Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 31, 2016.

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Thursday, March 31, 2016

11:00 am in 241 Altgeld Hall,Thursday, March 31, 2016

Overpartition analogues of partitions associated with the Ramanujan/Watson mock theta function $\omega(q)$

Atul Dixit (Indian Institute of Technology Gandhinagar)

Abstract: Let $\omega(q)$ denote the third order mock theta function of Ramanujan and Watson. Recently George E. Andrews, Ae Ja Yee and I showed that $q\omega(q)$ is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. We also studied the associated smallest parts partition function $\mathrm{spt}_{\omega}(n)$ and proved some congruences for the same. Very recently, we considered the overpartition analogue of $p_{\omega}(n)$, namely, $\overline{p}_{\omega}(n)$. Finding an alternate representation for the generating function of $\overline{p}_{\omega}(n)$ turns out to be difficult in this case. We devise a new seven parameter $q$-series identity which generalizes a deep identity of Andrews (as well as its generalization by R. P. Agarwal), and then specialize it, along with the use of some identities in basic hypergeometric series, to arrive at an alternate representation in terms of a ${}_3\phi_{2}$ and an infinite series involving the little $q$-Jacobi polynomials. We also prove some congruences for $\overline{p}_{\omega}(n)$ and for the overpartition analogue of $\mathrm{spt}_{\omega}(n)$. This is joint work with George E. Andrews, Daniel P. Schultz and Ae Ja Yee.

3:00 pm in 243 Altgeld Hall,Thursday, March 31, 2016

Singularities of cluster algebras and related algebras

Jenna Rajchgot (U Michigan)

Abstract: Cluster algebras are a family of combinatorially-defined commutative rings which were introduced by Fomin and Zelevinsky at the turn of the century. Cluster algebras have since found connections to multiple areas of mathematics, and many important varieties have a cluster algebra structure. Examples include double Bruhat cells and Grassmannians. While these varieties are smooth, a general cluster algebra can be pathological. I'll provide an introduction to cluster algebras and certain related algebras (i.e. upper cluster algebras, lower bound cluster algebras), and then discuss some results on the singularities of these algebras using both positive characteristic commutative algebra, as well as Grobner degenerations plus Stanley-Reisner combinatorics. This talk will include joint work with Angelica Benito, Greg Muller, Karen Smith, and Bradley Zykoski.

4:00 pm in 245 Altgeld Hall,Thursday, March 31, 2016

Arithmetically exceptional Jacobians

Rachel Pries (Colorado State University)

Abstract: Geometry plays a pivotal role in understanding the arithmetic of curves in positive characteristic. In the first part of the talk, I will give an introduction to invariants of curves defined over finite fields, with the goal of understanding what it means for a curve to be arithmetically exceptional. Secondly, I will describe the boundary of the moduli space of curves and how its geometry has played a key role in many foundational results in arithmetic geometry and Galois theory. Finally, I will discuss joint work with Ekin Ozman, in which we use the boundary of the moduli space of curves to prove the existence of arithmetically exceptional Prym varieties and shed new light on the geometry of the theta divisor.