Lecturer:Ren Jinbo

Abstract:

A fundamental question of Diophantine theory is that if a polynomial equation $F(x_1,...,X_m)=0$ contains many "special solutions", does it mean this equation have a special meaning? This question has a profound background. Let $V$ be an algebraic variety defined by the zeros of several polynomial equations mentioned above. When $V$ is a subvariety of the torus $\mathbb{G}_m^n$, the problem is essentially Roth's Fields Medal work on approximating irrational numbers with rational numbers and its generalization; When $V$ is a subvariety of abelian variety, this problem implies Faltings' Fields Medal work on Fermat's Last Theorem; When $V$ is a subvariety of Shimura variety, this problem is closely related to the theory of complex multiplication of elliptic curves and even abelian variety.

On the other hand, if an abstract group $\Gamma$ said to be generated with boundary in group theory can be written as the product of a finite number of cyclic subgroups, then $\Gamma= \langle g_1 rangle ... \langle g_r \rangle$. The study of bounded generation has important applications in many fields, such as Serre's congruence subgroup problem, Margulis rigidity and Kazhdan Property (T).

In my recent cooperation, we have used a theorem of Diophantine theory above to prove that an arithmetic group cannot be generated with boundary by Toral elements. In fact, we have proved that the growth rate of sets that can be generated with boundary by Toral elements is at most $O(\log T)$, where $T$ is a point function (called height) defined by Diophantine geometry.

I will introduce the general idea of proof and give some open questions in the report.

Introduction to the Lecturer:

RenJinbo, Institute for Advanced Study

Invitee:

Cui Weideng, Zhao Lilu

Time:

10:00-11:00, March 23 (Wednesday)

Venue:

Tencent Meeting, Meeting ID: 224 840 946

Sponsor: School of Mathematics of Shandong University