Georgia Institute of Technology College of Sciences

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed. We will talk about how to adapt the results from the standard Ginzburg-Landau theory into an inhomogeneous framework and present results from a recent work in collaboration with Carlos Roman (Pontificia Universidad Catolica de Chile).

The hard sphere model is a simple to define and long studied mathematical model of gas, in which the only interactions are the hard-core constraint that two spheres cannot overlap in space.  In three dimensions it is expected to exhibit a gas-to-crystal phase transition.  Despite its simplicty, rigorous results on the model are rather sparse.  I will introduce the model, discuss some of the main open questions, and present some results new and old.

We study differential inclusions of the type $A v=0$ and $v \in K$, where $v$ is a vector field satisfying a linear PDE system $A$ and $K$ is a compact set. We are particularly interested in the case when $K$ consists of two vectors (\textit{two-state problem}). We consider Dirichlet boundary conditions for $v$, in which case the differential inclusion typically has no solutions. We study a suitable relaxation of the system, in which we penalize the surface energy required to switch between the two states. We study the asymptotics of the regularized energy integral. We show that the asymptotics depend polynomially on the regularization parameter with a quantification which — somewhat surprisingly — depends on the order of the linear PDE system $A$. Joint work with A. R\”{u}land, C. Tissot, A. Tribuzio.

Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.  By analyzing the Gibbs distribution of various fixed-magnetization models from equilibrium statistical mechanics, we show that particles moving stochastically according to local affinities can solve various useful collective tasks. Finally, we will briefly introduce new tools that may prove fruitful in nonequilibrium settings as well.