From Knots to Number Theory I

Explore the fascinating connections between quantum invariants of knots and 3-manifolds and advanced number theory in this lecture by Don Zagier from Max Planck/ICTP. Delve into the rigidity theorems of 3-dimensional hyperbolic topology and their arithmetic implications, linking hyperbolic 3-manifold volumes to the Bloch group and algebraic K-theory through the dilogarithm. Discover the Kashaev invariant's relationship to hyperbolic volume and the Habiro ring, a intriguing number-theoretical object. Uncover recent developments in algebraic number theory, including the construction of non-trivial units and extensions of the Habiro ring to arbitrary algebraic number fields. Learn about the surprising "quantum modularity" properties of the Kashaev invariant and its generalizations, leading to new concepts in modular forms theory. Gain insights into collaborative research with Stavros Garoufalidis, Rinat Kashaev, and Peter Scholze. No prior knowledge of knot theory, K-theory, or modular forms theory is required for this accessible lecture designed for a general mathematical audience.