In the broadest terms, my research focuses on the synthesis of geometry and analysis and through it derives new insights concerning the behavior of physical systems.
Geometry deals with shapes and distances, and has been studied since the time of Euclid and Apollonius; analysis deals with limits and the concept of the infinitesimally small, and originated with ideas of Antiphon. Since ancient times the two have had fruitful cross-pollinations.
Archimedes used Antiphon's method of exhaustion to solve the geometric problem of finding the area of a circular disk.
Newton's Principia, in which his differential and integral calculus---the forefather of modern analysis---were described, owes much to the Apollonian studies of conic sections.
The mathematics in the Principia, however, was set out in service of the Philosophiae Naturalis, and geometry, analysis, and physics have been wedded ever since.
What differentiates my field of research from that of many other geometric analysts, is that the geometry I consider is not merely the geometry of space as first described by Riemann. Riemannian geometry is another synthesis of geometry and analysis, this time considering the notion of shape and distance on spaces which, only in the infinitesimally small limit, look like the spaces familiar to Euclid. The emphasis on spatial geometry, however, means that when it comes to applications to the natural sciences, Riemannian geometric methods can only illuminate systems in stasis. To understand dynamics, one has to incorporate time into the theory, leading us to the geometry of Minkowski and Lorentz, which underlies most famously, Einstein's special and general theories of relativity.
The introduction of dynamics significantly complicates the theory. In the static picture described by Riemannian geometry, equilibria can often be described as the optimum solution minimizing a cost function, and inherits strong rigidity properties aiding their study. In the dynamical setting, solutions are fluid and a main goal is understanding the effects due to the propagation of disturbances which are initially small. Of particular interest is whether one should expect a "butterfly effect" where initial disturbances can grow and lead to dynamical solutions diverging from either a equilibrium or a reference trajectory, or a "restorative effect" where small disturbance eventually die down. This question is the stability problem for evolutionary geometric partial differential equations and is crucial to our understanding of long-term behavior of physical systems; it also is my main field of research.
The relativistic membrane equation describes the world sheet of an extended test body following the theory of relativity. These world sheets are described as evolving strings or surfaces in space, subject to the constraint that their mean curvature vanishes. This idea of focusing on the mean curvature derives from two places:
Mathematically the equations of motion can be described by quasilinear wave equations. These equations are highly special in satisfying a totally non-resonant condition. (This is a unique feature of the membrane equation among nonlinear scalar Lagrangian field theories.) Intuitively, this non-resonant condition should prevent certain types of singularities from occurring, and this would have significant consequences toward the existence, regularity, and stability of solutions to the membrane equations.
My interest in the membrane equation mainly rests on understanding how this totally non-resonant condition manifests itself analytically and geometrically, and what impact this has on the qualitative behavior of solutions to the equation.
For additional overview of the subject, please see:
General relativity posits that gravity is described by the curvature of space-time. This curvature affects how objects move within the space-time, and yet is itself prescribed by how matter and energy is distributed within space-time. An interesting consequence is that, as the gravitational field itself can "carry energy"1, changes in curvature feed back into itself and propagates. Thus general relativity predicts that the gravitational field (i.e. curvature) can sustain wave-like motion independently of other forms of matter and energy, and this is what we call gravitational waves.
The presence of gravitational waves makes the dynamics of space-time in itself an interesting problem involving both geometry and analysis. Many of the open problems of the field concerns the global structure of space-time, especially as we move toward the "idealized boundary of space-time"--points that are either infinitely far away (either spatially or temporally) or points that cannot be in the space-time due to the presence of curvature singularities.
At our current level of understanding, many of the questions being asked are of the type of stability: given a known solution in general relativity, is the behavior it exhibits stable under small perturbations? This covers the known result of the stability of Minkowski space, the currently open question of the stability of Kerr(-Newman) black holes, and many ongoing works concerning the stability of cosmological space-times (both in the expanding direction and in the big-bang direction). (It is interesting to note that stability is not always "good": for example, the stability of the Kerr interior would actually disprove Penrose's strong cosmic censorship conjecture.)
The bigger picture that we hope some day to achieve, is to obtain an understanding of the generic behavior of space-times near such idealized boundaries, as well as to get a full classification of the possible limiting behaviors. In this vein some of the open questions are the various cosmic censorship conjectures, and the final state problem for isolated gravitational systems. Another question that I find extremely interesting is the idea that some sort of geometrization should take place as the limiting behavior of expanding cosmological space-times.
For some more info, some years ago I compiled a list of some open problems in mathematical general relativity; while I have not kept it up-to-date since 2014, most of the questions there are still very much open.
In the study of (inviscid) compressible fluids there are two types of dynamical structures. First, there is the transportation by the fluid flow: as the fluid moves, it carries along with it some of its properties, such as vorticity and entropy. Second, there is the propagation of pressure waves (sound waves) through the fluid.
The sound waves satisfy second order nonlinear wave equations, the principal parts of which is described by the so-called acoustic metric, which is a Lorentzian metric. Recent developments in the field brought to light that importance of understanding the geometry of this Lorentzian manifold.
The main analytic difficulty here is caused by the complication introduced by the coupling of acoustic wave dynamics to the transport dynamics. One has to marry disparate techniques developed separately for two different types of equations.