# Seminars & Events for Analysis Seminar

##### Non-uniform dependence for the Periodic CH equation

Using approximate solutions we show that the solution map for the periodic Camassa-Holm (CH) equation is not uniformly continuous in Sobolev spaces with exponent greater or equal to one. This extends earlier results to the whole range of Sobolev exponents.

##### Derivation of the Gross-Pitaevskii equation: the case of strong interaction potential

In this talk I am going to present recent results obtained in collaboration with L. Erdoes and H.-T. Yau concerning the derivation of a certain cubic nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, from first principle many body quantum dynamics. With respect to previous results, we can now relax the smallness condition on the interaction potential.

##### Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

In this talk I will present recent joint work with P. D'Ancona and D. Foschi on the classical Maxwell-Dirac system, which is the fundamental PDE in quantum electrodynamics. We show that the system has some special structural properties ("null" structure) which improve the regularity of solutions. To see this structure, however, one must consider the system as a whole: it cannot be seen in the individual component equations. For the multilinear forms that thus arise, we prove estimates in $X^{s,b}$ spaces at the scale invariant regularity up to some logarithmic losses, and as a consequence we obtain almost optimal local well-posedness by iteration.

##### The abstract concept of Duality and some related facts (part of a joint project with Shiri Artstein-Avidan)

We discuss in the talk an unexpected observation that very minimal basic properties essentially uniquely define some classical transforms which traditionally are defined in a concrete and quite involved form. We start with a characterization of a very basic concept in Convexity: Duality and the Legendre transform. We show that the Legendre transform is, up to linear terms, the only involution on the class of convex lower semi-continious functions in R^n which reverses the (partial) order of functions. This leads to a different understanding of the concept of duality, which we call an *abstract duality concept*, and which we then apply also to many other well known settings.

##### Mass inequalities for Cauchy data in general relativity

We will survey a number of inequalities involving the total mass and other invariants for initial data for the Einstein equations in general relativity.

##### The Composite Membrane Problem

We wish to build a body of prescribed shape, and of prescribed mass out of materials of varying density so as to minimize the first Dirichlet eigenvalue with fixed boundary of the body. Existence, uniqueness and regularity of the solution and the resulting free boundary problem will be discussed.

##### On singularity formation for certain geometric wave equations

We dicuss recent results, obtained jointly with W. Schlag and D. Tataru, on a new kind of singularity formation for certain critical nonlinear geometric wave equations.