|
10:00am to 11:00am - RH 340N - Mathematical Physics Reuben Drogin - (Yale University) Some ideas in Random Band Matrices and Random Permutations A surprising prediction is that the eigenfunctions of random band matrices have similar spatial spread as cycles in various models of random permutations. In this talk we discuss various ideas in the proofs of localization in these models. This talk is part of the 39th WSMP program. |
|
2:00pm to 3:00pm - 340P Rowland Hall - Combinatorics and Probability Dingding Dong - (Caltech) Structure of tight (k,0)-stable graphs We say that a graph G is (k,l)-stable if removing k vertices from it reduces its independence number by at most l. We say that G is tight (k,l)-stable if it is (k,l)-stable and its independence number equals ⌊(n−k+1)/2⌋+l, the maximum possible, where n is the vertex number of G. Answering a question of Dong and Wu, we show that every tight (2,0)-stable graph with odd vertex number must be an odd cycle. Moreover, we show that for all k≥3, every tight (k,0)-stable graph has at most k+6 vertices. This is joint work with Sammy Luo. |
|
4:00pm - RH 306 - Applied and Computational Mathematics Bohan Chen - (Caltech) Learnable Operators on Probability Measures for Ensemble Data Assimilation Data assimilation (DA) in hidden Markov models evolves the filtering distribution via a nonlinear, measure-valued recursion that propagates uncertainty through the dynamics and updates it with observations. Classical ensemble methods such as the ensemble Kalman filter remain efficient and robust, but their Gaussian update can be inaccurate in strongly nonlinear, non-Gaussian regimes. We introduce the measure neural mapping (MNM), a learnable operator acting directly on probability measures to approximate the filtering map, leading to an MNM-enhanced ensemble filter (MNMEF) formulated both at the mean-field level and as an interacting-particle algorithm. MNMEF is implemented with the set transformer, enabling a single parameterization to transfer across ensemble sizes and the permutation invariance. We provide a theoretical foundation for this transfer by establishing a continuum limit for attention on measures: under mild regularity or boundedness assumptions, attention applied to empirical measures converges in Wasserstein distance to its continuous-measure counterpart as sample size increases. This continuum limit explains the observed stability of MNMEF trained on one ensemble size and evaluated across varying ensemble sizes. Empirically, MNMEF yields improved RMSE over leading baselines on Lorenz '96 and Kuramoto–Sivashinsky benchmarks for the state estimation problem. |
|
11:00am to 12:00pm - - Mathematical Physics Xianzhe Li - (UC Berkeley) Projective action and fibered rotation number for Hermitian symplectic cocycles In this talk, we discuss a natural generalization of the fibered rotation number for higher dimensional HSp(2m)-cocycles. Applying this to 1D Schrödinger operators, we extend the famous gap labeling theorem to the strip model. If time permits, I will also talk about some applications related to this. This talk is part of the WSMP program. |
|
1:00pm to 2:00pm - RH 340N - Dynamical Systems Philipp Kunde - (Oregon State University) Flexibility of slow entropy Abstract: Measure-theoretical and topological entropy play a central role in structural questions for dynamical systems and serve as crucial tools in detecting chaoticity of a system. However, entropy is positive if and only if the system has exponential growth of distinguishable orbit types and it does not provide any information for systems with slower orbit growth. To measure the complexity of systems with subexponential orbit growth several different invariants have been studied. For instance, Anatole Katok and Jean-Paul Thouvenot introduced the concept of slow entropy. In this talk we discuss flexibility results on the values of measure-theoretical slow entropy for rigid transformations and finite-rank systems. |
|
3:00pm to 3:50pm - 440R - Logic Set Theory Julian Eshkol - (UC Irvine) Weak Threading Ideals This is a continuation of the lectures getting the consistency of a weak threading ideal on \omega_2 from a measurable cardinal. |
|
11:00am to 12:00pm - - Mathematical Physics Liyang Shao - (UC Berkeley) Continuity of the intersection spectrum of analytic periodic Schrödinger operators Given a periodic Schrödinger operator with analytic potential and rational frequency α, let S_- denote the intersection of its spectra taken over the phase x in a torus. We show that up to sets of Lebesgue measure zero, S_- associated with α could be obtained asymptotically from S_- associated with rationals approximating α that satisfy certain approximating properties. We will talk more about the proof details. This work is joint with Svetlana Jitomirskaya and Xianzhe Li. This talk is part of the WSMP program. |
|
1:00pm - RH 340N - Algebra Jaiung Jun - (SUNY New Paltz) Introduction to hyperstructures and related topics Hyperstructures extend classical algebraic frameworks by allowing algebraic operations to be multi-valued. After introducing the basic concepts and key examples, we discuss how these structures interact with and enrich classical theories, with emphasis on applications and appearances in algebraic geometry and combinatorics. |
|
4:00pm to 4:50pm - RH 406 - Colloquium Philipp Kunde - (Oregon State University) (Anti-)classification results in Topological Dynamical Systems and Ergodic Theory A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacypreserves the qualitative behavior of topological dynamical systems. Smale's celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy. In Ergodic Theory the isomorphism problem dates back to von Neumann's foundational paper and asks to classify measure-preserving transformations up to measure isomorphism. These classification problems not only turn out to be hard but sometimes even to be impossible. In this talk, we give an overview of classification as well as anti-classification results. |
|
4:00pm to 5:00pm - NS II 1201 - Special Colloquium Nicolas Monod - (École Polytechnique Fédérale de Lausanne) Geometry, harmonic analysis and some groups Geometry in non-positive curvature is a vast landscape. But some examples are familiar: Euclidean spaces, hyperbolic spaces, or more combinatorial objects like trees. What makes these classical geometries so special? In harmonic analysis, there are also very fundamental archetypes, such as the Fourier transform or spherical functions. We will discuss a bridge between these two worlds from the perspective of group theory. |