# Mathematical Sciences Research Institute

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# Seminar

DDC - Model Theory Seminar: Model theory and bi-algebraic geometry October 12, 2020 (08:00 AM PDT - 09:00 AM PDT)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) James Freitag (University of Illinois at Chicago)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

#### Model Theory And Bi-Algebraic Geometry

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

Let $X$ and $Y$ be algebraic varieties over $\mathbb C$ and let $\phi: X^{an} \rightarrow Y^{an}$ be a complex analytic map which is not algebraic. In this case, for most algebraic subvarieties $X_0 \subset X$, the image $\phi (X_0)$ is not algebraic. The pairs of algebraic subvarieties $(X_0 , Y_0)$ with $X_0 \subset X$ and $Y_0 \subset Y$ such that $\phi (X_0) = Y_0$ are called \emph{bi-algebraic} for $\phi$. Specific instances of bi-algebraic problems have played an important role in the resolution of conjectures in diophantine geometry and model theory over the past decade. In this talk, I will give a general overview of model-theoretic approaches to this type of problem coming from o-minimal and differential algebraic geometry. I will give a more detailed description of one approach to the problem in the case that $\phi$ is the uniformizing function of a discrete subgroup of $SL_2$.