In this talk, I will discuss several topics related to
transverse knots in contact 3-manifolds. I will introduce a
conjecture on the maximal self-linking number of a topological knot
in the standard contact 3-sphere. I will show how to apply
braid...
We formulate several new boundary value problems for the 2D
Navier-Stokes system. In all cases, we obtain quantitative decay
estimates of the Fourier modes for both the vorticity and the
velocity. In some special cases we found that the Fourier...
In this talk, I will first give an overview of recent progress
on the scattering conjecture and rigidity conjecture for mass
critical NLS. Then I will describe our recent results concerning
both conjectures for L_2 initial data. I will also discuss...
We construct new solutions to the $\sigma_k$-Yamabe problem on
compact manifolds by considering the connected sum of two
nondegenerate solutions $(M^n_1,g_1)$ and $(M^n_2,g_2)$, for any $2
\leq 2k < n$.
Dislocations are line defects in crystals, and can be modeled
using non-local operators. In this talk we will consider a related
reaction-diffusion equation with a half-Laplacian. We show that the
limit dynamics is characterized by a system of ODEs...
We study a class of fully nonlinear metric flow on K\"ahler
manifolds, which includes the J-flow as a special case. We provide
a sufficient and necessary condition for the long time convergence
of the flow, generalizing the result of Song-Weinkove...
The talk is devoted to the homogenization of a stochastic
evolution problem with non Lipschitz forcing. The problem is
considered in a sequence of perforated cylindrical domains obtained
from the removal of tiny cylinders from a fixed one (the...
The location of blowup points is often related to critical
points of Green function. In fact, Green function plays a important
role to understand the solutions structure of mean field equations.
We will show how to use the elliptic function and the...
A three dimensional strongly pseudoconvex Cauchy Riemann
manifold can be considered as a contact manifold with a conformal
structure on the hyperplane distribution. In this talk, we will
discuss CR structures near the standard one on S^3 , the...
