Hörmander auditorium, Centre for Mathematical Sciences, Sölvegatan 18A
Geometric function theory and vortex motion: the role of connections
We discuss point vortex dynamics on a closed two-dimensional Riemann manifold from the point of view of affine and other connections. The speed of a vortex then comes out as the difference between two affine connections, one derived from the coordinate Robin function and the other being the Levi-Civita connection associated to the Riemannian metric.
In a Hamiltonian formulation of the vortex dynamics, the Hamiltonian function consists of two main terms. One of them is a quadratic form based on a matrix whose entries are Green and Robin functions, while the other describes the energy contribution from those circulating flows which are not implicit in the Green functions. These two terms are not independent of each other, and one major aim is to try to understand the exchange of energy between them.
Remarks on the endpoint behaviour of some Littlewood-Paley operators: optimal asymptotic estimates and open problems
In the first part of this talk, we will review aspects of the classical Littlewood-Paley theory and briefly mention some of its applications in harmonic analysis. Then, motivated by some results of Bourgain and Pichorides, we will present optimal asymptotic estimates for the endpoint behaviour of certain classical Littlewood-Paley operators and discuss about some related open problems.
Weyl's law and Hecke eigenvalues
For compact Riemannian manifolds, the Weyl law is a classical result giving an asymptotic count for the number of Laplace eigenvalues on that manifold. In number theory non-compact arithmetic manifolds are of central importance, for example the locally symmetric spaces SL(n,Z)\SL(n,R)/SO(n). Counting Laplace eigenvalues on such non-compact spaces comes with the added difficulty of the existence of continuous spectrum.
Even more interestingly, such spaces come with many more operators, the Hecke operators, and we not only want to understand the distribution of the Laplace eigenvalues, but also the distribution of eigenvalues of the Hecke operators. I want to talk about recent joint work with T. Finis in which we establish an effective distribution result of the Hecke eigenvalues, including a Weyl law with error term. This has applications to various questions in number theory, for example, low-lying zeros in families of automorphic L-functions.
Unique continuation for wave equations and applications in general relativity
The fundamental PDEs in general relativity are wave equations. Close to event horizons of black holes and close to so-called Cauchy horizons, the natural coefficients for the wave equations degenerate. In particular, Hörmander's classical unique continuation theorem does not apply. In this talk, I will explain this issue and present a unique continuation theorem which does apply. The argument is based on new (singular) Carleman estimates for wave equations.