U. of Haifa Colloquium 4/5/21
|Speaker: Yakov Karasik (Technion)
Topic: Counting rational flags of $Z^n$
Place: This is an online seminar. Please email Yuval Ginosar “ginosar at math dot haifa dot ac dot il” for the Zoom ID and password.
Date: Tuesday, May 4, 2021
A subspace V of $ R^n$ is called rational (w.r.t. the lattice $Z^n$) if $ L=V \cap Z^n$ is a lattice of V which is, in turn, referred to as a primitive sub-lattice of $Z^n$ . In 68 Schmidt, using rather basic number-theoretical tools, was able to provide an asymptotic expression (with an error term) for the number of d dimensional primitive lattices of covolume no bigger than T . A much more refined question is to perform this kind of counting in a subset obtained by restricting different parameters associated to them. For instance, one considers the direction, the shape, the homothety class of L or of its “complement”. 30 years later(!), Schmidt was able to obtain partial results w.r.t. this refinement but not enough to deduce equidistribution of all of the aforementioned parameters.
In this talk, I will talk about a series of works together with T. Horesh where we use a dynamical approach to expand the results of Schmidt in two major ways. The first is more or less a full solution to the above question and the second one is a solution to an analogous question when we replace primitive sub-lattices with a more general object: rational flag.
The talk should be accessible to graduate students and no prior knowledge will be assumed.