Ultrametric analysis in Diophantine approximation.

Abstract: Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A breakthrough by Kleinbock and Margulis in 1998 was to study Diophantine approximations for manifolds using homogeneous dynamics. Deep down in the result of this dynamical type lies the property of showing a class of function being `good' with respect to `nice' measures. In recent work with Victor Beresnevich and Anish Ghosh, we show that such good properties hold in ultrametric spaces like p-adics. As a result, we answer a conjecture by Kleinbock and Tomanov assertively that extends previous works of Kleinbock, Lindenstrauss, and Weiss. In this talk, I plan to give an overview of this area, leading to the results I mentioned.