Georgia Institute of Technology College of Sciences

This talk is the story of an encounter of three distinct fields: non-Euclidean geometry, gas dynamics and economics. Some of the most fundamental mathematical tools behind these theories appear to have a close connection, which was revealed around the turn of the 21st century, and has developed strikingly since then.

Modelling and computational approaches provide powerful tools in the study of disease dynamics at both the micro- and macro-levels. Recent advances in information and communications technologies have opened up novel vistas and presented new challenges in mathematical epidemiology. These challenges are central to the understanding of the collective dynamics of heterogeneous ensembles of individuals, and analyzing pertinent data that are less coarse and more complex. The evolution of dynamic modelling is typified by the agent-based modelling (ABM) as a shifting paradigm, a lattice-distributed collection of autonomous decision-making entities (i.e., agents), the interactions of which unveil the dynamics and emergent properties of a real-life problem, such as an infectious disease outbreak. In this talk, we show a general framework for developing an ABM that can be used to computationally optimize intervention strategies for novel influenza viruses with pandemic potential. Our findings contrast previous results !

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.