Title: Fundamental Gap of Laplacian on Convex Domains

Speaker: Guofang Wei, UC Santa Barbara (10/30)

Abstract: The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. We will survey many recent fantastic results for convex domains in Euclidean spaces, spheres, hyperbolic spaces and surfaces with positive curvature with Dirichlet boundary conditions, starting with the breakthrough of Andrews-Clutterbuck. Then we will present a very recent work on horoconvex domains in the hyperbolic space. This last result is joint with Ling Xiao.

All necessary background information will be introduced in the talk.

Taibleson Colloquium: Quantum and classical low degree PAC learning, Harmonic analysis, and dimension-free Remez inequality

Speaker: Alexander Volberg, Michigan State University (10/23)

Abstract: The talk is based on recent works with Lars Becker, Ohad Klein, Joseph Slote, and Haonan Zhang. Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including noncommutative spaces (matrix algebras). Suppose you wish to find a 2^n by 2^n matrix by asking this matrix question that it honestly answers. For example you can ask questions "What is your (i,j) element?" Obviously you will need exponentially many questions. But if one knows some information on Fourier side one can ask only log n questions if they are carefully randomly chosen. One pays the price: one would find the matrix only with high confidence, and with a small error. Such learning is known as PAC; PAC stands for 'probably approximately correct'.
The origins of the problem are in theoretical computer science, but the methods are pure harmonic analysis and probability. I will explain how this can be done using harmonic analysis and probability. The main ingredient is dimension free Bernstein-Remez inequality.

This inequality is the key ingredient that allows us to extend to new spaces a recent series of algorithms for learning low-degree polynomials on the hypercube and low-degree quantum observables on qubits.

Title: From knots to 4-manifolds

Speaker: Ciprian Manolescu, Stanford University, ( 10/16)

Abstract: I will survey the connections between knot theory and four-dimensional topology.

Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to

progress in computing invariants of smooth four-manifolds that can detect exotic structures. I will explain how

this was done in two contexts: Heegaard Floer theory and skein lasagna modules.

Title: The Arduous Path of AI Models to the Bedside

Speaker: Thomas Kannampallil, Washington University in St. Louis (10/09)

Abstract: In this talk, I will describe the fundamental challenges for researcher-developed AI models (both deep learning and generative AI) to be brought to the point-of-care. Specifically, using examples from my own research in clinical medicine, I will discuss the pragmatic challenges model innovation, causal reasoning, and clinician-AI collaboration. I will also highlight key areas for computational innovation in clinical medicine, and collaborative opportunities for federal and philanthropic funding.

Bio: I am an Associate Professor of Anesthesiology at WashU Medicine, with affiliate appointments in in the Department of Computer Science and Engineering and the Institute for Informatics, Data Science, and Biostatistics (I2DB). I am also an Assistant Dean for Data Science and the Chief Data Scientist for WashU Medicine and BJC HealthCare.

My research interests lie at the intersection of computer science, cognitive science, and clinical informatics. Specifically, my research focuses on developing and evaluating intelligent computational tools for improving clinical decision making and patient safety. My research is currently funded by multiple R01s from the National Library of Medicine (NLM), National Institute of Aging (NIA), and the Agency for Healthcare Research and Quality (AHRQ).

I am currently the Deputy Editor-in-Chief for the Journal of Biomedical Informatics I was elected as a Fellow of the American College of Medical Informatics in 2025. Additional details can be found at: https://sites.wustl.edu/thomask/

Title:Incidence geometry and tiled surfaces

Speaker: Sergey Fomin, University of Michigan (09/18)

Abstract: Abstract: We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones. This is joint work with Pavlo Pylyavskyy.

Title: Spectral Learning for Dynamical Systems

Speaker: Matthew Colbrook (09/11)

Abstract: Koopman operators recast nonlinear dynamics as infinite-dimensional linear systems, enabling spectral analysis of time-series data. Over the past decade, they've found widespread use in fields ranging from robot control and climate variability to neural networks, epidemiology, and neuroscience. But which spectral features can actually be reliably computed from finite data-and which cannot? We show that the answer depends critically on the choice of observable space (e.g., L2 vs RKHS) and the nature of the underlying dynamics. In particular, we construct adversarial systems where any algorithm provably fails, even with unlimited data and randomized methods. By identifying when convergence is possible, we design algorithms that succeed both in theory and in practice, overcoming the notorious problem of spurious eigenvalues that undermine prediction. Applications to climate systems-including sea ice and ocean flow-demonstrate how spectral analysis reveals coherent structures and enables state-of-the-art forecasting. This leads to a classification theory that quantifies the difficulty of data-driven problems for dynamical systems and proves the optimality of certain algorithms. I'll conclude with a look ahead at where this field is going, and the exciting opportunities it presents for mathematicians.

Title: Transverse Spheres in Flag Manifolds

Speaker: Parker Evans (09/04)

Abstract: Flag manifolds are natural objects in algebraic and differential geometry, generalizing familiar examples like projective space and Grassmannians. A subset of a flag manifold is called transverse when every pair of points is in general position.

Transverse circles abound. Conversely, recent restriction results show that in certain cases transverse circles are maximally transverse. We shall detail a complementary result. Using spinors, we build arbitrarily large transverse spheres in real full flag manifolds. Previously, there were no known transverse subsets larger than a circle for these ambient spaces. The new spheres are, in fact, maximally transverse, as witnessed by topological K-theory. Time permitting, we discuss implications for Anosov representations. This project is joint with Max Riestenberg.