MATH SEMINAR: New Directions in Manin's Conjecture

Guest: Ratko Darda, Sabancı University

Title: New Directions in Manin's Conjecture (MATH)

Date/Time: March 21, 2025, 12.00 

Place: FENS G029 (reserved until 13.30)

Abstract: One of the central questions in number theory and Diophantine geometry is to understand how solutions to polynomial equations are distributed among the rational numbers. The behavior is closely related with the geometry of the space - the variety - defined by the equations. Depending on certain properties of the variety, different mathematical theories are used to analyze it. The case where rational solutions are abundant is addressed by Manin’s conjecture, proposed by Yuri Manin and collaborators in the early 1990s. This conjecture provides a precise prediction for the number of solutions when counted according to their arithmetic size. Over the years, it has been widely studied and has become one of the central problems in the field. Recently, we have investigated a new environment for the conjecture - that of stacks - the geometric objects generalizing varieties. This perspective has led to unexpected applications beyond its original scope, offering insights into phenomena related to the occurrence of Galois extensions—a topic that, at first glance, seems unrelated to counting solutions of polynomial equations.

Biography: Ratko Darda obtained his PhD from University of Paris Cité in 2021. He subsequently held a JSPS postdoctoral fellowship at Osaka University and a postdoctoral position at the University of Basel. He is currently a Marie Skłodowska-Curie postdoctoral fellow at Sabancı University. His research interests include number theory, arithmetic geometry, and Diophantine geometry.