Activities This Week

Diophantine approximation quantifies the density of the rational numbers in the real line. The extension of this theory to algebraic numbers raises many natural questions. I will focus on a dynamical resolution to Davenport’s problem and show that there are uncountably many badly approximable pairs on the parabola. The proof uses the Kleinbock–Margulis uniform estimate for nondivergence of nondegenerate curves in the space of lattices and a variant of Schmidt’s game. The same ideas applied to Ahlfors-regular measures show the existence of real numbers which are badly approximable by algebraic numbers. This talk is based on joint works with Victor Beresnevich and Lei Yang.

I will review some ALW-type results for the $j$ function and its derivative, following the paper Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

L-functions of motives over Spec Z encompass both L-functions associated to smooth projective varieties over Q and zeta functions of schemes over Spec Z. In this talk I will state a conjecture about special values of such L-functions. This conjecture shares a kinship with Poincaré duality or Artin-Verdier duality. It is equivalent to the conjunction of the conjectures of Beilinson, Soulé and Tate and allows to take advantage of the fact that motives over Spec Z form a triangulated category.