Nigel Higson

Penn State

Topology and Spectrum

Monday, September 28, 2020

6 p.m. EDT

Online talk using Zoom, Zoom ID 939 6822 1759

(please email Ina Petkova for passcode)

Abstract: Isaac Newton showed, while sheltering from the plague, that sunlight can be decomposed into a continuous spectrum of colors, or frequencies. Much later, at the dawn of quantum theory, matrices were used to explore cases in which this decomposition leads to a discrete — as opposed to continuous — range of frequencies. The term “spectrum” was from that point onwards co-opted for mathematical use. All three lectures in this series will be about the relationship between this mathematical concept of spectrum in matrix theory and topics in geometry and topology. The themes of the first lecture will be the spectrum of families of matrices, interactions with topology, and surprising applications to mechanical systems.

Integral Equations from Hilbert and Weyl to Connes and Beyond

Colloquium

Tuesday, September 29, 2020

4 p.m. EDT

Online talk using Zoom, Zoom ID 939 6822 1759

(please email Ina Petkova for passcode)

Abstract: David Hilbert’s theory of integral equations, developed in the early years of the last century, led to giant advances in a number of mathematical areas. The development of quantum mechanics soon after dramatically extended the reach of the theory, and then, towards the end of the last century, Alain Connes transformed Hilbert’s theory yet again into what is called noncommutative geometry. I shall present an introduction to noncommutative geometry from the perspective of Hilbert’s work, organized around one fascinating example related to the Atiyah-Singer index theorem.

A Counterfactual History of the Hypoelliptic Laplacian

Specialized lecture

Wednesday, September 30, 2020

4 p.m. EDT

Online talk using Zoom, Zoom ID 939 6822 1759

(please email Ina Petkova for passcode)

Abstract: Over the past two decades Jean-Michel Bismut has built powerful new machinery to obtain exact, rather than asymptotic, information about the spectrum of the Laplace operator on locally symmetric spaces. The simplest real-world application is a new approach to the Selberg trace formula for closed surfaces, but there are several other new and exciting applications in representation theory and beyond. The design of the machinery gradually emerged from Bismut’s extensive work on the Atiyah-Singer index theorem, but I shall try to describe more simply how the various parts of the design might have (but did not) come into being.