# Research

My primary research interest is Mathematical General Relativity (and Partial Differential Equations) and in particular long time behavior of Einstein's equations. Mathematical relativity is fascinating primarily because it includes both interesting mathematics  (such as analysis of partial differential equations, differential geometry etc.) as well as physics. This is an exciting time for general relativity due to the recent detection of gravitational waves and black holes. There are a large number of mathematically interesting problems that one might think of formulating. I am interested in problems that are physically motivating but also mathematically attractive. I am also interested in quantum field theory, fluid dynamics, gravity-fluid duality, Teichmuller theory.

Stability and Convergence issues in Mathematical Cosmology: Out of pure curiosity, an undergrad studying GR for the first time may simply ask: "Is our physical universe correctly predicted by Einstein's general relativity? To conclude that it is would seem to hinge on the proof that the current theoretical model (so-called an FLRW model) of the universe is dynamically stable. In other words, one would want to perturb these model solutions by a small amount (in a suitable sense) and study the temporal behavior (in particular asymptotic properties) of these disturbances. At the level of PDE, this translates to studying a 'small data' initial value problem. What makes it fascinating is that Einstein's equations are examples of non-linear partial differential equations (quasi-linear wave equation to be precise). The presence of the non-linearity may potentially focus gravity to form singularities. Therefore, proving stability amounts to showing that the opposing forces (such as dissipation caused by expansion or purely geometrical dispersion: wave amplitude decays with distance) suppress the non-linearity for sufficiently small initial data.

In another situation, an undergrad studying cosmology for the first time may ask "is the observed large scale approximate' homogeneity and isotropy of the physical universe an accident or a product of GR?". This is a perfectly fair question since, in textbook cosmology (or elsewhere), the description of the universe starts with the Cosmological Principle.  Answering such questions would require a fully rigorous analysis of Einstein's field equations (possibly including matter). Indeed it is remarkable that Einstein's GR has an in-built dynamical mechanism that drives the physical universe to an asymptotic state that is volume dominated by homogeneous and isotropic components. Of course, a definite conclusion would require 'large data' long-term existence of solutions i.e., the solutions should not blow up in finite time obstructing the time evolution. This essentially leads to the following long-standing problem of classical GR.

Global Existence Issue: In addition to the small data problems mentioned previously, there is also the so-called large data' problem where one essentially removes the restriction on the size (suitable function space norm) of the initial data. This could be formulated as an initial value problem: can the fields evolving from regular initial data extend uniquely and continuously to a globally defined singularity-free solution to Einstein's equations? This problem is extremely difficult if one does not assume some special characteristics of the solutions (e.g., symmetry). Once again, the problem is the formation of singularity (possibly naked) via concentration of energy by nonlinearity. However, in this case, the problem is much harder since one removes the smallness of the data. These problems are closely related to Penrose's Cosmic Censorship Conjecture that roughly states the validity of classical determinism i.e., given present we can uniquely determine the future. Together with Prof. Moncrief, I am studying the global existence issues associated with certain special spacetimes (but the `special' characteristics is less than the previous studies: the so-called U(1) problem in technical terms) .

Study of mass in GR: It is well known that the equivalence principle does not allow for a local definition of mass in general relativity. There have been several definitions of mass in a quasi-local sense i.e., the mass of a 2 surface embedded in the spacetime. The most promising definition that satisfies several good properties of a 'mass' is the one recently defined by Prof. Yau and Wang. I am interested in understanding the properties of Wang-Yau quasi-local mass for different spacetimes and in particular, black hole spacetimes. This is important since, if one wants to study the dynamics of a 2 body problem (e.g., the collision of two black holes, etc.) in the general relativistic settings, one must understand the mass. In addition, this mass recovers a certain component of the Bel-Robinson tensor (which encodes the pure gravitational degrees of freedom) at the small sphere limit i.e., when one evolves the topological 2-sphere along its incoming null direction and approaches the vertex of the associated null cone. Of course, for compact manifolds (negative Yamabe for a technical reason) foliating the spacetime, a suitably defined re-scaled volume functional may be an equivalent for the mass (one may extend this definition even to a quasi-local level) which do exhibit good monotonic property in a suitable choice of gauge. These additional inputs from the quasi-local mass may be used in the global existence problem where one is ultimately interested in understanding the focusing of gravitational energy (curvature). These problems are under intense investigation with Prof. Yau.

Study of Hyperbolic PDE: Since Einstein's equations when written in suitable gauge (e.g. spacetime or spatial harmonic) take the form of a quasi-linear hyperbolic PDE.  One would expect a hyperbolic characteristic of general relativity due to the finite speed of propagation or more roughly the presence of the light cones. Since the nonlinearities involved are rich, one needs to utilize novel techniques to study them. This led to a dramatic development in the field of general quasi-linear hyperbolic PDE and so many researchers took part in it in the last 30 years. This is a progress in mathematics motivated by physics. I am particularly interested in studying analog hyperbolic equations that can be handled by methods of GR. Here is an example. In vacuum GR, spacetime curvature satisfies a semi-linear wave equation as a consequence of Einstein's field equations and Bianchi identities. This equation essentially describes curvature propagation and is physical since spacetime curvature is a true geometric entity. One may write down an expression for the curvature in terms of integrals over the past light cone from an arbitrary spacetime point to an initial, Cauchy hypersurface and additional integrals over the intersection of this cone with the initial hypersurface. One may write down such useful integral formula for any field which satisfies a semilinear wave equation. Recently, I have proved the global existence of the critically non-linear wavefields propagating on a curved spacetime utilizing this integral equation supplemented by energy estimates. This new technique is motivated by the original work of my adviser Prof. Moncrief. We call it the light cone estimate technique and there is a wide variety of hyperbolic systems where this technique is applicable.

Teichmuller Theory and GR: Classical Teichmuller theory is primarily studied from algebraic topologic and complex analytic perspectives and therefore it does not have a straightforward connection to physics. But, when it is formulated in the Riemannian geometric framework, it naturally plays the role of the configuration space of gauge fixed 2+1 dimensional vacuum gravity. Several classical results of Teichmuller theory are easily derivable using the 'physical' picture of GR such as the space of initial singularities (big-bang) of vacuum spacetimes foliated by closed higher genus Riemann surfaces (number of genus is strictly greater than 1) is exactly the Thurston boundary. I am hopeful that these results are the 'tip' of an iceberg in a manner of speaking and much remains to be explored. This is currently under intense investigation with Prof. Yau.