Mathematics

Partitions associated with three third order mock theta functions

Atul Dixit   [email] (Tulane University)

Abstract: The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function $\phi(q)$. Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal theorem are also obtained. This is joint work with George E. Andrews and Ae Ja Yee.

On the Jones subgroup of Thompson's group $F$

Mark Sapir (Vanderbilt University)

Abstract: This is a joint work with Gili Golan. Recently Vaughan Jones showed that Thompson's group $F$ encodes in a natural way all knots and links in $\mathbb R^3$, and a certain subgroup $\overrightarrow F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\overrightarrow F$. In particular we prove that the subgroup $\overrightarrow F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i,i\in \mathbb N$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\overrightarrow F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\overrightarrow F$ is irreducible. Finally we show how to replace $3$ in the above results by an arbitrary $n$, and to construct a series of irreducible representations of $F$ defined in a similar way.

A modular transformation involving series of Hurwitz zeta function

Atul Dixit (Tulane University)

Abstract: In 2011, I proved that an integral identity involving the Riemann $\Xi$-function in one of Ramanujan's published papers leads to an elegant modular transformation between two infinite series of Hurwitz zeta function. Its special case $z=0$ occurs in Ramanujan's Lost Notebook. I also obtained another proof of this transformation using Mellin transforms. Both these proofs crucially use the invariance of the integral involving the Riemann $\Xi$-function under $\alpha\to 1/\alpha$. Recently, a need arose to give a proof of this transformation independent of the invariance property of this integral. We discuss this proof here. We show why Poisson's summation formula is inapplicable in this particular case. This proof requires an interesting self-reciprocal property of a generalization of function studied by N. S. Koshliakov that we obtained along the way. This is a part of joint work with Nicolas Robles, Arindam Roy and Alexandru Zaharescu.

Complex symmetric operators: a quick overview

Stephan Garcia (Pomona College)

Abstract: Complex symmetric operators are a surprisingly large class of (typically) non-normal operators that arise frequently at the intersection of complex analysis, operator theory, and matrix analysis. We highlight a number of examples and several key results. This survey talk will be accessible to graduate students.

D-modules

Yi Zhang (UIUC Math)

Abstract: We introduce the basics of D-modules, including filtration and dimension, and discuss the connection between D-modules and local cohomology modules.

Supercharacters and their superpowers: the graphic nature of exponential sums

Stephan Garcia (Pomona College)

Abstract: The theory of supercharacters was recently developed by P. Diaconis and I.M. Isaacs (based upon earlier work of C. André) to study previously intractable problems in combinatorial representation theory. When this machinery is applied to abelian groups, a wide variety of applications emerge. We develop a "super" version of the discrete Fourier transform and some combinatorial tools. This perspective illuminates several classes of exponential sums that are of interest in number theory while also producing complex-valued functions that display striking patterns of great complexity and subtlety. This talk will be accessible to students and there will be lots of attractive visuals. (Partially supported by NSF Grant DMS-1265973 and the Fletcher Jones Foundation)