Abstract: For roughly the first half of the semester we will study recent developments in the geometry of nonabelian N=2 supersymmetric gauge theories in four dimensions as well as their dimensional reductions to three dimensions. Some recent progress results from regarding these as compactifications of a six-dimensional theory which does not have a classical description but is somehow a theory of nonabelian gerbes. The first lecture or two will cover some background material focused on the abelian version of these theories. Note: After this first lecture the seminar will move to RLM 8.136.

What is K-homology good for? I'll try to answer with examples in this talk and the next. There is an obvious connection to the Atiyah-Singer index theorem, and roughly speaking K-homology provides a context in which to consider the index theorem's many elaborations. A general theme is that while two operators may look rather different, for example when studied using a symbol calculus, functional analysis and the framework of K-homology can sometimes give a means to identify their Fredholm index theories.

The Langlands program relates complex representations of GL_n(Q_p) to Galois representations. For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms. For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a p-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2. We survey some of the known story in the classical and in the mod p case, and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p).

We consider Klein-Gordon equations coupled to U(1)-invariant nonlinear oscillators. The solitary waves of the coupled nonlinear system form two-dimensional submanifold in the Hilbert phase space of finite energy solutions. Our main results read as follows: Theorem: Let all the oscillators be strictly nonlinear. Then any finite energy solution converges, in the long time limit, to the solitary manifold in the local energy seminorms. The investigation is inspired by Bohr’s postulates on transitions to quantum stationary states. The results are obtained for: • 1D KGE coupled to one oscillator; • 1D KGE coupled to finite number of oscillators; • nD KGE coupled to one oscillator via mean field interaction.

Matrix polynomial functions and which satisfy are called a Lax pair. I will present a construction of Hitchin which gives a concrete way of interpreting linear flows on the space of line bundles over a Riemann surface as solutions to Lax pair equations. This is a candidacy talk.

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported. This is a joint work with Xiangxiong Zhang.