MS Seminar (Mathematics - String Theory)

Speaker: Don Zagier (ICTP(Trieste, Italy) and SUSTech (Shenzhen, China))
Title: Arithmetic and modular properties of quantum knot invariants
Date (JST): Tue, Sep 26, 2023, 13:30 - 15:00
Place: Lecture Hall
Abstract: I will report on ongoing (for several years) joint work with Stavros Garoufalidis on the arithmetic properties of the Kashaev invariant of knots and other quantum invariants of 3-dimensinal manifolds. The starting point is the famous Kashaev Volume Conjecture and its extension to a kind of weak modularity statement relating the asymptotics of the Kashaev invariant near different roots of unity. Successive refinements of this lead to new invariants and new modularity properties. The ingredients on the number-theoretical side are the Bloch group of a number field, the Habiro ring, and the notion of quantum modular forms, all of which will be explained in the talk, and the final outcome is the (conjectural) existence of a certain matrix-valued "holomorphic quantum modular form" associated to any knot. This work also led to two nice applications in pure number theory, the construction of certain units in cyclotomic extensions of number fields and the proof of one direction of Nahm's conjecture on the modularity of certain q-hypergeometric series, both of which will also be described briefly.
Remarks: Lecture Hall + Zoom