Explore the fascinating connections between quantum invariants of knots and 3-manifolds and advanced number theory in this comprehensive lecture. 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. Investigate recent developments in algebraic number theory, including the construction of non-trivial units and extensions of the Habiro ring to arbitrary algebraic number fields. Uncover the surprising "quantum modularity" properties of the Kashaev invariant and its generalizations, leading to new concepts in modular forms theory. Learn about collaborative research with Stavros Garoufalidis, Rinat Kashaev, and Peter Scholze. Gain insights into topics such as regulators, inverse limits, congruence, Norman Sagir equations, the Gashith invariant, and power series with coefficients. No prior knowledge of knot theory, K-theory, or modular forms theory is required for this accessible lecture designed for a general mathematical audience. Read more