Blyth 2007

Department of Mathematics
University of Toronto

Thirteenth Annual R.A. Blyth Lectures in Mathematics

 

Professor Manjul Bhargava

Princeton University

Will give three lectures on Sums of Squares, and Generalizations

Manjul Bhargava

Lecture 1: Sums of Squares and the 290-Theorem
Monday, September 24, 2007, 4:10 p.m.
Bahen Centre for Information Technology, 40 St. George St.
Room 1170

Abstract: The classical "Four Squares Theorem" of Lagrange asserts that any positive integer can be expressed as the sum of four squares; that is, the quadratic form a2+b2+c2+d2 represents all (positive) integers. When does a general quadratic form represent all integers? When does it represent all odd integers? When does it represent all primes? We show how all these questions turn out to have very simple and surprising answers. In particular, we will describe the recent work (joint with J. Hanke, Duke University) in proving Conway's "290-Conjecture" for universal quadratic forms.

Lecture 2: Gauss composition, and generalizations
Tuesday, September 25, 2007, 4:10 p.m.
Bahen Centre for Information Technology, 40 St. George St.
Room 1170

Abstract: A centuries-old identity about sums of two squares asserts that (a2+b2)(c2+d2)=(ac-bd)2+(ad+bc)2. This implies that the product of two numbers, each the sum of two squares, is again the sum of two squares. In 1801, Gauss presented a general theory of "composition" of such binary quadratic forms, turning the set of equivalences classes of such forms of a given discriminant into a finite abelian group. His discovery helped lay the groundwork for modern algebraic number theory.

The question arises as to whether other spaces of forms might have similar laws of "composition". In this lecture we describe several such higher analogues of Gauss composition, and discuss some of their recent applications in number theory.

Lecture 3: The parametrization of rings of low rank
Wednesday, September 26, 2007, 4:10 p.m.
Bahen Centre for Information Technology, 40 St. George St.
Room 1170

Abstract: A ring of rank n is a commutative ring with identity that is free of rank n as a Z-module. (In other words, a ring of rank n is Zn with a ring structure on it!) The prototypical example of a ring of rank n is, of course, an order in a degree n number field. How can one explicitly describe all rings of rank n for small values of n? The answer plays an important role in developing the composition laws of Lecture 2, and also in understanding the distribution of number fields.


The Blyth Lecture Reception will follow the first lecture and will be held in the McLennan Physical Lab, Student Lounge, Room 110, 255 Huron Street (Entrance on Russell Street). All are invited to attend.