

(Oregon) 
Strong Cosmic Censorship.  The HawkingPenrose theorems tell us that solutions of Einstein's equations are generally singular, in the sense of the incompleteness of causal geodesics (the paths of physical observers). These singularities might be marked by the blowup of curvature and therefore crushing tidal forces, or by the breakdown of physical determinism. Penrose has conjectured (in his "Strong Cosmic Censorship Conjecture") that it is generically unbounded curvature that causes singularities, rather than causal breakdown. The verification that “AVTD behavior” (marked by the domination of time derivatives over space derivatives) is generically present in a family of solutions has proven to be a useful tool for studying model versions of Strong Cosmic Censorship in that family. I discuss some of the history of Strong Cosmic Censorship, and then discuss what is known about AVTD behavior and Strong Cosmic Censorship in families of solutions defined by varying degrees of isometry, and discuss recent results which we believe will extend this knowledge and provide new support for Strong Cosmic Censorship. I also comment on some of the recent work on “Weak Null Singularities”, and how this relates to Strong Cosmic Censorship. 

(U Syracuse) 
A warped product version of the CheegerGromoll splitting theorem.  In this talk I'll discuss the extension of the theory of weighted Ricci curvature to the range of "negative" synthetic dimension. Specifically, we prove a new generalization of the CheegerGromoll splitting theorem where we obtain a warped product splitting under the existence of a line. 

(UNC) 
Uniform lower bounds for restrictions of QE eigenfunctions.  Lower bounds on just about anything are elusive! Sometimes we can say something  in this talk I will discuss Quantum Ergodic (QE) eigenfunctions on Riemannian surfaces. The main result is that the $L^2$ norm of restrictions of QE eigenfunctions to a curve with nonvanishing curvature are uniformly bounded below as the eigenvalue tends to infinity. The proof uses a brand new eigenfunction "energy" method, which measures the propagation of tangential mass of restricted eigenfunctions. Joint work with Y. Canzani and J. Toth. 

(UT Austin) 
Lie groups and beyond: KunzeStein phenomena and Riesz potentials.  Sharp forms of KunzeStein phenomena on SL(2,R) are obtained by using symmetrization and SteinWeiss potentials. A new structural proof with remarkable simplicity can be given on SL(2,R) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and ndimensional hyperbolic space through application of the Riesz Sobolev rearrangement inequality. Relatively simple arguments are given to obtain endpoint convolution inequalities that strengthen KunzeStein phenomena. 

(U Penn) 
On the Singular Yamabe Problem on Spheres. 
The solution to the Yamabe problem of finding a constant scalar curvature metric in a prescribed conformal class on a closed manifold was a major achievement in Geometric Analysis. Among several interesting generalizations to open manifolds, great attention has been devoted to the socalled "singular Yamabe problem". Given a closed Riemannian manifold M and a submanifold S, this problem consists of finding a complete metric on the complement of S in M that has constant scalar curvature and is conformal to the original metric. In other words, these are solutions to the Yamabe problem on M that blow up along S. A particularly interesting case is the one in which M is a round sphere and S is a great circle. In this talk, I will describe how bifurcation techniques and spectral theory of hyperbolic surfaces can be used to prove the existence of uncountably many nontrivial solutions to this problem. This is based on joint work with B. Santoro and P. Piccione. 

(JHU) 
On KakeyaNikodym type maximal inequalities. 
In this talk, I will discuss some recent results related to the KakeyaNikodym problem. The main result is that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on KakeyaNikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key ingredient is to reduce to a 2dimensional $L^2$ estimate with an auxiliary maximal function. A similar argument can be applied to show that the same $L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $(M^d,g)$ with constant curvature, which generalizes Sogge's results for $d=3$ to any $d\ge3$. As in the 3dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function. 

(Chinese U of Hong Kong) 
Desingularizing minimal surfaces.  In this talk, we will give a brief survey on the desingularizations of minimal surfaces pioneered by the groundbreaking work of Kapouleas. These are related to complete minimal surfaces in Euclidean space, closed minimal surfaces in the threesphere and selfshrinkers for mean curvature flow. We will discuss some of the fundamental ideas in these desingularizing constructions. At the end of the talk, I will talk about some recent joint work with Kapouleas on applying these ideas in the construction of free boundary minimal surfaces in the unit ball.  
(U Chicago) 
Harmonic map flow with singular targets.  In this talk we consider two heat flow problems from Euclidean domains into nonsmooth spaces. The first problem, a nonlocal constrained heat flow into a metric tree, was originally motivated by a stationary eigenvalue partition problem. We prove spatial Lipschitz regularity of this flow, free interface regularity, and characterize the limit of the flow as time goes to infinity as a stationary solution of the partition problem. Next, we discuss the extension of this study to nonconstrained heat flows into Fconnected simplicial complexes, which are natural generalizations of trees to higher dimensions.  
(Aalto U) 
Nonlocal selfimproving properties.  The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a selfimproving property that socalled reverse Hölder inequality implies. In the case of nonlocal equations a selfimproving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. .  
(Brown) 
Some weighted L_pestimates for Elliptic and parabolic equations.  I will discuss a generalized FeffermanStein theorem on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. The result is applied to establish mixednorm weighted $L_p$estimates for elliptic and parabolic equations with partially BMO coefficients in regular or irregular domains. This is based on joint work with Doyoon Kim (Korea University).  
(UT Austin) 
Hypersurfaces with almost constant mean curvature.  Motivated by capillarity theory we review some geometric inequalities in quantitative form and we address the geometry of boundaries with almost constant mean curvature. The latter is a joint work with Giulio Ciraolo (U. Palermo). 
Archive of Analysis Seminar: