# Research

Here are some exciting areas of mathematics that I research:

Nonlinear Waves

We are surrounded by waves: water waves, sound waves, electromagnetic waves, gravitational waves etc. In mathematics, the motion of a wave is described by a partial differential equation (PDE). Although linear PDEs are sufficient to understand some simple wave phenomena, most physical systems are inherently nonlinear in nature. Nonlinear equations are relevant in engineering, physics, and mathematics, and are used to understand the breaking of waves, for example. They are notoriously difficult to solve and give rise to interesting phenomena such as chaos. An example of a nonlinear wave equation that I have studied is the nonlinear Schrödinger equation, which is a model for wave propagation in water and in fiber optics:

$i\frac{\partial q}{\partial t} + \frac{\partial^2 q}{\partial x^2} - 2 \lambda |q|^2 q = 0, \qquad \lambda = \pm 1.$

Integrable Systems

Most nonlinear systems are too complicated to be well-understood even when employing the most modern methods and computers. However, there is a special class of systems, called integrable systems, which can be solved exactly. The mathematics of integrable systems is full of surprising results, intriguing connections, and beautiful formulas. I am particularly interested in understanding geometric properties of solutions and the solutions of problems with a boundary. Many phenomena in nature naturally involve a boundary. For example, in the study of tsunami waves, the wave propagates across the ocean before it reaches the shore line, which constitutes a boundary.

The Riemann Zeta Function

The Riemann zeta function, which is defined by

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s},$

has applications in physics, probability, and statistics, and plays a pivotal role in number theory. The famous Riemann hypothesis is a conjecture that states that all nontrivial zeros of the Riemann zeta function have real part 1/2. Many mathematicians consider proving this hypothesis the most important unsolved problem in pure mathematics – in fact, if you solve it, the Clay Mathematics Institute will give you \$1,000,000. Together with Prof. A. S. Fokas at the University of Cambridge, UK, I have derived various asymptotic formulas for the Riemann zeta function, which perhaps one day will prove useful in establishing this hypothesis.

Black Holes in Einstein’s Theory of Relativity

Einstein’s theory of relativity tells us how space and time are curved in the vicinity of massive objects such as stars and galaxies. A black hole is an object so massive and so dense that nothing, not even light, can escape from it. It is currently believed that supermassive black holes exist at the centers of most galaxies. Mathematically, the study of relativistic bodies is exceedingly complicated because it involves the study of free boundary value problems for the Einstein equations, which are ten coupled nonlinear PDEs in four dimensions. However, in certain cases when the motion possesses additional symmetries, these equations reduce to a single nonlinear PDE called the Ernst equation. I have solved a class of problems for this equation which involve a disk rotating around a black hole. In the absence of a disk, the new solutions reduce to the celebrated Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.