# Collquium: Jinbo Ren, University of Virginia, Transcendental number theory in algebraic geometry

### Event Description:

Abstract: The classical Lindemann-Weierstrass theorem asserts that if

n algebraic numbers are linearly independent over$ \mathbb{Q}$, then

their exponentials are algebraically independent over $\mathbb{Q}$.

This theorem generalizes the fact that $e$ is a transcendental number.

More generally, the (very difficult) Schanuel's conjecture predicts

that if n complex numbers $a_1, a_2,...,a_n$ are linearly independent

over $\mathbb{Q}$, then the field $\mathbb{Q}(a_1,...,a_n,

e^{a_1},...,e^{a_n})$ has transcendence degree at least $n$ over

$\mathbb{Q}$. This assertion is very strong, for example, the

transcendence of $\pi$ is one of its simple consequences.

In my colloquium talk, I will explain how to formulate the analogues

of these classical conjectures in the context of algebraic tori,

abelian varieties and Shimura varieties. I will also discuss their

applications in Diophantine geometry.