge this gap, we introduce a new Bellman equation, PhiBE, which integrates discrete-time information into a PDE formulation. PhiBE allows us to skip the identification of the dynamics and directly evaluate the value function using discrete-time data. Additionally, it offers a more accurate approximation of the true value function, especially in scenarios where the underlying dynamics change slowly. Moreover, we extend PhiBE to higher orders, providing increasingly accurate approximations. |
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4:00pm to 5:30pm - RH 440R - Logic Set Theory Julian Talmor Eshkol - (UC Irvine) Stationary Partition Relations 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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4:00pm to 5:00pm - RH 306 - Differential Geometry Xuwen Zhu - (Northeastern University) Analysis and degenerations of ALH* gravitational instantons Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will focus on the ALH* type which has a non-compact end with inhomogeneous collapsing near infinity. I will talk about a joint project with Rafe Mazzeo on the Fredholm mapping property of the Laplacian and the Dirac operator, where the geometric microlocal analysis of fibered metrics plays a central role. Application of this Fredholm theory includes the L^2 Hodge theory, polyhomogeneous expansion and local perturbation theory. I will also discuss a joint project with Yu-Shen Lin and Sidharth Soundararajan on the degeneration of such metrics which gives a partial compactification of their moduli space. |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Robert Webber - (UCSD) Randomly sparsified Richardson iteration is really fast Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as 10^{108} x 10^{108}. So far, a complete mathematical explanation for their success has proven elusive. The family of random sparsification methods has not yet been extended to the important case of linear system solves. This talk proposes a new algorithm based on repeated random sparsification that is capable of solving linear systems in extremely high dimensions and provides a complete mathematical analysis of the new algorithm. The analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector is too large to store. |
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9:00am to 9:50am - Zoom - Inverse Problems Fadil Santosa - (Johns Hopkins University) EIT with minimal measurements |
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3:00pm to 4:00pm - Zoom: https://uci.zoom.us/j/98066115065 - Number Theory Heejong Lee - (Purdue University) Reciprocity Laws and Congruences in Number Theory Gauss's quadratic reciprocity law has been extensively generalized in multiple directions within number theory. This talk will begin with explicit examples of reciprocity laws, including an interpretation of the proof of Fermat’s Last Theorem by Wiles and Taylor-Wiles as a consequence of a reciprocity law. As part of this discussion, I will introduce modular forms, elliptic curves, and Galois representations, leading to an overview of the Langlands reciprocity. I will then discuss the role of congruences in the study of reciprocity laws, with a particular focus on the Serre weight conjectures. I will conclude by outlining the proof of the Serre weight conjectures for GSp4. This is partly based on joint work with Daniel Le and Bao Le Hung. |
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1:00pm to 2:00pm - RH 440R - Dynamical Systems Victor Kleptsyn - (CNRS, University of Rennes 1, France) Critical regularity for nilpotent group actions in dimension one My talk will follow a joint work with Maximiliano Escayola, devoted to the study of critical regularities for nilpotent group actions. The questions of critical regularities have been studied by many authors in many different contexts: starting from the classical Denjoy theorem and example, there are works by M. Herman, J.-C. Yoccoz, N. Kopell, B. Deroin, A. Navas, C. Rivas, E. Jorquera, K. Parkhe, S.-H. Kim, T. Koberda, any many others. Our main result allows to describe the critical regularity in algebraic terms; while proving it, we introduce some new technique for establishing the bounds. We also obtain a generalisation of Bass’ formula to the case of a relative growth of a nilpotent group with respect to its subgroup. |