Algebra, Geometry, and Number Theory Seminar

Abstracts

Zev Klagsbrun - Searching for elliptic curves of high rank

I will describe my recent work with Noam Elkies that led to our discovery of an elliptic curve E/Q having Mordell-Weil rank 29, which set a new record for the rank of an elliptic curve over Q.

Pankaj Singh - Classification of forms of tori using separable algebras

Algebraic tori are widely studied due to their connection with Galois cohomology and other algebraic invariants. The connection with Galois cohomology gives us a bijection between Severi-Brauer varieties and central simple algebras. Blunk has given a classification of the forms of two dimensional tori in del Pezzo surfaces of degree 6 in terms of separable algebras over an arbitrary field. Duncan has proved that there exists an injective map from the forms of retract rational tori to isomorphism classes of separable algebras but there is no explicit description of the image of this map as for Blunk. In this talk, we discuss how to use the structure of Mackey functors to address this question.

Rena Chu - Short character sums evaluated at homogeneous polynomials

Let $p$ be a prime. Bounding short Dirichlet character sums is a classical problem in analytic number theory, and the celebrated work of Burgess provides nontrivial bounds for sums as short as $p^{1/4+\varepsilon}$ for all $\varepsilon>0$. In this talk, we will first survey known bounds in the original and generalized settings. Then we discuss the so-called ``Burgess method'' and present new results that rely on bounds on the multiplicative energy of certain sets in products of finite fields. Grad student pretalk to precede at 2:30.

Alex Kalogirou - On disjoint covering systems

It was established fairly early in the history of covering systems that no disjoint coverings of the integers exist when all the moduli are required to be distinct. In recent work with Michael Filaseta we prove a related old conjecture of Erdos and Selfridge that claims that that the sum of the reciprocals of the moduli in a distinct covering system, is bounded away from 1, if the minimum modulus is at least 5. We conclude with further motivation of an open question pertaining to infinite covering systems. 

Spring Break - Spring Break

Last semester's seminar.