Alignments of Definable Groups and Explicit Bounds in General Elekes-Szabó

Explore advanced mathematical concepts in this 50-minute conference talk examining the intersections of algebraic varieties with finite grids and their connections to algebraic groups. Delve into the influential Elekes-Szabó theorem, which demonstrates that algebraic varieties can achieve maximal intersection sizes with large finite point grids only when closely related to algebraic groups. Learn how model theory techniques, including variants of Hrushovski's group configuration and Zilber's trichotomy principle, prove essential for identifying these groups and have enabled significant generalizations of the Elekes-Szabó theorem over the past decade. Focus on the o-minimal case while discovering a generalization to arbitrary co-dimension that extends previous results and provides explicit bounds in the Bays-Breuillard theorem. Gain insights into cutting-edge research at the intersection of algebraic geometry, model theory, and combinatorial geometry through this collaborative work with Kobi Peterzil and Sergei Starchenko.