Department of Mathematics
University of Toronto
Thirteenth Annual
R.A. Blyth Lectures in Mathematics
(Information on the Blyth
Lectures is available on the index page)
Professor Manjul Bhargava
Princeton University
Will give three lectures on
Sums of Squares, and Generalizations
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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.
