My Mathematical Work

My mathematical work spans integrable systems theory, nonlinear PDEs, random matrix theory and their interrelationship.

With regard to integrable systems theory, my work with Professor Kenneth McLaughlin involves the semi-classical analysis of the focusing nonlinear Schrödinger (NLS) equation. A good portion of the mathematics involved falls into what is generally described as "integrable systems theory." It includes a careful analysis of the direct spectral problem (the Zakharov-Shabat ode system), as well as attacking the inverse spectral problem via Riemann-Hilbert, and d-bar techniques.

The NLS equation describes, among other things, solitonic transmission in fiber optic communication, and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures. The initial value problem for the NLS equation is solvable by the method of inverse scattering. The initial spectral data of the Zakharov Shabhat (ZS) operator, a particular linear operator having the solution to NLS as the potential, are calculated from the initial data of the NLS; they evolve in a simple way as a result of the integrability of the problem, and produce the solution to NLS through the inverse spectral transformation. A good portion of the mathematics involved falls into what is generally described as "integrable systems theory." It includes a careful analysis of the direct spectral problem (the Zakharov-Shabat ode system), as well as attacking the inverse spectral problem, via Riemann-Hilbert and d-bar techniques.

My Ph.D. thesis was written in the area of random matrix theory (RMT), under the supervision of Professor Craig A. Tracy. RMT is a vibrant area of research at the intersection of probability theory and statistics, operator theory, combinatorics, number theory, integrable systems, quantum chaos to name only a few of a growing list of related areas within mathematics. The main object of study in RMT is the statistical behavior of eigenvalues in certain probability spaces of matrices. Asymptotic analysis via Riemann-Hilbert problems provides a powerful approach to important questions in RMT, and is an example of the growing and productive interaction between integrable systems and RMT. I am interested in exploring more applications of Riemann-Hilbert techniques, as well as other integrable systems theory techniques, in RMT.

The past decade has seen exciting developments in the area, and the realization of many deep connections to other branches of mathematics, as well as outside mathematics. One of the milestones has been the discovery of explicit analytic expressions for the distribution of the largest eigenvalue in the Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic ensembles (denoted GOE, GUE, and GSE respectively) by Tracy and Widom ([4], [5]). It is natural to ask about similar results for the m^th largest eigenvalue in general. This is relevant for theoretical reasons as well as for many applications (e. g. multivariate statistics). Tracy and Widom answered this question in the GUE case in [4]. They provide a recursive formula which involves a Fredholm determinant D₂(s, λ) whose m^th derivative with respect to λ is evaluated at λ = 1. I solve the problem in the GOE and GSE cases in [1], which has been submitted for publication in the International Mathematics Research Notices. My work generalizes the GOE and GSE Tracy-Widom distributions. Together with the work of Johnstone and Soshnikov (see [2], [3]), the formulas in [1] also give the asymptotic behavior of the m^th largest eigenvalue of the appropriate Wishart distribution, which is of interest in multivariate statistics. A corollary in [1] also provides an alternate, purely random matrix theoretical proof of an intriguing interlacing property between GOE and GSE eigenvalues in the edge-scaling limit.

[1] M. Dieng. Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations. [pdf]. ArXiv:math. PR/ 0411421 (accepted for publication in the International Mathematics Research Notices).
[2] I. M. Johnstone. On the distribution of the largest eigenvalue in principal component analysis. Ann. Stats., 29( 2): 295-327, 2001.
[3] A. Soshnikov. A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys., 108(5-6): 1033-1056, 2002.
[4] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Commun. Math. Physics, 159: 151-174, 1994.
[5] C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Commun. Math. Physics, 177: 727-754, 1996.