This installment of the Brown-Yale Analysis/PDE Working Seminar will be held on Thursday, May 8, at 10 a.m. - 5 p.m. at Yale with the following speakers. 

Benoit Pausader, Brown
Title:  TBA

Title:  On the Mathematical Foundations of Infinite-Dimensional Spectral Computations
Abstract:  Spectral computations involving infinite-dimensional operators have both fascinated and frustrated mathematicians since the seminal work of Goldstine, Murray, and von Neumann in the 1950s. In the 1990s, Arveson observed that despite extensive literature on computing spectra, the general computational spectral problem remained unsolved. In this talk, we present a resolution of this problem through the Solvability Complexity Index (SCI) hierarchy. This framework allows us to classify which problems are algorithmically possible and which are provably impossible. A key difficulty is that many spectral problems require multiple limiting processes, a phenomenon first shown by McMullen in polynomial root-finding. Upper bounds are demonstrated via explicit algorithms with controlled error using injection moduli, applicable to partial differential operators on $L^2(\mathbb{R}^d)$ with locally bounded total variation coefficients and to discrete infinite matrix operators. Lower bounds are obtained using reductions from descriptive set theory, exposing intrinsic barriers that hold in any model of computation. Applications include Koopman spectral analysis in dynamical systems, leading to improved forecasting of Arctic sea ice and the construction of adversarial dynamical systems designed to thwart spectral computation. The framework also connects to broader areas such as optimization, neural networks, PDEs, and computer-assisted proofs.

Reuben Drogin, Yale
Title: Random Matrix Theory and Diffusive Transport in the Anderson Model

Mengyi Xie, Yale
Title:  Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations