Research

I am primarily interested in number theory, in particular in p-adic L-functions, Bloch-Kato conjecture and automorphic forms. My secondary research interests are Data Science, Geometry and Math Education.

Number Theory:

Yoshida lifts and the Bloch-Kato conjecture for the convolution L-function (Journal of Number Theory, 133 (2013), 2496-2537,joint with Kris Klosin)

Abstract: Let f1 (resp. f2) denote two (elliptic) newforms of prime level N, trivial character and weight 2 (resp. k + 2, where k=8 or 12). We provide evidence for the Bloch-Kato conjecture for the motive M = rho_f1x rho_f2(-k/2-1) by proving that under some assumptions the p-valuation of the order of the Bloch-Kato Selmer group of M is bounded from below by the p-valuation of the relevant L-value (a special value of the convolution L-function of f1 and f2). We achieve this by constructing congruences between the Yoshida lift Y(f1xf2) of f1 and f2 and Siegel modular forms whose p-adic Galois representations are irreducible. Our result is conditional upon a conjectural formula for the Petersson norm of Y(f1x f2).

On the Bloch-Kato conjecture for elliptic modular forms of square free level (Mathematische Zeitschrift 276 (2014), no 3-4, 889-924,joint with Jim Brown)

Abstract: Let k \ge 6 be an even integer, M an odd square-free integer, and f in S_{2k-2}(Gamma_0(M)) be a newform. We prove that under some reasonable assumptions that half of the \lambda-part of the Bloch-Kato conjecture for the near central critical value L(k,f) is true. We do this by bounding the l-valuation of the order of the appropriate Bloch-Kato Selmer group below by the l-valuation of algebraic part of L(k,f). We prove this by constructing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel modular form

Saito-Kurokarwa lifts of square free level (Kyoto J. Math. 55(3): 641-662, joint with Jim Brown)

In this paper we survey the construction of the Saito-Kurokawa lifting from the classical point of view. We also provide some arithmetic results on the Fourier coefficients of Saito-Kurokawa liftings. We then calculate the norm of the Saito-Kurokawa lift in the process correct some results in the literature.

p-adic L-function for GSp(4)xGL(2)(preprint)

In this paper we construct a p-adic analog of a degree eight L-function L(s, Fxf) where F is an ordinary holomorphic degree two Siegel eigencusp form of level a power of p and f is an ordinary eigen cuspform of level a power of p. Our method makes use of the work of M. Furusawa which gives an integral representation for this L-function. By suitably interpreting this integral representation in the context of inner products of automorphic forms, we show that it p-adically interpolates the L-values as the forms F and f vary in ordinary families (with the weights varying p-adically). This interpolation is carried out by exploiting a pull-back formula of P. Garrett and G. Shimura.

Geometry

Inheritance relations of hexagons and ellipses (The College Mathematics Journal Vol. 47, No. 3 (May 2016), pp. 208-214, joint with N Natarajan)

We show that a hexagon can be circumscribed by an ellipse if and only if the child hexagon (diagonal hexagon) has an ellipse inscribed inside it. This parent-child relationship is symmetric in that a hexagon has an ellipse inscribed inside it if and only if the child can be circumscribed by an ellipse. In short, Circumscribed begets inscribed and inscribed begets circumscribed. This shows that inscribing and circumscribing are recessive traits, in the sense that it can skip a generation but certainly resurfaces in the next.

Duality and Inscribed Ellipses (Computational Methods and Function Theory volume 15, pages 635-644(2015), joint with John Clifford and Michael Lachance)

We give a constructive proof for the existence of inscribed family of ellipses in convex n-gons for 3 \le n \le 5 using duality. In the case of a pentagon, we also exhibit the simultaneous existence of two ellipses, one inscribed in the pentagon and the other inscribed in its diagonal pentagon. The two ellipses are as intrinsically linked as are the pentagon and its diagonal pentagon. Our method uses the theory of dual curves.

Elliptic curves do arise from ellipses (under revision, joint with N. Natarajan)

We show that the loci of the foci of a family of ellipses that are inscribed in a triangle with one point of prescribed tangency is an elliptic curve. We also get some nice geometric properties as a consequence.

Data Science

Hybrid bag of approaches to characterize selection criteria for cohort identification(Journal of the American Medical Informatics Association, Volume 26, Issue 11, November 2019, Pages 1172-1180)

The 2018 National NLP Clinical Challenge (2018 n2c2) focused on the task of cohort selection for clinical trials, where participating systems were tasked with analyzing longitudinal patient records to determine if the patients met or did not meet any of the 13 selection criteria. This article describes our participation in this shared task.

Curriculum Guidelines for Undergraduate Programs in Data Science(Annu. Rev. Stat. Appl. 2017. 4:2.1-2.16, joint with Veaux R., et. al.)

The Park City Math Institute 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in data science. The group consisted of 25 undergraduate faculty from a variety of institutions in the United States, primarily from the disciplines of mathematics, statistics, and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in data science.

Books

Data Science for Mathematicians (Chapman and Hall/CRC Press,2020, (Chapter-Machine Learning), edited by Nathan Carter)

Mathematicians have skills that, if deepened in the right ways, would enable them to use data to answer questions important to them and others, and report those answers in compelling ways. Data science combines parts of mathematics, statistics, computer science. Gaining such power and the ability to teach has reinvigorated the careers of mathematicians. This handbook will assist mathematicians to better understand the opportunities presented by data science. As it applies to the curriculum, research, and career opportunities, data science is a fast-growing field. Contributors from both academics and industry present their views on these opportunities and how to advantage them.

Papers in preparation:

Analogue of Hida's projection operators for GSp(4)(joint with Kris Klosin)

2-variable p-adic L-function for GSp(4)xGL(2)