Zeta Functions and Their Zeros
Student No.:50
Time:Part One: Nov.12/14/19/21/26 Time:15:10-17:00; Part Two: Dec. 09/11/16/18 Time: 10:10-12:00
Instructor:Lin Weng  [Kyushu University, Japan]
Place:Conference Room 1, Floor 1, Jin Chun Yuan West Building
Starting Date:2013-11-12
Ending Date:2013-12-18




Zeta functions, a central theme in mathematics, have been proved to be very mysterious. In this lecture series, we explain some genuine constructions of both abelian and non-abelian zetas for number fields and function fields over any base fields, using stability. We will expose some reasons via examples why they satisfy the Riemann Hypothesis. The topics covered range from motivic Euler products and motivic K group, Atiyah-Bott versus Harder-Narasimhan for various masses of moduli stacks, Arthur and Lafforgue truncations, Langlands-Slegel Eisenstein series, etc to the Riemann Hypothesis. Some exciting developments, jointly obtained with Zagier on zetas for elliptic curves will also be included. The lectures end with a conjectural structures on moduli stacks of principal bundles, exposing how non-abelian invariants can be obtained via abelian ones with the help of Lie structure guided by stability.



Some understanding of languages of modern mathematics



References (Roughly):






1) Atiyah-Bott: Ying-Mills on Riemann surfaces

2) Harder-Narasimhan: Cohomology of moduli spaces

3) Lafforgue: Ramanujan conjecture

4) Ki, Komori, Suzuki: Riemann Hypothesis for Weng zetas

5) Moglin-Waldspurger: Spectrum decomposition and Eisenstein series

6) Weng: Geometric Approach to L functions,

7) Weng&Zagier: Riemann Hypothesis for Higher Rank Zetas of Elliptic Curves


Course Notes: