# Algebra & Geometry Seminar, Speaker Jinbo Ren, University of Virginia, Mathematical logic and its applications in number theory

### Event Description:

Abstract: A large family of classical problems in number theory such as:

a) Finding rational solutions of the so-called trigonometric

Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is

an irreducible multivariate polynomial with rational coefficients;

b) Determining all $\lambda \in \mathbb{C}$ such that

$(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both

torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$;

c) Studying algebraicity of values of hypergeometric functions at

algebraic numbers

can be regarded as special cases of the Zilber-Pink conjecture in

Diophantine geometry. In this talk, I will explain how we use tools

from mathematical logic to attack this conjecture. In particular, I

will present some partial results toward the Zilber-Pink conjecture,

including those proved by Christopher Daw and myself

Abstract: A large family of classical problems in number theory such as:

a) Finding rational solutions of the so-called trigonometric

Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is

an irreducible multivariate polynomial with rational coefficients;

b) Determining all $\lambda \in \mathbb{C}$ such that

$(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both

torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$;

c) Studying algebraicity of values of hypergeometric functions at

algebraic numbers

can be regarded as special cases of the Zilber-Pink conjecture in

Diophantine geometry. In this talk, I will explain how we use tools

from mathematical logic to attack this conjecture. In particular, I

will present some partial results toward the Zilber-Pink conjecture,

including those proved by Christopher Daw and myself.