Hello and welcome to my webpage!

I am Nishanth Gudapati, a mathematics PhD student at the Albert Einstein Institute (AEI) at Golm within the program
of the International Max-Planck Research School (IMPRS).

## Research Summary

The general area of my research is geometric analysis, in particular the mathematical study of hyperbolic equations arising in
Einstein's equations of general relativity.
My PhD project is on global regularity of wave maps. Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which
are critical points of a Lagrangian which is a natural geometrical generalization of the free wave Lagrangian. I work on a conjecture
that associates the formation of blow-up of a wave map to the existence of a nontrivial solution of the corresponding static
equation i.e the harmonic maps equation - thereby establishing a blow up criterion for the Cauchy problem of wave maps. Such a result
has been established for flat spacetime by Struwe for the equivariant and spherically symmetric case and, Tataru and Sterbenz for the
general case. In my PhD project I attempt to extend this result to wave maps on dynamical curved backgrounds. My PhD advisors are Lars
Andersson and Gerhard Huisken.

In addition, even though I don't claim by any means to be an expert, I keenly inform myself about the basic ideas and developments in the following topics:
regularity of linear and nonlinear wave equations on various backgrounds, formation of black holes, trapped surfaces, stability of black holes,
general relativistic interpretation of gravitational collapse, Penrose inequalities, Hamiltonians in general relativity and (a bit far fetched topic of )
propagation of electromagnetic waves through different nanophotonic structures.