Analysis, Geometry and Dynamics Seminar - UW

There is an emerging interest in understanding the behavior of partial differential equations on graphs G=(V,E).  The classic approach is to think of a graph as a (discretized) compact manifold without boundary (since there is no `complement', no place where the domain/graph ends, no boundary conditions are imposed).  I will discuss an axiomatic definition of boundary on graphs that interacts well with classic ideas from Analysis and Probability Theory (including the isoperimetric inequality, exit time estimates for Brownian motion, the Faber-Krahn theorem, Hardy's inequality and the Alexandrov-Bakelman-Pucci estimate).   I will not assume any prior knowledge and introduce all ideas from scratch, there will also be many pretty pictures.

I will review a topic in mathematical physics, called Lieb-Robinson bounds, that exemplifies analysis, geometry, and dynamics.  These bounds describe the dynamics of a physical system of many interacting quantum degrees of freedom: they control how rapidly a disturbance can propagate through the system.  The speed of the propagation depends on the geometry of the system.  Then, using elementary ideas in analysis, one can obtain some powerful results on the static properties of the system, such as decay of correlations or quantization of the Hall conductance.  This will be an introductory talk with no specific background required. 

Given a Jordan curve in the plane, we can associate a circle homeomorphism (conformal welding) via the conformal welding correspondence. These homeomorphisms arise naturally in Teichmuller theory, Mathematical physics and dynamics. In this talk, we will study this correspondence and we will show that given a flexible curve there is a homeomorphism of the plane, conformal off the curve, that maps the curve to positive area. We will also state a more general result involving Hausdorff dimension and explain connections to some open problems.

We will review different notions of anti-concentration, arising from geometry, analysis, and probability. One focus of the talk will be on understanding how these definitions relate to each other, sometimes leading to surprising and nontrivial consequences. The other focus of the talk will be on potential applications, maybe even involving some dynamics if time allows.

We discuss a differential geometric construction of distinguished holomorphic 2-spheres inside a K3 surface. These 2-spheres degenerate to a line on an affine 3-manifold. The example illustrates a general principle in the SYZ program, where graphs on an affine space B should correspond to limits of calibrated submanifolds along a sequence of compact Ricci-flat manifolds collapsing to B. This is joint work with Federico Trinca. 

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.