Quantum topology studies manifolds using invariants coming from quantum field theory. This program focuses on quantum invariants and their application to questions in geometry and low-dimensional topology such mapping class group representations, hyperbolic structures on three-manifolds, and smooth structures on four-manifolds.
The program will be centered around four streams: 1) character varieties and their quantum deformations 2) hyperbolic geometry and quantum invariants 3) contact geometry and cluster algebras 4) categorification in quantum topology.
Unifying these four streams is the formalism known as skein theory. Skein relations are elementary algebraic relations, typically encoded by embeddings of 1- and 2-dimensional singular submanifolds into ambient 2-, 3- or 4-manifolds, and equipped with various decorations informed by representation theory. Skein invariants provide an accessible point of entry: without a lot of background, a lot of the intuition in these areas can be conveyed by hand-drawn pictures. The streams all involve hands-on geometric and combinatorial constructions, cutting/gluing/TQFT-like properties, and computation via diagrams, which we hope will facilitate Rosetta-stone like translations between previously disjoint fields
Daniel Douglas
(Virginia Tech)
Ko Honda
(University of California, Los Angeles (UCLA))
David Jordan
(University of Edinburgh)
Effie Kalfagianni
(Michigan State University)
Aaron Lauda
(University of Southern California (USC))
Ian Le
(Australian National University)
Jessica Purcell
(Monash University)
Paul Wedrich
(University of Hamburg)
Copyright. All Rights Reserved.