Analysis, Geometry and Dynamics Seminar - UW

The capillary energy functional models the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between $90^{\circ}$ and $120^{\circ}$ must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between $90^{\circ}$ and $180^{\circ}$. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions.

A central question in quantum chaos is how classical chaotic dynamics influence quantum behavior. On compact Riemannian manifolds, pure quantum states correspond to Laplacian eigenfunctions. The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak predicts that on hyperbolic manifolds, all high-energy eigenfunctions become uniformly distributed. The asymptotic behavior of eigenfunctions can be formulated in terms of semiclassical measures, which describe the microlocal distribution of eigenfunction mass. One approach towards the QUE conjecture applies microlocal analysis and uncertainty principles to characterize the support of semiclassical measures. I will discuss recent work that uses the breakthrough higher-dimensional fractal uncertainty principle of Cohen. Using this uncertainty principle, we prove the first result on the support of semiclassical measures on real hyperbolic n-manifolds.  To explain some of the main proof ideas, we will discuss work on the toy model of quantum cat maps. This is joint work with Nicholas Miller. 

In this talk we will discuss recent results concerning stochastic (and deterministic) free boundary problems, particularly arising in fluid structure interaction (FSI). We will begin by considering a nonlinearly coupled FSI system perturbed by stochastic effects involving a viscous fluid modeled by the Navier-Stokes equations in a 2D/3D domain, where part of the fluid domain boundary consists of an elastic deformable structure modeled by plate equations. The fluid and the structure are coupled via two sets of coupling conditions (balance of stress and kinematic entities) imposed at the fluid-structure interface. We will present results for both incompressible and compressible fluid flows. We will also discuss various kinematic coupling conditions and their implications for degeneracies in the problem. Finally, we will discuss long-time behavior of the solutions to a reduced stochastic FSI model obtained by spatially averaging a component of the fluid velocity in long compliant tubes.

Abstract: A unitary connection on the 2-sphere is called transparent, if its parallel along all great circles is the identity. In the scalar case this is equivalent to the connection being odd up to gauge, but for higher ranks the situation is more intricate. Mason proposed a classification of transparent connections on the 2-sphere in terms of complex geometric data on . In the talk I will discuss a generalisation of this classification that incorporates unitary pairs (connection + matrix field), as well as other closed Riemannian surfaces. The role of is then played by transport twistor space, a degenerate complex surface tailored to the geodesic flow, or (when available) its desingularisation.

We survey some new and old, positive and negative results on a priori estimate, regularity, rigidity, and constant rank results for special Lagrangian and quadratic Hessian equations. These equations originate in calibrated, convex, conformal, and complex geometries among other fields. Unlike all other order Hessian equations such as the linear (Laplace) and top order (Monge-Ampere) equations, the regularity/irregularity and a priori Hessian estimates for these two equations have not been settled yet in five and higher dimensions.