IPAM "Quantum Topology, Character Varieties and Low-Dimensional Geometry" Fall 2026 Program Overview
Apply to IPAM's Fall 2026 Programs: https://www.ipam.ucla.edu/programs/long-programs/quantum-topology-character-varieties-and-low-dimensional-geometry/?tab=activities"Quantum Topology, Character Varieties and Low-Dimensional Geometry" is the Fall 2026 long program at the Institute for Pure and Applied Mathematics at UCLA. In this webinar, Selenne Bañuelos, the Associate Director of IPAM, explains the resources provided to long program resident scholars. She is joined by Ian Le of the Australian National University for an overview of the long program content, beginning at 06:44. The themes of the three main week-long workshops are introduced by:
10:27 - Jessica Purcell of Monash University, "Hyperbolic Structures and Quantum Invariants"
12:03 - Ko Honda of the University of California, Los Angeles, "Contact Geometry, Cluster Algebras and Skein Theory"
13:37 - Paul Wedrich of the University of Hamburg, "Categorification in Quantum Topology"
Long Program Overview:
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.
Apply to IPAM's Fall 2026 Programs: https://www.ipam.ucla.edu/programs/long-programs/quantum-topology-character-varieties-and-low-dimensional-geometry/?tab=activities Receive SMS online on sms24.me
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