Isadore Singer Memorial Conference

Abstract

Following Sophus Lie's pioneering work on continuous transformation groups and their applications to differential equations, the structure theory of the finite (dimensional) case was well-developed, but the story in the infinite (dimensional) case was much more obscure. In the first decade of the 20th century, Élie Cartan's research on the infinite case was profound but beset with significant difficulties. By mid-century, geometers were ready to begin addressing these difficulties. In 1965, Singer and Sternberg revisited Cartan's classification in an influential paper published in Journal d'Analyse Mathématique, completing and clarifying Cartan's work and laying the foundation for our modern understanding of the so-called `infinite groups'. In this lecture, I will explain what the interesting issues were and still are and explain how they relate to our current understanding of diffeomorphism groups.

Abstract

I will recall the work of Atiyah, Patodi and Singer introducing the eta-invariant and the effect it had my thinking, including the development of algebras of pseudodifferential operators. From this prospective I will describe the later work by Singer, Varghese and myself on projective versions of the Atiyah-Singer theorem.

Abstract

Isadore Singer and Michael Atiyah introduced the Dirac operator into differential geometry 60 years ago and it was the basis of my first interaction with him. The talk will focus on some of the ways in which the Dirac operator has appeared in the topics I have worked on and Singer’s influence on them.

Abstract

I will explain how a holomorphic version of the Ray-Singer Torsion in the context of Calabi-Yau 3-folds is related to topological string amplitudes via mirror symmetry, and captures Gromov-Witten invariants at genus 1. Moreover I explain how this is deeply related to properties of black holes in N=2 supersymmetric quantum gravity theories in 4 dimensions.