Speaker: William Duke, UCLA<\/p>\n
Date: Tuesday, April 14, Location: SC1404, Time: 10:00-11:00<\/p>\n
Title: New directions in the geometry of numbers<\/p>\n
Abstract:\u00a0 The geometry of numbers is a relatively old and highly developed area of number theory.
\nOne 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\u00a0 estimate is given
\nin terms of $\\mathrm{SL}(n,\\R)$ invariants of the polynomial.
\nA major refinement is to understand the {\\it spectrum}, the\u00a0 set of minima of all of those polynomials in the $\\mathrm{SL}(n,\\R)$-orbit of $P$.<\/p>\n
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.
\nA different kind of generalization\u00a0 occurs\u00a0 when we replace the lattice
\n$\\Z^n$\u00a0 by the set of\u00a0 integer points on an affine homogeneous variety and seek to minimize the absolute value of a polynomial\u00a0 restricted to these integers.<\/p>\n
In one of the simplest interesting cases the integer points are those on an (affine) quadric surface.
\nIn this talk, after going over the background, I will describe\u00a0 analogues of some well-known results about the minima of\u00a0 \u00a0quadratic and\u00a0 cubic\u00a0 forms and their spectra, when we restrict them to the integer points on this quadric surface.<\/p>\n","protected":false},"excerpt":{"rendered":"
Speaker: William Duke, UCLA Date: Tuesday, April 14, Location: SC1404, Time: 10:00-11:00 Title: New directions in the geometry of numbers Abstract:\u00a0 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 $\\R^n$…<\/p>\n","protected":false},"author":219,"featured_media":749,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[11],"tags":[],"acf":[],"jetpack_featured_media_url":"https:\/\/as.vanderbilt.edu\/math\/wp-content\/uploads\/sites\/69\/2025\/03\/number-theory.png","_links":{"self":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/1821"}],"collection":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/users\/219"}],"replies":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/comments?post=1821"}],"version-history":[{"count":5,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/1821\/revisions"}],"predecessor-version":[{"id":1827,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/1821\/revisions\/1827"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/media\/749"}],"wp:attachment":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/media?parent=1821"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/categories?post=1821"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/tags?post=1821"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}