Georgia Institute of Technology College of Sciences

An array of powerful mathematical tools can be used to identify the key underlying components and interactions that determine the mechanics of biological systems such as cancer and its interaction with various treatments. In this talk, we describe a mathematical model of tumor growth and the effectiveness of combined chemotherapy and anti-angiogenic therapy (drugs that prevent blood vessel growth). An array of mathematical tools is used in these studies including dynamical systems, linear stability analysis, numerical differential equations, SAEM (Stochastic Approximation of the Expectation Maximization) parameter estimation, and optimal control. We will develop the model using preclinical mouse data and discuss the optimal combination of these cancer treatments. The hope being that accurate modeling/understanding of experimental data will thus help in the development of evidence-based treatment protocols designed to optimize the effectiveness of combined cancer therapies.

Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial? I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).

Dr. Hurth will talk about two relatively simple, related switching systems: one in 1D, the other in 2D. For both systems, he will sketch how to analyse the density of the associated invariant measure. This is straightforward for the 1D-example, but proves somewhat unexpectedly difficult for the 2D one.

We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.