The lectures will be in-person, and they will be preceded by the Workshop on Categorical Methods in Representation Theory and Quantum Topology.
Abstract: Topological quantum field theories (TQFT) convert topological information about cobordisms between manifolds to linear maps between vector spaces. They have become a fundamental structure in modern mathematics. We will discuss a method, called universal construction, of building TQFTs starting with numbers associated to closed cobordisms. It allows to recover familiar TQFTs up to dimension three and produces new ones subject to a weaker form of Atiyah’s tensor product axiom.
Interestingly, this construction is nontrivial even in dimension one, especially if allowing 0-dimensional defects with labels inside 1-cobordisms. We explain how universal construction in this case, and taking values in the Boolean semiring, recovers the classical notions of a regular language and a finite state automaton and provides a new viewpoint on these well-known structures in the foundation of language theory and computer science.
Lecture 1 is accessible to mathematics majors.
Abstract: Quantum invariants of knots and links relate low-dimensional topology to representation theory of quantum groups and to a wealth of other structures in representation theory and mathematical physics. About twenty years ago it’s been discovered that many of these invariants can be realized as Euler characteristics of exotic homology theories of links. We will review these constructions in the simplest cases and how extending them to links with boundary points (tangles) connects the theory with homological algebra of very special rings that can be found throughout representation theory and algebraic geometry.
Abstract: Foams are two-dimensional CW-complexes embedded in 3-space. Known link homology theories are based on foams subject to suitable skein relations as part of the package. We’ll talk about a relatively recent advance (2017) by Louis-Hadrien Robert and Robert Wagner allowing to develop a clean theory for GL(N) foams bypassing previous categorical constructions and its connections to link homology, to Kronheimer-Mrowka gauge theory for 3-orbifolds, and to the Four-Color Theorem.
Virginia Mathematics Lectures archive