Research

Puskar is intersted in mathematical physics.

( Harvard College Students interested in general relativity, quantum field theory, or geometric analysis, in general, are welcome to get in touch. Ongoing Harvard college student projects : (a) Marrs Griggs: "Non-linear stability of cosmological spacetimes under coupled Einstein-Yang-Mills perturbations", (b) Oswaldo Vazquez: "Continuation criteria for Einstein's equations and the issue of cosmic censorship"- Notice that this is the first use of Moncrief's exact integral equation for the spacetime Riemann curvarture in vacuum. This is hoped to provide new ways to tackle large data problems in GR. Previous works on this topic involve using an `approximate' integral equation which may not be very useful in many applications)

Geometry of Configuration spaces and quantum field theory: Classical field theory can have a particle interpretation in the sense that  "a particle is moving in an infinite dimensional space". The very infinite dimensional characteristics of the configuration space turns the problem much more difficult to handle than ordinary finite-dimensional systems since infinite dimensional spaces do exhibit several non-trivial properties starting with closed bounded balls being only weakly compact. Sophisticated techniques of functional and harmonic analysis can be invoked to handle the classical field theory. However, one might ask "does the geometry of this configuration space have any consequence in the corresponding quantum theory?" As we know interacting quantum field theories in 3+1 dimensions are extremely difficult to handle in a mathematically rigorous way and as such very little is known about them. One important example is the pure Yang-Mills theory in 3+1 dimensions. It is expected that the Hamiltonian of the theory (suitably defined such that it makes sense) exhibits a strictly positive gap in its spectrum i.e., the lowest glueball state has a non-zero mass. If one assumes that a quantum field theory makes sense, then one can explicitly show that this mass gap may be a consequence of the positive definiteness of the Bakry-Emery Ricci curvature (if it verifies positive definiteness after regularization) of the orbit space of the theory. In finite-dimensional quantum mechanical systems, rigorous results regarding spectral gap of Schrodinger operator are available due to Lichnerowicz, S-T Yau, Singer, etc. This issue of the geometry of the configuration space and its role in quantum theory is under intense investigation and expected to shed light on constructing rigorous quantum theories. 

Formation of blackholes:  Since black holes are observed recently, one would like to understand the exact mechanism of their formation. Seminal work of D. Christodoulou marked the initiation of such study in the context of pure gravity. Recently, I became interested in understanding the detail mechanism behind the formation of Einstein-Yang-Mills and Einstein-Vlasov black holes. Another important question to ask might be in the context of Einstein-Euler equations. Since we already know Euler's equations develop shock singularities on Minkowski background in finite time, it would be interesting to see if a traped surface can form before the shock formation in the gravity-fluid coupled system i.e., can gravity hide pathologies such as shock and merge these with the black-hole? These issues are under intense investigation with Prof. Yau and N. Athanasiou.

Stability and Convergence issues in Mathematical General Relativity (Ph.D work)

Global Existence Issue: U(1) problem is under intense investigation with Prof. Moncrief.

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.   

Check out this recent article where we give a complete `quasi-local' proof of the positivity of Wang-Yau and other quasi-local masses that appear in GR and Riemannian geometry.

Study of Non-linear evolutionary PDE: Incompressible Euler equations in a compact domain with appropriate boundary conditions  since these are geometric in a sense of geodesic equations on an infinite dimensional Lie group (group of volume preserving diffeomorphisms)

Teichmuller Theory: 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 the Thurston boundary. This is currently under intense investigation with Prof. Yau.