Georgia Institute of Technology College of Sciences

We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow variables. Assuming a steady state persists, we construct the stable, unstable, center-stable, center-unstable, and center manifolds of the steady state of a size of order $O(1)$ and give their leading order approximations. Finally, using these tools, we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems.

We are interested in the cubic to tetragonal phase transition in a shape memory alloy. We consider geometrically linear elasticity. In this framework, Dolzmann and Mueller have shown the following rigidity result:The only stress-free configurations are (locally) twins (i.e. laminates of just two of the three Martensitic variants).However, configurations with arbitrarily small elastic energy are not necessarily close to these twins: The formation of microstructure allows to mix all three Martensitic variants at arbitrary volume fractions. We take an interfacial energy into account and establish a (local) lower bound on elastic + interfacial energy in terms of the Martensitic volume fractions. The model depends on a non-dimensional parameter that measures the strength of the interfacial energy. Our lower, ansatz-free bound has optimal scaling in this parameter. It is the scaling predicted by a reduced model introduced and analyzed by Kohn and Mueller with the purpose to describe the microstructure near an interface between Austenite and twinned Martensite. The optimal construction features branching of the Martensitic twins when approaching this interface.

The changing status of knowledge from descriptive to analytic, from empirical to theoretical and from intuitive to mathematical has to be one of the most striking adventures of the human spirit. The changes often occur in small steps and can be lost from view. In this lecture we will review vignettes drawn from the speaker's personal knowledge that illustrate this transformation in thinking. Examples include not only the traditional areas of physics and engineering, but also newer topics, as in biology and medicine, in the social sciences, in commerce, and in the arts. We also review some of the forces driving these changes, which ultimately have a profound effect on the organization of human life.

New technologies have been introduced into the front tracking method to improve its performance in extreme applications, those dominated by a high density of interfacial area. New mathematical theories have been developed to understand the meaning of numerical convergence in this regime. In view of the scientific difficulties of such problems, careful verifaction, validation and uncertainty quantification studies have been conducted. A number of interface dominated flows occur within practical problems of high consequence, and in these cases, we are able to contribute to ongoing scientific studies. We include here turbulent mixing and combustion, chemical processing, design of high energy accelerators, nuclear fusion related studies, studies of nuclear power reactors and studies of flow in porous media. In this lecture, we will review some of the above topics.

I will talk about how tropical geometry can be used for implicitization and elimination problems. Implicitization is the problem of finding the defining equations (implicit equations) of an algebraic variety from a given parameterization. Elimination is the problem of finding the defining equations of a projection of an algebraic variety. In some instances such as the case when the polynomials involved have generic coefficients, we give a combinatorial construction of the tropical varieties without actually computing the defining polynomials. Tropical varieties can then be used to compute invariants of the original varieties.