Georgia Institute of Technology College of Sciences

We prove stochastic representation formulae for solutions to elliptic boundary value and obstacle problems associated with a degenerate Markov diffusion process on the half-plane. The degeneracy in the diffusion coefficient is proportional to the \alpha-power of the distance to the boundary of the half-plane, where 0 < \alpha < 1 . This generalizes the well-known Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process. The generator of this degenerate diffusion process with killing, is a second-order, degenerate-elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the 2\alpha-power of the distance to the boundary of the half-plane. Our stochastic representation formulae provides the unique solution to the degenerate partial differential equation with partial Dirichlet condition, when we seek solutions which are suitably smooth up to the boundary portion \Gamma_0 contained in the boundary of the half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulae provides the solutions which are not guaranteed to be any more than continuous up to the boundary portion \Gamma_0 .

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erdos and H.-T. Yau, which yields universality for log-gases at arbitrary temperature at the microscopic scale. A main step consists in the optimal localization of the particles, and the involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

We introduce the admissible risk class as the set of possible aggregate risks when the marginal distributions of individual risks are given but the dependence structure among them is unspecified. The convex ordering upper bound on this class is known to be attained by the comonotonic scenario, but a sharp lower bound is a mystery for dimension larger than 2. In this talk we give a general convex ordering lower bound over this class. In the case of identical marginal distributions, we give a sufficient condition for this lower bound to be sharp. The results are used to identify extreme scenarios and calculate bounds on convex risk measures and other quantities of interest, such as expected utilities, stop-loss premiums, prices of European options and TVaR. Numerical illustrations are provided for different settings and commonly-used distributions of risks.