Homogeneous Dynamics and Probabilistic Diophantine Approximations - Cauchy Limit Laws for Linear Forms

Explore a comprehensive lecture on homogeneous dynamics and probabilistic Diophantine approximations, focusing on Cauchy limit laws for linear forms with random coefficients. Delve into a survey of classical and recent results in lattice counting and Diophantine approximation problems from a statistical perspective. Examine the distribution of linear forms with random coefficients evaluated mod 1 on integers, and discover the Cauchy limit law for the associated discrepancy function normalized by ln^d N. Gain insights into the key proof ingredient: a Poisson limit theorem for visits to the cusp under the Cartan action of lattices distributed uniformly on a positive codimension leaf of the horospheres. This hour-long talk, presented by Bassam Fayad at BIMSA, offers a deep dive into advanced mathematical concepts at the intersection of dynamics and number theory.