Metric number theory via geometry and dynamics: Mahler to Margulis

Abstract: There are two well-known approaches in solving the measure theoretic problems in Diophantine approximation.  The metrical approach arise from the geometry of numbers and the ergodic theoretic approach arise from the dynamics on the space of lattices. One of the main ingredients in the geometry of numbers is the usage of Borel-Cantelli lemmas from probability theory. Dynamics on the space of lattices rely on the Dani correspondence principle (1985) which was extensively  developed further by Margulis and Kleinbock.  I will discuss both of these approaches and along the way discuss some well-known results such as the resolutions of Oppenheim (1929), Mahler (1932) and  Sprindzuk (1965) conjectures which influenced my research in the last few years.