*** HYBRID: Speaker will be on zoom. Join by zoom or join the watch party in Math 220 ***

https://cuboulder.zoom.us/j/91928134679

The passcode is the j-invariant of the elliptic curve with CM by the Gaussians (or email Kate).

Abstract:

In the last three and a half decades, starting with the pioneering solution of the Oppenheim conjecture on the values of quadratic forms by Margulis, the interplay between the ergodic theory of homogeneous spaces and number theory solidified. Moreover, exciting new connections between the fields keep arising even today. The aim of my talk will be to demonstrate this with a circle of problems which might be best described as problems in the "geometry of integral vectors".

To each integral vector v in Z^d one can associate several natural geometric and arithmetic objects. For example: (1) The "direction" of v, namely its radial projection to the unit sphere. (2) The residue class of v modulo a fixed integer k. (3) The "orthogonal lattice" to v, defined by intersecting Z^d with the hyperplane orthogonal to v. Each of these objects resides in a natural probability space which allows one to ask statistical questions as the integral vector varies.

I will survey classical and recent results describing the limit statistics of these objects as the integral vector varies in certain sets. For example: (1) A ball of radius T as T goes to infinity (Schmidt). (2) A sphere of radius T as T goes to infinity (Linnik/Duke/Aka - Einsiedler - S) (3) The integral points on certain non-compact quadratic surfaces (S - Sargent). (4) The integral points approximating an irrational line in R^d (S - Weiss).

Focussing on one of the above examples, I will try to explain how the corresponding problem can be casted as a question in homogeneous dynamics.