Mahbub Alam: Homogeneous dynamics and some number-theoretic applications

Abstract: I will provide an overview of the quantitative and distributional aspects of homogeneous dynamics, focusing on Diophantine approximation and lattice point counting as key examples. In this talk, I will introduce a dynamical approach for counting the number of solutions to Diophantine inequalities over number fields. This method links the count of solutions to the problem of counting lattice points within specific regions, and to the moments of the lattice point counting function. I will discuss these moments in the broader framework of adeles, highlighting their significance in understanding the underlying structures. If time permits, I will also touch upon related topics, such as the sphere packing problem and the covering radius problem, emphasizing their connections to the aforementioned topics.