We introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in space-time or, equivalently, foliations of a null cone. We also show how certain convexity assumptions along a foliation of a null cone give rise to the Penrose inequality. For monotonicity from a MOTS, one requires these convexity assumptions be sharp, so we explore existence within a Weakly Isolated Horizon. Building on a lemma of S. Alexakis, this allows us to show that the Penrose Inequality is stable under small perturbations of the standard Schwarzschild null cone.

TBA

In 2013, Gromov proposed a dihedral rigidity conjecture, aiming at establishing a geometric comparison theory for metrics with positive scalar curvature. The conjecture states that if a Riemannian polyhedron has nonnegative scalar curvature in the interior, and weakly mean convex faces, then the dihedral angle between adjacent faces cannot be everywhere less than the corresponding Euclidean model. I will prove this conjecture for a large collection of polytopes. The strategy is to relate this conjecture with a geometric variational problem of capillary type, and apply the Schoen-Yau minimal slicing technique for manifolds with boundary.

In recent works with N. Wickramasekera we develop a regularity and compactness theory for stable hypersurfaces (more precisely, integral varifolds) whose generalized mean curvature is prescribed by a (smooth enough) function on the ambient Riemannian manifold. I will describe the relevance of the theory to analytic and geometric problems, and some GMT and PDE challenges.

During the talk I will introduce renormalized volume for convex co-compact hyperbolic 3-manifolds and will also describe bounds for Schottky manifolds in term of extremal lengths in the conformal surface at infinity. This will be used to partially answer a question by Maldacena about comparing renormalized volume for Schottky and Fuchsian manifolds with the same conformal boundaries.

On a complex manifold, the (0,1)-part of the Chern connection corresponds to the Cauchy-Riemann operator on the tangent bundle. In this talk we compare the Chern scalar curvature to the Riemannian one. Moreover, we study an analog of the Yamabe problem by looking for an almost-Hermitian metric with constant Chern scalar curvature in a conformal class. We also study the Chern-Yamabe flow and get convergence to a solution in certain cases. These are joint works with Markus Upmeier and Ali Maalaoui.

The goal of this talk is to give a characterization of hyperbolic space using a geometric invariant that is defined for a class of manifolds asymptotic to hyperbolic space. This result is inspired by a characterization of Euclidean space called the rigidity of the Riemannian positive mass theorem in the context of general relativity. We discuss the history and context of our result, as well as some of the interesting elements of its proof

TThe asymptotic expansion of the complete K-E metric on the punctured sphere is the dimension one case of the asymptotic expansion on a quasi-projective manifold M\D, which was proposed by S. T. Yau. Several people have worked on this problem by using techniques from partial differential equations. In this talk, I will use analytic properties of the covering map, the Schwarzian derivative and the modular form to derive a precise asymptotic expansion on the punctured sphere. More precisely, the coefficients in the expansion will be uniquely determined up to two parameters. In a recent work joint with G. Cho, the Kobayashi-Royden metric can be given as a consequence.

The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics.
In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question.

There are 1-dimensional models inspired by equations of fluid mechanic
which exhibit quite interesting behavior. Lesson learned from some of
the 1d models can be important for the original equations. We will
discuss a few examples, focusing mostly on the De Gregorio
modification of the Constantin-Lax-Majda mod

Reaction-diffusion systems have been primary tools for studying pattern formation. A skew-gradient system generalizes an important class of activator-inhibitor type reaction-diffusion systems that exhibit localized patterns such as fronts and pulses. In this talk we study the standing pulse solutions to a skew-gradient system of the form
\[ u_t = du_{xx} + f(u) - v, \quad \quad v_t= v_{xx} - \gamma v - v^3 + u,\]
on the domain $(-\infty, \infty)$. Using a variational approach, we establish the existence of standing pulse solutions with a sign change. In addition, we explore some qualitative properties of the standing pulse solutions.

A convex real projective structure on a surface is given by writing it a quotient of a convex domain D in the real projective plane by a representation of the fundamental group into PGL(3,R) which acts properly discontinuously on D. The geometry is determined an invariant convex surface in the cone over D, a hyperbolic affine sphere H, as studied by Cheng-Yau and others. H in turn is determined by two elliptic PDEs, first by a real Monge-Ampere equation on D whose solution determines H directly, and second by a semilinear equation on S in terms of a conformal structure and holomorphic cubic differential C. This semilinear equation serves as an integrability condition for H with an appropriate initial frame. C=0 is equivalent to H being a hyperboloid and the real projective structure being Fuchsian under the Klein model of hyperbolic space.

We will use these cubic differentials to explain the moduli space of convex real projective structures as a vector bundle over the moduli space of Riemann surfaces (this is also independently due to Labourie), and also to extend this picture to the boundary of the Deligne-Mumford compactification of moduli space.

On symplectic manifolds, there is a novel A-infinity algebra of
differential forms constructed by Tsai-Tseng-Yau. We will review this and
introduce a twist to this algebra, motivated by consideration of fiber
bundles over symplectic manifolds and sections on them. Interestingly,
the twist leads to a natural flatness condition for such bundles. The
flatness condition is in general a non-linear PDE system dependent on the
symplectic structure, and we will describe a few well-known solutions of
this system. This talk is based on a joint work with Jiawei Zhou.

n this talk, we will discuss the uniqueness of K-polystable log Fano cone degeneration of
any K-semistable log Fano cone. This confirms a conjecture of Donaldson-Sun stating that
the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kaehler-
Einstein Fano manifolds depends only on the algebraic structure of the singularity.
(this is a joint work with Chi Li and Chenyang Xu)