We define a class of L-functions that properly contains the Selberg class and which has a natural ring structure. We prove some properties of this ring, in particular that it is non Noetherian. This is joint work with Anup Dixit.

The central limit theorem in probability theory expanded its influence into number theory in the middle of the 20th century. This began with the celebrated Erdos-Kac theorem which generalized the classical theorem of Hardy and Ramanujan regarding the "normal" number of prime divisors of a random integer. Since then, probabilistic number theory has blossomed into various branches resulting in spectacular foliage including unexpected applications in algebra. In particular, one can combine the study of Artin L-series and probabilistic number theory to derive a central limit theorem for the normal number of prime factors of Fourier coefficients of modular forms. We will report on this research along with recent (and not so recent) results obtained in joint work with V. Kumar Murty, Arpita Kar and Neha Prabhu.

Verones codes are projective Reed-Muller codes of order 2 in P^2. They are obtained by looking at the Veronese map from P^2 to P^5. We already know the higher weight hierarchy of such codes, but in this talk, we will obtain higher weight spectra of these codes, that is, how many subcodes of given dimension and support there are. We will use the machinery of matroids, resolutions of Stanley-Reisner rings, and generalized weight polynomials to give our result.

I will talk about the convex hulls of trajectories of polynomial dynamical systems. Such trajectories also include real algebraic curves. The boundary of the resulting convex bodies are stratified into families of faces. I will discuss the numerical algorithms we developed for identifying these patches of faces. This work is also a step towards computing the attainable region of a trajectory. This is a joint work with Daniel Ciripoi, Andreas Loehne, and Bernd Sturmfels.

attached

In this study, our primary focus is on examining if using optional RRT models as opposed to non-optional RRT models for mean estimation of a sensitive variable when measurement errors are present, produces more efficient estimators. A unified measure of model efficiency and respondent privacy will be used in this comparison. The models discussed have data security implications as well.

(Abstract currently not available)

We discuss ongoing work where we relate Brill-Noether theory on a finite graph to homological invariants of certain modules associated to it. Our approach resembles Stanley's commutative algebraic approach to enumeration of magic squares.