The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez (Universidad de Sevilla), Javier Gomez-Serrano (Brown University) and Benoit Pausader (Brown University)

We will discuss applications of a discrepancy trick, which sometimes can be used to prove quantitative disjointness results. As an illustration, consider the following example. Fix an imaginary quadratic number field and consider a "diagonal" measure on a product of the unit tangent bundles of two modular curves given by the push-forward under two multivariate monomials of the uniform measure on two copies of the class group. If the monomials are independent, assuming GRH for the Dedekind zeta function, this diagonal measure is quantitatively equidistributed in the product in terms of the size of the discriminant of the quadratic field.

This version of the result implies (conditional) bounds on the size of torsion subgroups of the class group, which was proven previously by Ellenberg and Venkatesh. The main dynamical theorem has several applications and, for example, can be used to prove for most dimensions quantitatively the equidistribution of the triples consisting of a rational subspace, the shape of the integer lattice in this subspace, and the shape of the integer lattice in the orthogonal complement.

This is joint work with Menny Aka, Manfred Einsiedler, Philippe Michel, and Andreas Wieser.

We study Schrodinger equations which are periodic in space and time. These models are inspired by recent experimental progress in the study of wave propagation and quantum mechanics under time-periodic forcing. Time-periodic Hamiltonians, however, are not as well understood as their static (autonomous) analogs.

In particular, many discrete models of periodic materials (e.g., of graphene) are known to develop spectral gaps under a time-periodic driving. In PDEs, however, no such gaps are conjectured to form. How do we reconcile these two facts? Using periodic homogenization, we prove that the driven Schrodinger equation has an “effective gap” - a new and physically-relevant relaxation of a spectral gap.

Taking a broader perspective, we ask - how does time-periodic forcing affect a general band structure? We will show that a spectrally-local notion of stability can be formulated and proved, in a way which again corresponds with periodic homogenization theory.

In this talk I will give an introduction to the mathematical theory of strongly disordered topological insulators. These are novel materials which insulate in their bulk but (may) conduct along their edge; the quintessential example is that of the integer quantum Hall effect. What characterizes these materials is the existence of a topological index, experimentally measurable and macroscopically quantized. Mathematically this is explained by applying algebraic topology to the space of appropriate quantum mechanical Hamiltonians. I will survey some recent results regarding bulk-edge correspondence, calculation of the path-connected components and the connection to Anderson localization.