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Brown University
Mathematics Department
2012 Distinguished Lecture Series


Manjul Bhargava, Princeton University
Monday, March 5, 2012
Wednesday, March 7, 2012
(M) 4:15 PM
(W) 3:00 PM
(W) 4:30 PM
{Receptions will be held at 3:30 PM Monday and
4:00 PM Wednesday in the MacMillan lobby}
(M-4:15) MacMillan 115
(W-3:00) MacMillan 115
(W-4:30) MacMillan 115
Lecture I:
From Gauss composition to prehomogeneous spaces
In 1801, Gauss presented a law of composition on the space of "binary quadratic forms", i.e., homogeneous quadratic polynomials in two variables. This law of composition has played a very important role in the development of algebraic number theory and group theory.

In this lecture, we present several higher analogues of Gauss composition on more general spaces of forms, namely, on spaces of forms that are "prehomogeneous" (i.e., have only one polynomial invariant). We examine the connection between these prehomogeneous spaces of forms and some important objects in number theory, such as number fields and their class groups. We also describe some applications to counting problems.
Lecture II:
Coregular spaces and algebraic curves
We next turn to representations of algebraic groups that are "coregular" (i.e., have a free ring of invariants) but are not "prehomogeneous". It turns out that such spaces parametrize algebraic curves rather than number fields and their class groups!

This also leads to a number of applications, e.g., to counting problems involving the arithmetic of elliptic curves. (This is joint work with Wei Ho.)
Colloqium (Lecture III):
The average rank of elliptic curves
A rational elliptic curve may be viewed as the set of solutions to an equation of the form y2 = x3+Ax+B, where A and B are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The rank of a rational elliptic curve measures how many rational points are needed to generate all the rational points on the curve. There is a standard conjecture, originating in work of Goldfeld, that states that the average rank of all elliptic curves should be 1/2; however, it has not previously been known that the average rank is even finite! In this lecture, we describe recent work that shows that the average rank is finite; in fact, the average rank is less than 1. (This is joint work with Arul Shankar.)

Previous events in the Distinguished Lecture Series