Georgia Institute of Technology College of Sciences

In the preceeding talk, I outlined a framework for variational problems and some of the basic tools and results. In this talk I will attempt describe several problems of current interest.

 A graph G contains a graph H as a minor if a graph isomorphic to H can be obtained from a subgraph of G bycontracting edges. One of the central results of the rich theory of graph minors developed by Robertson and Seymour is an approximate description of graphs that do not contain a fixed graph as a minor. An exact description is only known in a few cases when the excluded minor is quite small.In recent joint work with Robin Thomas we have proved a conjecture of his, giving an exact characterization of all large, t-connected graphs G that do not contain K_t, the complete graph on t vertices, as a minor. Namely, we have shown that for every integer t there exists an integer N=N(t) such that a t-connected graph G on at least N vertices has no K_t minor if and only if G contains a set of at most t- 5 vertices whose deletion makes G planar. In this talk I will describe the motivation behind this result, outline its proof and mention potential applications of our methods to other problems.

Hajos' conjecture is false, and it seems that graphs without a subdivision of a big complete graph do not behave as well as those without a minor of a big complete graph. In fact, the graph minor theorem (a proof of Wagner's conjecture) is not true if we replace the minor relation by the subdivision relation. I.e, For every infinite sequence G_1,G_2, ... of graphs, there exist distinct integers i < j such that G_i is a minor of G_j, but if we replace ''minor" by ''subdivision", this is no longer true. This is partially because we do not really know what the graphs without a subdivision of a big complete graph look like. In this talk, we shall discuss this issue. In particular, assuming some moderate connectivity condition, we can say something, which we will present in this talk. Topics also include coloring graphs without a subdivision of a large complete graph, and some algorithmic aspects. Some of the results are joint work with Theo Muller.

In this talk, I will first discuss several chemotaxis models includingthe classical Keller-Segel model.Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. In our work we propose a family of stable (even at times near blow up) and highly accurate numerical methods, based on interior penalty discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. This model is the basic step in the modeling of many real biological processes and it is described by a system of a convection-diffusion equation for the cell density, coupled with a reaction-diffusion equation for the chemoattractant concentration.We prove theoretical hp error estimates for the proposed discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.Numerical experiments to demonstrate the stability and accuracy of the proposed methods for chemotaxis models and comparison with other methods will be presented. Ongoing research projects will be discussed as well.