продолжаются Harvard CMSA Math Science Literature Lecture Series

в среду (13.01) в 17:00 по Москве — Don Zagier, Quantum topology and new types of modularity

The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory.

A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry.

In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory.

These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old (…). About 10 years ago, I was led by numerical experiments to the discovery (…) the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property (…). This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story (…)