Date: Feb 6th
Speaker: Colette LaPointe
Title: Some dynatomic modular curves in positive characteristic
Abstract: In arithmetic dynamics, the smoothness and irreducibility of the dynatomic modular curves $Y_1(n)$ and $Y_0(n)$ have frequently been studied for the polynomial family $f_c(x)=x^d+c$ in both char 0 and positive char $p$, but less is known about the dynamical behavior of other families, in particular in char $p$. So I am studying the smoothness and irreducibility of $Y_1(n)$ and $Y_0(n)$ for various families of the form $f_c(x)=x^{p^k}+bx+c$, $b\neq 0$, in char $p$. For example, it has been shown that in char $p$, the curve $Y_1(n)$ for the family $f_c(x)=x^p+x+c$ is smooth if and only if $p\nmid n$, except in the case $n=p=2$ where $Y_1(n)$ is smooth. It also appears -- though it hasn't been fully shown -- that $Y_1(n)$ is only irreducible in a handful of cases of $n$ and $p$. More generally, I am interested in discovering whether maps in the family $f_c(x)=x^{p^k}+bx+c$, $b\neq 0$, share some dynamical properties attributed to them all having a unique critical point at infinity.
Date: Mar 6th
Speaker: Owen Sweeney
Title: On the second case of Fermat's Last Theorem for cyclotomic fields.
Abstract: A proof of Fermat's Last Theorem (FLT) over the rationals was completed by Wiles in the 90s. The first substantial progress made in the study of FLT was made by Kummer in the mid nineteenth century when he proved FLT for regular primes p, not just over Q, but over the p-th cyclotomic field. Since not all primes are regular, FLT for all primes p and the associated cyclotomic field is still an open problem. The index of irregularity i(p) of a prime p is a natural measure of a prime's failure to be regular. Eichler/Kolyvagin proved that if i(p)<p^(1/2)-2, then the first case of FLT for exponent p holds. There are no known counterexamples to this criteria. Combining this with heuristic estimates on i(p) provides nice evidence that at least the first case of FLT in the cyclotomic field always holds. In this talk, I will discuss my work on an analogous criteria for the second case.
Date: April 17th
TIME CHANGE: 5 - 6 p.m.
LOCATION CHANGE: C197
Speaker: James Myer
Title: Esquisses des quelques programmes
Abstract: I’ll sketch two programs regarding progress toward the problem of resolution of singularities.
One perspective is that of a frog (in the words of Freeman Dyson ~ this is not meant with any offense: I happen to adore frogs): given a hyperelliptic curve over a (insert pleasant adjectives here) field whose ring of integers is of mixed characteristic (0, 2), produce a regular model. This project is advised by Andrew Obus.
Another, more akin to a bird: establish topological obstructions to the existence of a resolution of the singularities of a variety (over a field of any characteristic, e.g. positive characteristic). In fact, the existence of alterations suggests there are no "étale obstructions”. This project is advised by Dennis Sullivan.
Date: May 1st
Speaker: Ajith Nair
Title: Higher Composition Laws for Quadratic and Cubic Fields
Abstract: In 2001, Manjul Bhargava found a remarkable generalization of Gauss composition of binary quadratic forms which made it possible to define several new composition laws on spaces of forms of higher degree. These higher composition laws share the property with Gauss's composition of containing information about class groups of number fields.
Starting with Gauss composition, I will describe the composition laws related to the quadratic and cubic cases.