20C - Multivariable Calculus,
20D - Differential Equations,
20E - Vector Calculus,
20F, 102 - Linear Algebra
130A,
B - Ordinary Differential Eq.,
110,
132A - Partial Differential Eq.,
150A,
B - Differential Geometry.
210B,
C - Math Physics & Engin.,
231A,
- Partial Differential Eq.,
250A,
B - Differential Geometry
257B - Topics in Differential Geometry: General Relativity
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
Local 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 astrophysical bodies like stars or galaxies.
This relates to the question 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.
E.g. Einstein's equations have global solutions for
initial data close to Minkowski space-time, which.
relates to the question if our universe is stable
or if we can expect space-time to break down.
This requires an understanding
of the geometry of space-time and light cones,
as well as of wave eq..
Another model is the minimal surface eq. in Lorentzian geometry
and membranes in field theory.
Blowup or formation of singularities of solutions to nonlinear wave
equations.
Examples in nature are
black-holes in general relativity or shockwaves in gas-dynamics,
the sound-bang after a supersonic airplane.
A related phenomena is instability or illposedness. This in turn may be
related to counterexamples to local existence with rough data.
