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Math Colloquium Alice Medvedev
University of California, Berkeley
Title: Algebraic Dynamics, Model Theory, and Number Theory
Abstract: Model Theory, a branch of mathematical logic, is a new and useful way to approach number-theoretic conjectures about ``special points'' and ``special subvarieties'', such as the Manin-Mumford Conjecture. Algebraic Dynamics, the study of discrete dynamical systems in the category of algebraic geometry, supplies natural generalizations of these conjectures. We use model-theoretic ideas to settle some cases of these dynamical generalizations of number-theoretic conjectures. Our key result is a complete characterization of invariant subvarieties for coordinate-wise polynomial dynamical systems on affine space, in characteristic zero. That is, we work over a field $K$ of characteristic zero (such as the complex numbers) and study the iteration of a function $$F(x_1, x_2, ... x_n) = ( f_1(x_1), f_2(x_2), ... f_n(x_n) )$$ for some univariate polynomials $f_i$ over $K$. Model-theoretic ideas reduce the question of invariant subvarieties in cartesian powers of $K$ to the question of invariant curves in $K \times K$. Refining Ritt's Theorem about composition of polynomials allows us to classify these invariant plane curves. Our classification implies that, barring obvious obstructions from linear $f_i$, such dynamical systems always have $K$-rational points with Zariski-dense forward orbits, a generalization of a case of Zhang's Conjecture. We also prove the dynamical analog of the Manin-Mumford conjecture for the very special case when all $f_i$ are defined over the integers, and for some prime $p$ all $f_i$ are congruent to $x^p$ modulo $p$.
Date: Wednesday, February 20, 2013
Time: 4:30 pm
Location: Wean Hall 7500
Submitted by: Bohman
Note: Refreshments 4:00 pm, Wean Hall 6220.