Mathematics Colloquium: Recent Progress on Mahler’s Problem in Diophantine Approximation

Speaker:  Osama Khalil, University of Illinois Chicago

Title:  Recent Progress on Mahler’s Problem in Diophantine Approximation

Abstract :  A classical result of Khintchine’s provides a zero-one law for the Lebesgue measure of points in Euclidean space with a given quality of approximation by rational points. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question is part of a long history of results and conjectures aiming at showing that unlikely intersections between Diophantine sets and natural subsets of Euclidean space must occur for simple algebraic reasons. Some of these elementary Diophantine questions ultimately lead to difficult problems at the interface of dynamics and spectral theory of automorphic forms. The goal of this colloquium is to describe recent advances in homogeneous dynamics leading to progress on Mahler’s problem and how it is linked to a notion of sparse Hecke operators. The talk will not assume prior knowledge of these topics. This is joint work with Manuel Luethi.

Time:  Friday, February 16, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106

Zoom and In-person