Abstract:

Modular forms are one of the most beautiful and fundamental notions in modern number theory and play a role in almost every domain of pure mathematics and physics. The famous specialist Martin Eichler once said, "There are five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms."  In the first talk I will explain this notion and present some highlights of the theory and its applications.  The last part, about the relatively new notion of "quantum modular forms" and its variants, is based on joint work with Stavros Garoufalidis (SUSTech) during the last several years in which many beautiful interconnections between number theory and 3-dimensional topology, and in particular the so-called quantum invariants of 3-dimensional topology, were found.

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