Automata and pseudocharacters from one-dimensional TQFTs with defects

Speaker

Mikhail Khovanov is a recognized leader of the “categorification” program, which plays an important role in modern mathematics and physics.

Khovanov received his Ph.D. in 1997 under the supervision of Professor Igor Frenkel at Yale University. Shortly thereafter he came up with his famous idea of categorifying the Kauffman bracket, which is a version of the celebrated Jones polynomial of links in a 3-sphere. This was the first example of the categorification which interprets polynomial invariants as Poincare polynomials of new homology theories. The construction of Khovanovhomology was amazingly fruitful and very unexpected.

A further categorification of the HOMFLY-PT polynomial of links and a categorification of quantum groups are other major achievments of Khovanov which have now important implications in low-dimensional topology, algebraic and symplectic geometry, geometric representation theory and string theory.

Abstract

TQFTs in dimensions two, three, and four have been at the forefront of developments in mathematics in the past three decades. In this light talk we will point out that even in dimension one TQFTs are interesting if one introduces defects. We will tell two stories:

I. Changing constants from a ground field to the Boolean semiring allows to interpret finite state automata as 1D TQFTs with defects with the free Boolean semimodule spanned by the states of an automaton as the value of the TQFT on a boundary point (joint work with P.Gustafson, M.S.Im, R.Kaldawy, and Z.Lihn).

II. Pseudocharacters and pseudorepresentations are an essential tool in modern number theory, discovered by A.Wiles and R.Taylor thirty years ago. We will explain that the pseudocharacter condition can be interpreted as lifting a 1D topological theory (a lax TQFT) to a TQFT and discuss an extension of that condition to higher dimensions (joint work with M.S.Im and V.Ostrik).