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2:00pm to 3:00pm - 340P Rowland Hall - Combinatorics and Probability Georg Menz - (UCLA) Data-driven Breakpoint detection with the QML estimator We study quasi-maximum likelihood (QML) breakpoint estimation for Joint work with Hubeyb Gurdogan (UCLA) |
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3:00pm to 4:00pm - RH 340P - Mathematical Physics Anton Kapustin - (Caltech) Chaos and thermalization in many-body systems Since the times of Ludwig Boltzmann, most physicists take it for granted that generic non-integrable closed dynamical systems, whether classical or quantum, thermalize at long times for generic initial conditions. Whether this belief is true or false depends on the type of systems one is willing to consider and the meaning of “generic” and “thermalizes”. In this talk I will discuss two types of many-body dynamical systems where thermalization (i.e. weak convergence to the state of maximal entropy) can be established for a large class of initial states. The first one is a system of an infinite number of spins with evolution generated by a repeated application of a Clifford Cellular Automaton. The second one is the classical counterpart of the first one and can be thought of as a mixing automorphism of an infinite-dimensional torus. |
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4:00pm - RH 306 - Applied and Computational Mathematics Xianjin Yang - (Caltech) Gaussian Processes for PDEs: Bilevel Hyperparameter Learning and Functional PDE Solvers In this talk, we present Gaussian process (GP) frameworks for PDE-related problems. We first discuss bilevel hyperparameter learning for GP solvers in PDEs and inverse problems, since the accuracy, stability, and generalization of kernel- and neural network-based methods depend strongly on hyperparameter choices. We evaluate the approach on nonlinear PDEs and PDE inverse problems, where the results indicate improved accuracy and robustness compared with random initialization. We then turn to GP operator learning for non-perturbative functional renormalization group equations, which are integro-differential equations defined on functionals. The method learns operators directly on function space, producing a functional representation that does not depend on a specific discretization, while still allowing physical priors to be incorporated through the prior mean or kernel design. We illustrate the approach on examples including the Wetterich and Wilson–Polchinski equations. |
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1:00pm - 340P - Harmonic Analysis Seewoo Lee - (UC Berkeley) Mathematics, AI, and Formalization AI is now impacting mathematics through tools for generating conjectures, searching examples, proving theorems, and formalizing theorems, but it is often unclear what “AI doing mathematics” actually means. This talk surveys recent developments and uses examples to distinguish these modes of assistance, emphasizing current capabilities, limitations, and practical workflows. |
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3:00pm - RH 440R - Logic Set Theory Nigel Pynn-Coates - (University of Vienna) Transserial tame pairs Hardy fields are differential fields of (germs at infinity of) real-valued functions. Interest in them comes from several areas of mathematics, including asymptotic analysis, dynamical systems, and o-minimality. The first-order theory of existentially closed Hardy fields is completely axiomatizable and model complete in the language of ordered valued differential fields, as M. Aschenbrenner, L. van den Dries, and J. van der Hoeven have shown in a long and impressive series of works; in particular, all maximal Hardy fields are elementarily equivalent. Moreover, each maximal Hardy field can be equipped with an elementary differential subfield that is Dedekind complete in the maximal Hardy field. Along the lines of tame pairs of real closed fields (or tame pairs of o-minimal fields, more generally), the theory of such pairs is axiomatized by the notion of a transserial tame pair, the subject of this talk. After introducing these objects, I will summarize some of their properties. For example, they are model complete and topologically tame in the sense of being locally o-minimal and d-minimal, as well as satisfying a definable Baire Category Theorem. |
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4:00pm to 5:00pm - 306 Rowland Hall - Differential Geometry Malik Tuerkoen - (UC Irvine) Concavity Properties of Dirichlet Eigenfunctions in Hyperbolic Space On convex domains in R^n and S^n, the first Dirichlet eigenfunction is known to be log concave, a fact that is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. It is known that the first Dirichlet eigenfunction is in general not log-concave for convex domains in H^n. I will discuss concavity estimates on horoconvex domains in hyperbolic space - which are domains whose boundaries second fundamental form is greater than 1 - which yield new spectral-gap bounds in H^n. In doing so, we resolve a conjecture by Nguyen, Stancu and Wei. This is based on joint work with G. Khan and on joint work with S. Saha and G. Khan. |
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1:00pm to 1:50pm - RH 340N - Algebra Daniel Bustamante - (UC Irvine) An Introduction to Artin-Schelter Regular Algebras In this talk, I will give a survey on Artin-Schelter (AS) Regular Algebras. AS Regular Algebras are often seen as good candidates for noncommutative polynomials. The importance of each condition present in the definition of an AS Regular Algebra will be explored. I will present some examples as well as mention how classification of these algebras has progressed. Their importance as noncommutative polynomials comes from realizing AS Regular Algebras as coordinate rings for “noncommutative spaces” in order to develop a theory of noncommutative algebraic geometry. I will present the work done for AS Regular Algebras of small dimension. |
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4:00pm to 5:00pm - RH 306 - Colloquium Dmitry Dolgopyat - (University of Maryland) Mixing properties of random transformations We will discuss methods to study statistical properties of random dynamical spaces concentrating on homogeneous systems and their small perturbations. |