Explicit moduli for genus 2 surfaces
I will explicitly describe the Teichmüller and moduli spaces for closed surfaces of genus 2, following the path suggested by Rivin, Leibon, and Springborn: Compactified Teichmüller space is tiled by copies of 10 explicit (relatively simple) 6-dimensional polyhedra, each parametrizing the possible realizations in hyperbolic geometry of a Delauney triangulation/circle pattern with one of 10 underlying graphs. Coordinates for the for the polyhedra allow the surface to be reconstructed as a hyperbolic surface .....
This answers, in the genus 2 case, questions raised by Sullivan and Witten in recent years: that Weierstraß points may help to describe moduli for closed surfaces ....
This is the first explicit cell decomposition of the compactified moduli space of any closed hyperbolic surface....

Second order structure for integral varifolds of locally bounded first variation
The main result is the existence of an approximate second order structure for integral varifolds of locally bounded first variation. The proof relies on a new differentiability criterion for functions in Lebesgue spaces phrased in terms of approximability by harmonic functions. Throughout the talk concepts from geometric measure theory will be illustrated by many examples.

Mean Curvature Flow of Entire Graphs
We study mean curvature flow of entire graphs in Euclidean space, in particular the work of Ecker and Huisken, who have shown that given some initial growth condition at infinity, such graphs become self-similar under the evolution and this convergence is exponential in time. In this talk we will motivate this condition at infinity and introduce a new logarithmic condition and show how it leads to a slower polynomial rate of convergence.

Horizons of Prescribed intrinsic and extrinsic Geometry
In this talk we discuss how to construct asymptotically flat time symmetric initial data in general relativity with one horizon of prescribed intrinsic and extrinsic geometry. Given a metric $h$ on $\Sphere^2$ together with a trace free second rank tensor $\chi$, the main theorem gives a sufficient, and `almost' necessary condition on $(h,\chi)$ for when $(\Sphere^2,h)$ can be realized as a horizon with second fundamental form $\chi$.

Some blow-up and quenching problems on semilinear parabolic equations
In this talk, we discuss two kinds of parabolic problems, i.e., blow-up and quenching problems. Firstly, a quasi-monotonicity formula for the solution to a semilinear parabolic equation is obtained. As an application, we establish the partial reguarity theory for this problem. And we show the positive borderline solution will blow up in finite time as long as the domain is convex and $p>(n+2)/(n-2)$. Secondly, the universal quenching of a porous medium equation is established. We also obtain some universal quenching properties for some semilinear parabolic problems and dichotomy properties for some general one-dimensional problems.

Rotationally symmetric surfaces evolving under mean curvature flow
In this talk, we consider mean curvature flow of immersed surfaces in 4-dimensional euclidean space which are generated via rotating closed curves such that the surfaces are invariant under all rotations in the $x^1x^4$-plane. The flow of the surfaces leads to a flow for the generating curves which do not evolve under the mean curvature flow but under a perturbed one. The singular behaviour of the perturbed flow is analyzed in the case that the curvature of the curves blows up with the same rate as the mean curvature of the surfaces. In this setting, we show that after rescaling along an essential blow-up sequence the generating curves (and therefore the surfaces) converge along an subsequence to some nonempty limit. If the singularity occurs away from the plane of rotation, each limit of the rescaled curves is a family of convex planar curves evolving under mean curvature flow.

Canonical solitons and Harnack inequalities for the Ricci Flow
Given a Ricci flow on a manifold M  over a time interval I, we introduce a second time parameter, and define  gradient Ricci solitons on the space-time M x I.
We shall explain how  the theory of optimal transport gives us the geometric intuition for the construction of the Canonical Shrinking Soliton, which encodes various of the monotonic quantities that underpin Perelman's work on Ricci flow. About the expanding case, we will show how to use it to produce different Harnack inequalities as simple curvature conditions on the space-time soliton.

The non-linear Plateau Problem in Hadamard Manifolds:
We prove the existence of hypersurfaces of constant special Lagrangian curvature with certain prescribed properties inside manifolds of negative sectional curvature.