Organizers / 组织者

Caucher Birkar，曲三太， 陈炳仪

Time / 时间

Thursday, 15:30-16:30

March 16, 2023

Online / 线上

Zoom Meeting ID: 455 260 1552

Passcode: YMSC

Zoom link:

https://zoom.us/j/4552601552?pwd=cWxBUjlIN3dxclgrZWFEOC9jcmlwUT09

Abstract

I will start with a brief introduction of the Riemann-Hilbert problem for periods of certain universal families of Calabi-Yau varieties. Two applications are discussed.

The first application is about the describing the variety of zeros for certain special functions, which have a rather long history. Zeros of certain special functions have been of interests to people since the times of Riemann, who of course famously studied zeros of his zeta function. Stieltjes, Hilbert, Klein, Hurwitz and others studied the number of zeros of the Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ for real $a,b,c$. Subsequently, many authors generalized their results to confluent hypergeometric functions. Eichler and Zagier gave a complete description of the zeros of the Weierstrass P-function in terms of a classical Eisenstein series. Duke and Imamo later used it to prove transcendence of values of certain classical generalized hypergeometric functions at algebraic arguments. Following Hille, Ki and Kim studied the zeros of generalized hypergeometric functions of the form ${}_pF_p$. For real parameters for such a function. In this lecture, I will discuss certain higher dimensional analogues of these functions and zeros of their derivatives. Examples include Gel'fand-Kapranov-Zelevinsky hypergeometric functions and period integrals of hypersurfaces in a homogeneous variety.

Our second application of the Riemann-Hilbert problem is to confirm the hyperplane conjecture in mirror symmetry in a special case. This conjecture gives a simple cohomological description the periods of a universal family of Calabi-Yau in a toric manifold. This talk is based on joint work with J. Chen, A. Huang, S.-T. Yau, and M. Zhu. Some earlier joint work with S. Bloch, R. Song, V. Srinivas, and X. Zhu will also be discussed.

Speaker

My research lies at the interface between algebra, geometry, and physics. I study the interplay between representations theory, Calabi-Yau geometry and string theory. Intuitions and insights from one theory can often manifest themselves in a powerful and sometimes surprising ways to help solve problems in another theory.

Personal Homepage:

https://people.brandeis.edu/~lian/