## The 290-Theorem and Representing Numbers by Quadratic Forms

- Date: 10/11/2006

Jonathan Hanke (Duke University)

University of British Columbia

This talk will describe several finiteness theorems for quadratic

forms, and progress on the question: "Which positive definite

integer-valued quadratic forms represent all positive integers?". The

answer to this question depends on settling the related question "Which

integers are represented by a given quadratic form?" for finitely many

forms. The answer to this question can involve both arithmetic and

analytic techniques, though only recently has the analytic approach

become practical. We will describe the theory of quadratic forms as it

relates to answering these questions, its connections with the theory

of modular forms, and give an idea of how one can obtain explicit

bounds to describe which numbers are represented by a given quadratic

form.

Algebraic Geometry Seminar 2006