Georgia Institute of Technology College of Sciences

There are many interesting patterns observed in activator-inhibitor systems. A well-known model is the FitzHugh-Nagumo system. In conjunction with calculus of variations, there is a close relation between the stability of a steady state and its relative Morse index. We give a sufficient condition in diffusivity for the existence of standing wavefronts joining with Turing patterns.

We improved a computational model of leukemia development from stem cells to terminally differentiated cells by replacing the probabilistic, agent-based model of Roeder et al. (2006) with a system of deterministic, difference equations. The model is based on the relatively recent theory that cancer originates from cancer stem cells that reside in a microenvironment, called the stem cell niche. Depending on a stem cell’s location within the stem cell niche, the stem cell may remain quiescent or begin proliferating. This emerging theory states that leukemia (and potentially other cancers) is caused by the misregulation of the cycle ofproliferation and quiescence within the stem cell niche.Unlike the original agent-based model, which required seven hours per simulation, our model could be numerically evaluated in less than five minutes. The results of our numerical simulations showed that our model closely replicated the average behavior of the original agent-based model. We then extended our difference equation model to a system of age-structured partial differential equations (PDEs), which also reproduced the behavior of the Roeder model. Furthermore, the PDE model was amenable to mathematical stability analysis, which revealed three modes of behavior: stability at 0 (cancer dies out), stability at a nonzero equilibrium (a scenario akin to chronic myelogenous leukemia), and periodic oscillations (a scenario akin to accelerated myelogenous leukemia).The PDE formulation not only makes the model suitable for analysis, but also provides an effective mathematical framework for extending the model to include other aspects, such as the spatial distribution of stem cells within the niche.

We present a new theory on Hamiltonian PDE. The linear theory solves an old spectral problem on boundedness of L infinity norm of the eigenfunctions of the Schroedinger operator on the 2-torus. The nonlinear theory develops Fourier geometry, eliminates the convexity condition on the (infinite dimension) Hamiltonian and is natural for PDE.

Stable sets and unstable sets of a dynamical system with positive entropy are investigated. It is shown that in any invertible system with positive entropy, there is a measure-theoretically ?rather big? set such that for any point from the set, the intersection of the closure of the stable set and the closure of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems, the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.