Georgia Institute of Technology College of Sciences

We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof. The talk is based on joint work with C. Hainzl, R. Seiringer and J. P. Solovej.

A contact manifold with boundary naturally gives rise to a sutured manifold, as defined by Gabai. Honda, Kazez and Matic have used this relationship to define an invariant of contact manifolds with boundary in sutured Floer homology, a Heegaard-Floer-type invariant of sutured manifolds developed by Juhasz. More recently, Kronheimer and Mrowka have defined an invariant of sutured manifolds in the setting of monopole Floer homology. In this talk, I'll describe work-in-progress to define an invariant of contact manifolds with boundary in their sutured monopole theory. If time permits, I'll talk about analogues of Juhasz' sutured cobordism maps and the Honda-Kazez-Matic gluing maps in the monopole setting. Likely applications of this work include an obstruction to the existence of Lagrangian cobordisms between Legendrian knots in S^3. Other potential applications include the construction of a bordered monopole theory, following an outline of Zarev. This is joint work with Steven Sivek.

Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular,  rigorous error estimates as well as efficient numerical algorithms for  elliptic equations in both bounded and unbounded domains.As an application, we shall use the proposed sparse spectral method to solve the N-particle electronic  Schrodinger equation.

Immersion is a containment relation between graphs (or digraphs) which is defined similarly to the more familiar notion of minors, but is incomparable to it. Of particular interest is to find conditions on a graph (or digraph) G which guarantee that G contains a clique (or bidirected clique) of order t as an immersion. This talk will begin with a gentle introduction, and will then share two new results of this form, one for graphs and one for digraphs. In the former case, we find that minimum degree 200t is sufficient, and in the later case, we find that minimum degree t(t-1) is sufficient, provided that G is Eulerian. These results come from joint work with Matt DeVos, Jacob Fox, Zdenek Dvorak, Bojan Mohar and Diego Scheide.