Topology Seminar

Equivariant multiplicative infinite loop space theory is interesting, but hard. I will describe definitions in terms of symmetric groups which promise to give an easy new way to build the theory. As I shall describe, the main obstruction to making the new theory rigorous is the elementary categorical combinatorics of symmetric groups.

I will give an overview of time series data (that is, data sampled at different points in time) and spatially autocorrelated data (e.g., home prices, where the price of the house next to yours contains information about the likely price of your house). I'll then discuss the most common statistical models for such data, including ARIMA models that factor in what the past knows about the present, spectral models based on Fourier analysis, spatially weighted models, and the general linear mixed model for data with both spatial and temporal autocorrelation. I'll illustrate these models with vignettes from my research, on problems related to drug overdose and to the policing of protests in the USA. If there is time, I will explain some new research applying topological data analysis (TDA) to model the spatiotemporal spread of overdoses.

Topology, a cornerstone of modern mathematics, provides a universal framework for classifying defects in physical systems, e.g., nematic disclinations, spin/orbital angular momentum, and spatial vortices etc. Its integration with band theory has revolutionized our understanding of momentum space as a parameter space, where band degeneracies (Dirac/Weyl points) are characterized by topological invariants constructed from Berry phase, exemplified by winding numbers and Chern classes. The recent incorporation of algebraic topology tools, particularly homotopy and homology theories, has further enriched the classification of topological phases. In this talk, we introduce the application of catastrophe theory–a distinct mathematical branch–in band theories. Focusing on PT-symmetric non-Hermitian systems, we demonstrate that catastrophe singularities in the ADE classification (cuspoids, umbilics, etc.) universally manifest as symmetry protected band degeneracies, defining unprecedented topological phase classes. Crucially, these catastrophe-engineered degeneracies host novel topological edge states that persist in gapless regimes, challenging the conventional bulk–boundary correspondence. Our findings establish connections between catastrophe theory and topological matter. This cross-disciplinary synthesis opens new avenues for topological physics and enriches the fundamental understanding of singularity-associated topological phenomena.

Many fundamental concepts from higher algebra–such as finitely presented, flat, and étale morphisms of ${\mathbb E}_n$-rings–admit natural generalizations to the setting of $t$-structured tensor triangulated $\infty$-categories (ttt-$\infty$-categories). Under a natural structural condition we term projective rigidity, we establish $\infty$-categorical analogues of key classical results, including Lazard's theorem, étale rigidity, and the universal property of the derived category. We demonstrate that projective rigidity is satisfied in a broad range of examples, such as

• spectra (with the Postnikov $t$-structure),
• filtered and graded spectra,
• genuine $G$-spectra for finite groups $G$,
• Artin–Tate motivic spectra over a perfect field.

A particularly illuminating example is the 1-dimensional framed cobordism category, which serves as a "generator" for ttt-$\infty$-categories with projective rigidity. This connection highlights the deep interplay between geometric topology and higher algebra in our framework. This is joint work with Xiangrui Shen.