Summer 2002 Graduate Student Seminar

The seminar meets Wednesdays at 3:30 in room 302 of the Boyd Graduate Studies Research Center, preceded by refreshments in room 409.

June 12 - Xander Faber, Probability theory can almost certainly provide tools for number theorists

Given an integer n, one may want to know how many prime factors it has without actually factorizing it. Using some ideas from probability theory, we can assert in an elementary way that "almost all" positive integers n have "approximately" log log n prime factors. We will make all of these concepts rigorous and prove this result originally due to Hardy and Ramanujan.

June 19 - Tom Tucker, Some simple questions about Diophantine equations

This talk will deal with various questions about integer and rational solutions to polynomial equations in two variables with integer coefficients. The Mordell conjecture, proved by Faltings in 1984, implies that a polynomial equation in two variables of degree at least 4 has finitely many rational solutions, unless the curve defined by the equation is singular. There are still a great many unanswered questions about these equations, however. Two of the most important are the following. Is there an algorithm for finding all of the solutions to these equations? Is there a bound on the number of solutions to these equations depending only on the degree? We will discuss recent results concerning (but not answering) these questions and explain how these questions connect with other open problems in number theory.

June 26 - Rene-Michel Shumbusho, Good reduction of elliptic curves

In this talk we will define elliptic curves over a field and what it means for an elliptic curve to have good reduction at a prime p of the field of rational numbers, or more generally, at a prime ideal P of a number field. Then we will consider the problem of finding all elliptic curves over Q, or more generally, over a number field, that have good reduction outside a given finite set of primes. We will see how this can lead to a problem of solving some diophantine equations.

July 3 - Stephen Donnelly, What's so great about doing number theory?

Take the set of all parabolas that are concave down, have integer coefficients and discriminant 5. What function do you get when you add the positive parts of the graphs of all those parabolas? This problem has all the elements that make number theory a delightful subject: it is simple to state, the answer is a beautiful suprise, and ultimately there is a completely satisfying explanation that leads one into deeper waters.

July 10 - Eric Pine, Amateur Vigre (before the $$$)

I'll give a brief overview of what is sometimes referred to as the 0th vigre project. The summer before receiving the VIGRE grant, four grad students worked with several professors on a summer project. We studied the integers which can be represented as the sum of three cubes. For example 6 = 2³ + (-1)³ + (-1)³. I'll describe previous work on the problem as well as our approach which uses nothing more advanced than modular arithmetic and multi-variable calculus yet provided new results.

July 17 - Janice Winner, The dth Symmetric Product of a Curve of Degree d

Given a curve of degree d, we can construct C^d as the direct product of C d-times, C x ... x C. Then we can consider this curve modulo the symmetric group on d elements. We will discuss this space and a natural stratification we can put on this space.

July 24 - Peter Petrov, Formal groups and elliptic curves

Formal groups are a useful tool in arithmetic and algebraic geometry, and in algebraic number theory. We will introduce them briefly, and then we'll attach to each elliptic curve a formal group law, and we'll show how it can be used over fields of positive characteristic. The needed facts about elliptic curves will be reviewed briefly.

July 31 - Tanya Cofer, A combinatorial approach to classifying tight contact structures on handlebodies

I will define and discuss tight and overtwisted contact structures and introduce some techniques and major results in contact topology. I will end with a description of a classification algorithm in the case of the solid torus.