数学科学学院学术报告[2023] 073号

(高水平大学建设系列报告845号)

报告题目:“Complex Analytic Methods in Functional Transcendence Theory and Related ProblemsI”

报告人：黄国坚博士（香港大学）

报告时间：11月2日，7日，10日；14:30-16:00（短期课程）

讲座地点：汇文楼1420

报告内容：Recently, the Ax-Schanuel conjecture for Shimura varieties has been resolved in Mok-Pila-Tsimerman [3]. In this mini-course, we discuss the strategy in [3]. Special focuses will be placed on the complex differential and algebraic techniques involved in some of the main steps of the proof. We will also discuss some related problems and results from complex analytic perspective.

Content 1. Functional transcendence. Classical transcendence theory in number theory is one of the most important motivation of modern functional transcendence theory. We recall some classical results and relate them to the function field analogue of the transcendence theory. The relationship with Hodge theory and atypical intersections will also be considered.

2.Hermitian symmetric spaces. Shimura varieties are arithmetic quotients of bounded symmetric domains. We introduce bounded symmetric domains first from the point of view of several complex variables. Then we indicate how these domains serve as bounded domain realization of Hermitian symmetric spaces of noncompact type, together with a brief review of the important properties.

3. Volume estimates on Shimura varieties. One of the key ingredients for the proof of Ax-Schanuel for Shimura varieties is the volume estimates of Hwang-To [1]. We discuss some differential geometric aspects of bounded symmetric domains and their finite volume quotients, which give necessary preparations for the proof of Hwang-To’s estimates.

References

[1]Hwang, Jun-Muk; To, Wing-Keung: Volumes of complex analytic subvarieties of Hermitian symmetric spaces. Amer. J. Math.124(2002), no.6, 1221–1246.

[2]Klingler, B.; Ullmo, E.; Yafaev, A.: The hyperbolic Ax-Lindemann-Weierstrass conjecture. Publ. Math. Inst. Hautes Etudes Sci.123(2016), 333–360. ´

[3]Mok, Ngaiming; Pila, Jonathan; Tsimerman, Jacob: Ax-Schanuel for Shimura varieties .Ann. of Math. (2)189(2019), no.3, 945–978

报告人简介：黄国坚博士，现于香港大学从事博士后研究。研究方向为复代数几何，复微分几何，尤其是霍奇理论，函数超越性理论。在IMRN，Math Z等期刊发表多篇论文。

欢迎师生参加！

邀请人：丁聪

数学科学学院

2023年11月02日