712
Topics in Mathematical Physics: Energy estimates for the wave equation on curved background.
637
Functional Analysis.
741
Topics in Partial Differential Equations:
Linear Stability of Black Holes.
201
Linear Algebra.
712
Topics in Mathematical Physics: Scattering for Nonlinear Klein Gordon.
211
Honors Multivariable Calculus.
633
Harmonic Analysis: Fourier Analysis
742
Topics in Partial Differential Equations:
The Analysis of Black Holes
712
Topics in Mathematical Physics:
Fluid Mechanics
439
Introduction To Differential Geometry
712
Topics in Mathematical Physics:
Existence for Einstein's equations.
302 Differential Equations.
631
Partial differential equations I:
Linear Equations mostly Elliptic.
632
Partial differential equations II:
Variable coefficient and nonlinear Equations mostly hyperbolic.
Research:
My research concerns basic mathematical questions about nonlinear
wave equations arising in Physics.
I am interested in existence, stability and behavior
of solutions to hyperbolic
differential equations.
Many important equations in physics can be written as systems of nonlinear
wave equations, e.g.
equations of continuum mechanics and Euler's equations,
describing the motion of elastic bodies and fluids,
Einstein's equations of general relativity, that relate
the geometry of space-time to the motion of matter,
Yang-Mills' equations that generalize
Maxwell's equations of electromagnetism.
Specifically I work on
Existence for Free Boundary Problems of continuous media,
describing the motion of a fluid or elastic body in vacuum or inside another fluid,
e.g. the motion of the surface of the ocean or water drop,
or the motion of stars or galaxies.
One question is if the water wave is unstable when it turns over.
The regularity and geometry of the free surfaces enters to highest order.
Global existence and stability for nonlinear wave equations with initial data
close to a given solution.
Among other things I
study if Einstein's equations of general relativity have global solutions
and the universe is stable or
if space time breaks down and black holes form.
This requires an understanding
of the geometry of space-time and light cones,
as well as of wave eq..
Blowup or formation of singularities of solutions to nonlinear wave
equations. Istability, illposedness and counterexamples to local existence.
Examples in nature are
black-holes in general relativity or shockwaves in gas-dynamics,
the sound-bang after a supersonic airplane.
