curriculum vitae
publications

### Teaching:

212 Honors Linear Algebra.
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.