I will discuss two important objects from geometric measure theory: rectifiable varifolds and currents. Rectifiable varifolds and currents can be used to study geometric problems such as the Plateau Problem. I will describe the basic properties of rectifiable varifolds and currents by relating them to the properties of C1 submanifolds and by using examples. This talk is intended to be accessible to those unfamiliar with geometric measure theory.

Area exam practice talk.

I will first introduce the concept of initial data sets in general relativity. An initial data set which is asymptotically flat (AF) at infinity is a model of an isolated system. We will discuss the concept of center of mass in AF manifolds via both Hamiltonian and geometric aspects. Then the rest of talk is devoted to the geometric aspect: the existence and uniqueness of stable spheres with constant mean curvature at infinity.
Whether a constant mean curvature foliation uniquely exists in an AF manifold was a question proposed by Yau in 80's. For manifolds with special asymptotics, some results have been obtained by Huisken-Yau, Ye, and Metzger. We generalize their results to AF manifolds with the parity condition at infinity. It is known that the Hamiltonian notion of center of mass is defined for those manifolds. Moreover, we show the geometric center of the foliation is indeed center of mass, so the foliation provides a geometric description of center of mass.

Abstract: Minimal surfaces have been studied for a long time. In fact, one of the first fields medals was awarded for a solution to the Plateau problem: finding area minimizing disk with given boundary. Since then, minimal surface theory has continually undergone substantial development, yet, many fundamental questions remain unanswered. In this talk, I will focus on the existence theory of minimal surfaces. Starting with the classic Douglas-Rado solution to the Plateau problem, I will describe some of the developments in the 70's-80's, where GMT (Geometric Measure Theory) came into play. In particular, I would give a sketch of proof that "any closed 3-manifold admits an embedded smooth minimal surface". This uses some tools from GMT, on which Brian has already given a excellent introduction. Other than that, I would try to make my talk as self-contained as possible.

The well-known uniformization theorem for Riemann surfaces asserts that a complete simply-connected Riemann surface is biholomorphic to either (i) the Riemann sphere (when M is compact), (ii) the complex plane (when M is noncompact and the curvature is positive) or (iii) the open unit disc (when M is noncompact and the curvature is negative). On natural question to ask is whether we can generalize this beautiful result to higher dimensional Kahler manifolds. It is by no mean an easy task, and has yet not been fully settled. In the compact case, some notable results include the Frankel's conjecture proven by Siu-Yau and Mori in early 1980s, and a splitting theorem proven by Mok in late 1980s. Recently there are some fruitful results done by Chau, Ni, Shi, Tam et al which discuss the noncompact case.

In this talk, I will give a survey of what have been done so far towards this big project in the case of non-negative bisectional curvature. To make the talk to be as self-contained as possible, I will first discuss some basic Kahler geometry relevant to the uniformization problem. Then I will outline the proofs of the Frankel's conjecture as well as Mok's splitting result. If time permits, I will also talk about some recent works done on the noncompact case. It will be a general talk and I promise to minimize the technicalities.

The proof of the Yamabe Problem establishes that for any compact Riemannian manifold $(M,g)$, there exists a conformal metric $\bar{g}$ with constant scalar curvature. The sign of this constant is a conformal invariant.
Metrics for which this invariant is nonpositive admit a unique solution to the Yamabe problem. For Yamabe-positive metrics, it is known that conformal classes containing multiple metrics with scalar curvature equal to $1$ exist, and authors have studied the strongest topology on metrics where conformal classes with multiple solutions are dense.
We show that given a smooth, Yamabe-positive metric $(M,g)$ with dimension $n \ge 9$, there exists a smooth metric $\tilde{g}$ which is arbitrarily close in the $C^{1,\alpha}$ topology whose conformal class contains an arbitrary number of distinct metrics with constant scalar curvature equal to $1$. If we assume, in addition, that $(M,g)$ is locally conformally flat, we may take $\tilde{g}$ to be close to $g$ in the $C^s$ topology for any $s<\frac{n}{2}$.
These results generalize, in dimensions $n \ge 9$, earlier results of Ambrosetti, Ambrosetti and Malchiodi, Berti and Malchiodi, and Pollack. Our proof constructs parameterized perturbations of an explicit approximate solution. The conformal class containing the constant scalar curvature metrics is obtained in this manner, and so has a well-understood geometry.

We study the method of Method of Modulus of Continuity engineered by Kiselev, Nazarov and Volberg in 2006 which led to the breakthrough in showing the existence of global solution to the 2-D critical Quasi-geostrophic Equation which is a toy model for the 3-D Navier-Stokes Equation.

Lax-Phillips scattering theory is concerned with, among other things, translation representations of the wave group of a Hamiltonian. In this talk, I will present work of Friedlander (1980 and 2001) and S{\'a} Barreto (2003) that constructs an explicit translation representation of the wave group on asymptotically Euclidean manifolds. This representation is known as a radiation field and is found as a restriction of the fundamental solution of the wave equation to an appropriate partial compactification of the manifold.

Let M be a compact, smooth Riemannian manifold. Varadhan proved that $t\log*p_t(x,y)$, where p is heat kernel, converges to -E(x,y) where E(x,y) is an energy function defined by distance between x and y. Malliavin and Stroock showed that under some condition, the hessian is asymptotic to -1/t times the variance of a random variable as t goes to zero. Neel more recently described a method to compute the hessian as simply an integral over the set of midpoints of a minimal geodesic from x to y. I will determine such set and compute the hessian of several symmetric spaces such as sphere, projective spaces and Lie Groups.

To study singularities of area minimizing submanifolds, one often studies the tangent cones to the submanifold. Tangent cones are defined a the limit of sequences of blow-ups of the submanifolds at a point on the submanifolds. The existence of tangent cones is a well-known consequence of compactness theorems; however, its still unknown whether the tangent cones are unique independent of the sequence of blow-ups. There are proofs of the uniques of tangent cones in some special cases, such as a proof by Brian White of the uniques of tangent cones to surfaces using the epiperimetric inequality. The natural way to discuss these tangent cones is from the perspective of geometric measure theory, so I will introduce some of the basic concepts from geometric measure theory. I will then discuss the key ideas from Brian White's proof.

A Kahler metric is called Kahler-Einstein if the Ricci tensor is a real multiple of the metric. Existence of such a metric has been a long-standing problem in complex geometry. The case where the first Chern class is nonpositive was settled by Aubin and Yau in 1976, and later in 1985, H.D. Cao provided another proof for the negative case using Kahler-Ricci flow. In this talk, I will introduce some basic Ricci flow on Kahler manifolds and I will go through Cao's proof for the existence theorem.

I would like present part of the Yamabe problem paper by John M. Lee and Thomas H.Parker. The Yamabe problem : Let $(M_n,g)$ be a $C^\infty$ compact Riemannian manifold of dimension $n\geq 3$, $S$ its scalar curvature. We would like to know if there is a metric conformal to $g$ with constant scalar curvature? I will present the result by Yamabe, Trudinger, Aubin which establishes that the Yamabe problem can be solved on any compact manifold $M$ with $\lambda(M) ‹ \lambda(S^n)$, where $S^n$ is the sphere with its standard metric and $\lambda(M)$ being the Yamabe invariant, which will be defined in the talk.

It is a beautiful theorem of Chazarain and Duistermaat-Guillemin that, on a compact Riemannian manifold, the singular support of the wave trace is contained in the length spectrum. I will explain what these words mean and give an idea of what the proof entails. I will do my best to avoid rigorously proving anything.

The asymptotically flat spacetime is an important model in general relativity which represents an isolated system. I will first review some basic concepts in general relativity and then talk about one notion to define asymptotically flat spacetimes using conformal compactification. If time allows, I will discuss a couple of notions of energy for asymptotically flat spacetimes.

Characteristic classes are topological invariants of smooth manifolds (vector bundles even). I'll motivate the role they should play and describe one/possibly two, explicit differential-geometric constructions of them in terms of curvature/self-intersection respectively. One example of a characteristic class measures the existence of a non-vanishing vector field thus having something to do with euler characteristic and a generalized Gauss-Bonet theorem. Another example measures the existence of an orientation.

More than 80 years ago, Caratheodory proposed a conjecture on the number of umbilic points on a convex surface in R^3. This was proven between 1940-59 by different authors for surfaces which are strictly convex and real analytic. The proof was very hard and it seems that rigor was varying. Very recently, Guilfoyle and Klingenberg announced a proof which deals with the most general case. They use various modern techniques, like mean curvature flow in spacetime, neutral Kahler metrics and J-holomorphic disk, in their proof. In this talk, I will try to describe an overview of their plan towards this conjecture, skipping ALL the details. This paper is just half a year ago so it may not even be correct. Nonetheless, I think their ideas behind are so brilliant that I can't resist telling you guys, even though I do not really know many of the details myself.

Everyone knows there are eight three dimensional geometries. There's spherical, euclidean, hyperbolic, and well, I guess there's gotta be $S^2 \times \mathbb{R}$ and $H^2 \times \mathbb{R}$, and... what are the rest of them? The remaining three geometries may not be as well known, but they're important too! In this talk we'll get to know these less popular geometries and some manifolds that exhibit them. We might also explain why there aren't any other model geometries.

Even though the eigenvalues of the Laplacian on a Riemannian manifold approach infinity, there is still a way to define a "determinant" of the Laplacian. I'll talk about this method, which uses zeta functions (though not in a scary way). Then I hope to sketch a proof of a result, due to Osgood, Phillips and Sarnak, that states: in the case of surfaces, among all metrics in a given conformal class and of a given area, the uniform metric(s) has maximum determinant. If time permits, I'll mention some other cool results of those three authors in this area.