Scott Armstrong (NYU)
Title: Quantitative results in stochastic homogenization
I will give an overview of some recent progress in elliptic homogenization with random coefficients with a focus on the precise scaling of the correctors. In the first lecture I will introduce the model, give some connections to problems in probability theory and try to give a feeling for some of the mathematical difficulties encountered in analyzing it.
In the second lecture I will give an outline of a "bootstrap" or "renormalization" scheme (developed jointly with Kuusi and Mourrat) for obtaining an optimal quantitative understanding of the correctors, emphasizing the important role played by the higher regularity theory in homogenization (based on previous joint work with Smart).
Thomas Chen (UT Austin)
Title: On the dynamics of Bose gases and Bose-Einstein condensates
In the first talk, we discuss the mean field limit of a system of N interacting bosons in Gross-Pitaevskii (BP) scaling. We explain how the nonlinear Schrodinger equation or Hartree equation is derived via the associated BBGKY hierarchy and the GP hierarchy equations as N tends to infinity, and how the NLS linked to the dynamics of Bose-Einstein condensates. In particular, the proof of uniqueness of solutions to the GP hierarchy via the quantum de Finetti theorems is discussed. This is based on joint work with N. Pavlovic, C. Hainzl, and R. Seiringer.
In this second talk, we discuss an extension to the Hartree equation, which describes thermal fluctuations around the Bose-Einstein condensate. Using quasifree reduction, we derive the Hartree-Fock-Bogoliubov (HFB) equations, and discuss the well-posedness of the corresponding Cauchy problem. In particular, the emergence of Bose-Einstein condensates at positive temperature via a self-consistent Gibbs state is addressed
This is based on joint work with V. Bach, S. Breteaux, J. Froehlich, and I.M. Sigal.
Jim Kelliher (UC Riverside)
Title: "Does the vanishing viscosity limit hold?"
A classical constant-viscosity incompressible fluid is modeled by the Navier-Stokes equations, derived by Navier in 1821 and refined by Stokes in the 1840s. Dropping the term that incorporates the effect of viscosity yields the Euler equations, derived by Euler much earlier in 1757. Despite the age of the equations, it is still not known whether solutions to the Navier-Stokes equations converge to that of the Euler equations as the viscosity vanishes when the fluids are interacting with a boundary.
In these talks I will focus on the central problem of the theory: a fluid contained in a fixed domain with the velocity of the Navier-Stokes solution vanishing on the boundary. To keep things simple, I will consider only 2D fluids, though most of what I present applies as well to higher dimensions. In the first talk, I will give some of the basic background needed to appreciate the problem. In the second talk I will present some recent results on the problem."
Inwon Kim (UCLA)
Title: Capillary Drops on Rough Surface
In the first lecture, we will discuss the problem of a liquid droplet resting on a solid surface, described with the interfacial free energy. We will discuss existing literature on both equilibrium drops and dynamic models. The main challenge arises due to possible singularities at the ``contact line", where the droplet boundary meets the surface.
In the second lecture, we will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises either by chemical composition or by roughness. Various physical models, the Cassie-Baxter or Wenzel Laws can be given rigorous sense via homogenization theory. This is joint work with William Feldman.
Wilhelm Schlag (UChicago)
Title: Quasiperiodic Schroedinger operators with multiple frequencies
We describe joint work with Michael Goldstein and Mircea Voda on discrete Schroedinger operators on the line with potentials defined on higher-dimensional tori. We describe properties of these operators related to Anderson localization, the separation of the eigenvalues, and the shape of the spectrum. The first talk will be a review of the field, whereas the second talk will present some more recent developments.