I will discuss the dynamics of the horocycle flow on a stratum of translation surfaces (which is an invariant subvariety of the bundle Omega M_g of holomorphic one forms over the moduli space of genus g Riemann surfaces). This flow can be defined as the action of upper triangular matrices with eigenvalue 1, acting linearly on flat charts. Work of Ratner on unipotent flows on homogeneous spaces leads to the question of whether the orbit-closures and invariant measures for this action can be meaningfully classified. I will quickly survey both positive and negative results in this direction. The talk will be based on joint work with Bainbridge, Chaika, Smillie, and Ygouf (in various combinations).

The physical properties and structure of materials under nanoscopic confinement can significantly differ from their bulk counterparts. This talk focuses on a class of estimators used to determine the position-dependent diffusivity tensor for a confined fluid. The estimators are suitable for trajectory data obtained from confocal microscopy experiments and molecular dynamics (MD) simulations. We first demonstrate the use of the estimators on trajectories generated from a few toy problems, to highlight some numerical issues associated with them. Specifically, we show that these estimators may lose accuracy near hard boundaries of the simulation domain. To address this issue, we introduce a correction scheme using Diffusion maps. We demonstrate the use of this correction on trajectories obtained from MD simulations of a model system consisting of a pure liquid within a slit pore. The resulting diffusivity profiles yield empirical distributions that are in good agreement with those obtained directly from MD simulations.

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.