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12:00pm to 1:00pm - RH 340N - Mathematical Physics Zhenghe Zhang - (UC Riverside ) u-States and u-Gibbs Measures and Their Relation to Lyapunov Exponents Abstract: In this talk, I will introduce the notions of u-states and u-Gibbs measures, and discuss their relationship for linear cocycles over hyperbolic base dynamics. I will then present applications to Lyapunov exponents, including results on properties such as continuity and large deviations of the Lyapunov exponents. |
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2:00pm to 3:00pm - 340P Rowland Hall - Combinatorics and Probability Siddharth Vishwanath - (UC San Diego) A Statistical Framework for Multidimensional Scaling From Noisy Data Multidimensional scaling (MDS) has a long history in statistics and underpins a broad class of unsupervised learning and spectral/nonlinear dimension reduction techniques. The objective of MDS is to extract meaningful information from relational data (e.g., distances between sensors, correlations between brain regions, or disagreement scores between individuals) by embedding these relationships into a Euclidean space. In practice, the observed relational information is often subject to measurement errors and/or corrupted by noise. However, the resulting embeddings are typically interpreted as exploratory visualizations without accounting for these variations. This talk presents recent work developing a principled statistical framework for MDS. We show that the classical MDS algorithm achieves minimax-optimal performance across a wide range of noise models and loss functions. Building on this, we develop a framework for constructing valid confidence sets for the embedded points obtained via MDS, enabling formal uncertainty quantification for geometric structure inferred from noisy relational data. These results provide a theoretical foundation for interpreting MDS embeddings, and extend naturally to a wide range of unsupervised learning techniques in modern data science. |
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4:00pm - RH 340N - Geometry and Topology Debin Liu - (UC Santa Barbara) Adiabatic Limit and Analytic Torsion of Vector Bundles Analytic torsion is a secondary topological invariant that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic. It can be defined analytically in terms of the determinant of Hodge Laplacian. In this talk, I will explain how Witten Laplacian can be used to generalize this construction to vector bundles over closed manifolds. I will also discuss how to relate the index and the analytic torsion of the total space to those of the base manifold. This is a joint work with Xianzhe Dai.
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4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Dmitriy Drusvyatskiy - (UCSD) When do spectral gradient updates help in deep learning? Abstract: Spectral gradient methods, such as the recently proposed Muon optimizer, are a promising alternative to standard gradient descent for training deep neural networks and transformers. Yet, it remains unclear in which regimes these spectral methods are expected to perform better. In this talk, I will present a simple condition that predicts when a spectral update yields a larger decrease in the loss than a standard gradient step. Informally, this criterion holds when, on the one hand, the gradient of the loss with respect to each parameter block has a nearly uniform spectrum—measured by its nuclear-to-Frobenius ratio—while, on the other hand, the incoming activation matrix has low stable rank. It is this mismatch in the spectral behavior of the gradient and the propagated data that underlies the advantage of spectral updates. Reassuringly, this condition naturally arises in a variety of settings, including random feature models, neural networks, and transformer architectures. I will conclude by showing that these predictions align with empirical results in synthetic regression problems and in small-scale language model training. Biosketch: Dmitriy Drusvyatskiy received his PhD from Cornell University in 2013, followed by a post doctoral appointment at University of Waterloo, 2013-2014. He joined the Mathematics department at University of Washington as an Assistant Professor in 2014 and was promoted to Full Professor in 2022. Since 2025, Dmitriy is a Professor at the Halıcıoğlu Data Science Institute (HDSI) at UC San Diego. Dmitriy's research broadly focuses on designing and analyzing algorithms for large-scale optimization problems, primarily motivated by applications in data science. Dmitriy has received a number of awards, including the Air Force Office of Scientific Research (AFOSR) Young Investigator Program (YIP) Award, NSF CAREER, SIAG/OPT Best Paper Prize 2023, Paul Tseng Faculty fellowship 2022-2026, INFORMS Optimization Society Young Researcher Prize 2019, and finalist citations for the Tucker Prize 2015 and the Young Researcher Best Paper Prize at ICCOPT 2019. Research currently supported by NSF DMS-2552323, NSF CCF 1740551, and AFOSR FA9550-24-1-0092 awards. |
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3:00pm to 4:00pm - 340N - Inverse Problems and Imaging Sebastien Bossu - (University of North Carolina at Charlotte ) Spanning Multi-Asset Payoffs with ReLUs We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to exploit numerically. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges. |
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1:00pm - RH 340N - Algebra Reginald Anderson - (UC Irvine) Example computation of Joyce-style sheaf-theoretic invariant for \beta = L \cong CP^1 is a line in X=\PP^3 Here we consider the moduli space of Gieseker semistable sheaves on $X=\mathbb{P}^3$ (with respect to the ample divisor $H$) with Chern character $(0,0,L,n-2)$. We show that in this case, semistable=stable so that Joyce's sheaf-theoretic invariant agrees with the Behrend-Fantechi virtual fundamental class, using the trace-free obstruction theory of a complex in $D^b \operatorname{coh}(X)$. We compute the Behrend-Fantechi virtual fundamental class in the rational Chow ring of $\operatorname{Gr}(2,4)$, and construct Joyce's invariant in the Lie algebra of the homology of rigidified piecewise-linear higher stack of objects as mentioned in the previous talk, via pushforward. |
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3:00pm to 4:00pm - RH 306 - Number Theory Robert Cass - (Claremont McKenna College) Hecke Algebras and Motives Hecke algebras play a central role in both number theory and representation theory. While some Hecke algebras have explicit descriptions in terms of generators and relations, others are understood through structure constants that encode multiplicities in tensor products of representations. In this talk, I will discuss several projects with Thibaud van den Hove and Jakob Scholbach aimed at using geometry and motives to give a uniform categorification of Hecke algebras. Along the way, we will encounter the geometric Satake equivalence, Gaitsgory's central functor, and Iwahori-Whittaker models. |
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4:00pm to 5:00pm - RH 306 - Colloquium Brian Conrey - (American Institute of Math) Critical zeros of L-functions L-functions are ubiquitous objects in number theory that encode important information about prime numbers or other arithmetic objects I'll talk about results on zeros of L-functions on the 1/2-line and problems that are just out of reach. There will be no discussion of whether AI can solve the Riemann Hypothesis. |