20C
Multivariable Calculus,
20D
Differential Equations,
20E
Vector Calculus,
20F102
Linear Algebra
130AB
Ordinary Differential Eq.,
110AB
Partial Differential Eq.,
150AB
Differential Geometry
210BC
Math Physics,
231AC
Partial Differential Equations,
250AB
Differential Geometry
237.Rel257.Rel
Topics in Partial Differential Equations / Differential Geometry:
General Relativity
237.Flu
Topics in Partial Differential Equations:
Fluid Mechanics
168.Med
Topics in Applied Mathematics: Medical Imaging
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 and illposedness.
Examples in nature are
black-holes in general relativity or shockwaves in gas-dynamics,
the sound-bang after a supersonic airplane.
