Number Theory Seminar: April 14, 2026
Speaker: William Duke, UCLA
Date: Tuesday, April 14, Location: SC1404, Time: 10:00-11:00
Title: New directions in the geometry of numbers
Abstract: The geometry of numbers is a relatively old and highly developed area of number theory.
One basic problem here is to estimate the minimal absolute value of a real homogeneous polynomial $P$ on $\R^n$ when evaluated at non-zero points of $\Z^n$. Typically, such an estimate is given
in terms of $\mathrm{SL}(n,\R)$ invariants of the polynomial.
A major refinement is to understand the {\it spectrum}, the set of minima of all of those polynomials in the $\mathrm{SL}(n,\R)$-orbit of $P$.
In his recent lectures, Sarnak explains how these problems are part of a much more general program that studies the “bass note” spectra of certain linear differential operators.
A different kind of generalization occurs when we replace the lattice
$\Z^n$ by the set of integer points on an affine homogeneous variety and seek to minimize the absolute value of a polynomial restricted to these integers.
In one of the simplest interesting cases the integer points are those on an (affine) quadric surface.
In this talk, after going over the background, I will describe analogues of some well-known results about the minima of quadratic and cubic forms and their spectra, when we restrict them to the integer points on this quadric surface.