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.