Speaker: Walter Strauss (Brown University)
Title: Stability of a Hot Plasma in a Torus
Abstract: In a tokamak huge numbers of charged particles whiz around a torus
at relativistic speeds. Finding stable particle configurations is
the holy grail of fusion energy research. We model a collisionless
plasma by the relativistic Vlasov-Maxwell system. There are many
equilibria, of which some are stable and some unstable. In this talk I
will present recent work with Toan Nguyen where the particles reflect
specularly and the field is a perfect conductor. These are however not
the physical boundary conditions. Given an equilibrium of a certain type,
we reduce linear stability to the positivity of a certain non-local linear
operator which is much less complicated than the generator of the full
Speaker: Raphael Ponge (UC Berkeley / Seoul National University)
Title: On the singularities of the Green functions of the conformal powers of the Laplacian.
Abstract: Green functions play a major role in PDEs and conformal geometry. In this talk, I will explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. This includes the Yamabe and Paneitz operators, as well as the conformal fractional powers originating from the work of Graham-Zworski on scattering theory for AH metrics. The results are formulated in terms of explicit conformal invariants arising from the ambient Lorentzian metric of Fefferman-Graham. As applications we obtain a new characterization of locally conformally flat manifolds and a spectral-theoretic characterization of the conformal class of the round sphere.