Course overview -- Sommersemester 2014 (Archive)

Monstrous moonshine started in the 1970s, when McKay and some others found apparent numerical coincidences relating modular functions on the complex upper half plane and linear representations of the monster simple group. We now know that the numerical relationships that were initially seen can be explained by the existence of the monster vertex algebra. Later computations by Queen and Norton suggested that certain subgroups of the monster also yielded modular functions from their representation theory, and this was codified by Norton in his generalized moonshine conjecture. I will describe this conjecture, a mathematical interpretation arising from orbifold conformal field theory, and some recent progress.

In this talk I will discuss some existence results for periodic orbits on fixed energy levels of mechanical Hamiltonian systems. In particular, these energy hypersurfaces are allowed to be non-compact. I will focus on the analytical and geometrical problems posed by the lack of compactness and discuss advantages and limitations of our method of proof ("linking"). This is joint work with T. Rot, J.B. van den Berg and R. Vandervorst.

There are many estimates in symplectic geometry which provide a lower bound for the number of, e.g., periodic orbits of Hamiltonian systems or Hamiltonian chords of Lagrangian submanifolds. In this talk, I will describe various Floer theories related to these estimates and sketch a Morse theoretic argument which can be applied to the Floer settings to give a unifoed proof of these estimates. Moreover, this argument is also applicable in other cases, where the corresponding cuplength estimates was not known before.

In this talk we review certain generalizations of Mathai-Quillen forms. Ordinary Mathai-Quillen forms are representations of Thom classes, and play an important role in topological field theories. We shall discuss analogues appearing in more recent pseudo-topological field theories, involving representations of certain sheaf cohomology classes which reduce to ordinary Mathai-Quillen forms in special cases. We discuss some basic properties and make a few conjectures.

Recently there has been much interest in studying a two-dimensional model of quantum mechanics, where the two components of position, and possibly also momentum, are made non-commutative. The understanding here is that such a model might mimic a non-standard structure of space at very short distances. In this talk we discuss some group theoretical aspects of such a model and some generalized complex Hermite polynomials associated with it.