Veranstaltungen

https://www.math.uzh.ch/mat074

Since the work of Minkowski in the early twentieth century, the space of lattices has been a fundamental tool in the study of natural or rational numbers. Then, Margulis and his followers, in particular Dani, showed that methods from ergodic theory could be used very efficiently in that setting. More recently, Schmidt and Summerer started the "parametric geometry of numbers", which is a way to describe diagonal orbits in the space of lattices, using a simple combinatorial coding. The goal of this mini-course is to introduce the main concepts of parametric geometry of numbers, and to use them to study two problems going back to Jarník and Schmidt: - (Jarník) Given r>(n+1)/n, does there exist x in R^n such that the inequality |x-p/q| < q^(-r) has infinitely many solutions p/q in Q^n, but for all c<1, the inequality |x-p/q| < c q^(-r) has only finitely many solutions p/q in Q^n? (And what is the Hausdorff dimension of the set of such points?) - (Schmidt) Given an l-dimensional subspace x in R^d, for what values of r can one always find an l-dimensional rational subspace v in Q^d arbitrarily close to x and such that the distance to x satisfies d(v,x) < H(v)^(-r)? (Height and distance on Grassmann varieties will be defined in the first lecture -- note that this talk will last 1h30 and it will be introductory lecture for the ensuing mini-course that will take place on Tuesday 19th and Wednesday 20th of December. Please contact organizers for more info.)

We consider a hypoelliptic Kolmogorov-type PDE corresponding to a particle under white noise force. We are interested in imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain, and investigate a boundary-layer kernel. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of the heat kernels, and then construct a Volterra equation.This Volterra equation has a periodic structure resulting from the novel jump condition.