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.

About the speaker：
Don Zagier is a pure mathematician of broad interests, but working primarily in the domain of number theory and the theory of modular forms and their applications in other areas ranging from knot theory to mathematical physics.He is an emeritus director at the Max Planck Institute for Mathematics in Bonn and holds the Ramanujan International Chair at the International Center for Theoretical Physics in Trieste. Among his many scientific recognitions, Zagier won the Frank Nelson Cole Prize in 1987, became a member of the U.S National Academy of Sciences in 2017, and received the Fudan-Zhongzhi Science Award in 2021.