L-function: Its Past and Future

Abstract

In this lecture, the speaker will first review a brief history of 2,500 years in searching a formula for finite sums

1^k + 2^k + ... n^k

which leads to work of Euler on infinite sums

1^k + 2^k + ... n^k + ...

He will also explain how this game of calculus unfolds some secrets of primes in a series of work by Euler, Dirichlet, and Riemann in the 18th and 19th century. Finally, he will describe how the study of L-function becomes a central topic of number theory under the framework of Langlands program and arithmetic geometry in the 20th and 21st century.

About the speaker

Prof Shou-Wu Zhang received his PhD from Columbia University in 1991. He was a member of the Institute for Advanced Study in Princeton and an Assistant Professor at Princeton University from 1991 to 1996. He has been tenured at Columbia University since 1996 and at Princeton University since 2011.

Prof Zhang’s research areas include number theory and arithmetic algebraic geometry. He is on the editorial boards of the Research in Number Theory, the International Journal of Number Theory, the Journal of Differential Geometry, and Science in China, among other publications.

Prof Zhang was an invited speaker of the International Congress of Mathematicians at Berlin in 1998 and was awarded a Morningside Gold Medal of Mathematics in the same year by the International Congress of Chinese Mathematicians for his work on the Bogomolov conjecture and Gross-Zagier formula. He was a Sloan Research Fellow, a Guggenheim Fellow, a L.-K. Hua Chair Professor at Chinese Academy of Sciences, a Changjiang Chair Professor at Tsinghua University, and a Prize Fellow at Clay Mathematical Institute. In 2011 and 2016, he was elected Fellow of the American Academy of Arts and Sciences and of the American Mathematical Society respectively.