## The Ax-Lindemann theorem and its applications

**Speaker(s): ** 高紫阳（巴黎第十一大学）

**Time: **July 28 - August 1, 2014

**Venue: ** Room 09 at Quan Zhai, BICMR

Time：7月28,30号,8月1号下午2:00-3:00.

Venue：全9

报告者：高紫阳（巴黎第十一大学）

Titles:

1. History and motivation of the Ax-Lindemann theorem;

2. Sketch of the proof;

3. Diophantine estimate and the use of o-minimality

abstract：The Ax-Lindemann theorem is a functional algebraic independence statement, which generalizes (the analog of) the classical Lindemann-Weierstrass theorem. This theorem plays a key role in the Pila-Zannier method of proving the mixed Andre-Oort conjecture. Klingler-Ullmo-Yafaev have recently proved the Ax-Lindemann theorem for all pure Shimura varieties. Using their result, I proved it for all mixed Shimura varieties. The o-minimal theory, in particular the counting theorems of Pila-Wilkie, is very important for all the proofs.

In the first talk, I will focus on the history of the Ax-Lindemann theorem. I will explain how this theorem is a natural generalization of the classical Lindemann-Weierstrass theorem. The basic definitions will be given during this review of history. At the end I will explain how this theorem is related to the Andre-Oort conjecture.

In the second talk, I will present the strategy of the proof assuming some results which I shall explain in the third talk.?The similarities and differences between my proof (for the mixed case) and the proof of KUY (for the pure case) will also be explained.

In the third talk we deal with the Diophantine estimates which we assumed last time. I will introduce (from the basic definition) the o-minimal theory, explain how it comes into the proof and then finish the proof.