Abstract:

Topology seems like the antithesis (opposite) of number theory: the former is continuous and deformable, the latter is discrete and rigid.  Surprisingly, however, thanks to results and ideas of Mostow and Thurston we know that for 3-dimensional objects (closed 3-manifolds, knot complements) the situation changes: they have intrinsic geometries that cannot be deformed. This leads to many connections with deep topics in number theory such as algebraic K-theory (Bloch group, special values of dilogarithms, construction of units in cyclotomic extensions of algebraic number fields), various types of modularity (classical and mock modular forms, quantum modular forms and holomorphic quantum modular forms), or the Habiro ring and a generalization of it associated to any algebraic number field.  All of these connections arise through invariants (Witten-Reshetikhin-Turaev invariant, Kashaev invariant) coming from quantum field theory.  A theme running through everything is that the quantum invariants always have a kind of very weak modularity property (quantum modular form, holomorphic quantum modular form) and that the obstruction to their actual modularity is given by a class in algebraic K-theory canonically associated to the 3-manifold in question.  This is still far from proved in general, but if time permits I will describe examples coming both from topology and from special functions ("Nahm sums") that support this picture.

The lectures are aimed at non-specialists and the many unfamiliar words in the above text, like "Bloch group", "quantum modular forms" and "Habiro rings", will all be explained.

All of the work described is joint with Stavros Garoufalidis and parts of it also with other co-authors (Frank Calegari, Campbell Wheeler, Peter Scholze).

Personal Website：https://www.mpim-bonn.mpg.de/node/97 

Registration: https://www.wjx.top/vm/rhru6xL.aspx#