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# TBA

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This is a joint Nonlinear PDEs seminar with Analysis seminar

# Bergman-Einstein metrics on Strongly pseudoconvex domains in a complex space.

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I will give a proof of S.Y. Cheng's conjecture that a bounded strongly pseudoconvex domain in C^n has its Bergman metric being Einstein if and only if it is holomorphically equivalent to the ball.

# Local Energy Decay and the Vector Field Method

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We discuss some recent work on stability of solutions to linear and nonlinear wave equations on asymptotically flat space times. This is joint work with Jason Metcalfe, Jesus Oliver, and Daniel Tataru.

# Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori of the 2D cubic NLS

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We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))