## Topology Seminar (Main Talk): Genus Two Generalization of $A_1$ spherical Double Affine Hecke Algebra

Seminar | September 5 | 4-5 p.m. | 3 Evans Hall

Semeon Artamonov, University of California, Berkeley

Department of Mathematics

Spherical Double Affine Hecke Algebra can be viewed as a noncommutative $(q,t)$-deformation of the $SL(N,C)$ character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus $SL(2,Z)$ acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of $A_1$ spherical DAHA and show that the Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters $q,t$ satisfying $q^n t^2=1$ for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.

Based on arXiv:1704.02947 joint with Sh. Shakirov

c_abbott@berkeley.edu