Georgia Institute of Technology College of Sciences

Image segmentation has been widely studied, specially since Mumford-Shah functional was been proposed. Many theoretical works as well as numerous extensions have been studied rough out the years. In this talk, I will focus on couple of variational models for multi-phase segmentation. For the first model, we propose a model built upon the phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. For the second model, we propose a variational functional for an unsupervised multiphase segmentation, by adding scale information of each phase. This model is able to deal with the instability issue associated with choosing the number of phases for multiphase segmentation.

I will present a generalization of a classical within-host model of a viral infection that includes multiple strains of the virus. The strains are allowed to mutate into each other. In the absence of mutations, the fittest strain drives all other strains to extinction. Treating mutations as a small perturbation, I will present a global stability result of the perturbed equilibrium. Whether a particular strain survives is determined by the connectivity of the graph describing all possible mutations.

h-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the C^1 isometric embedding theorem of Nash.

$h$-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the $C^1$ isometric embedding theorem of Nash.