In this talk, I will report a recent joint work with Mu-Tao Wang. We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones are obstructions to the existence problems of special Lagragians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow, and thus provide a canonical way to resolve the union of two such cone singularities. Our theorem is analogus to the Feldman-Ilmanen-Knopf gluing construction for the K\"ahler-Ricci flows.

I'll discuss the papers "Berger's isoperimetric problem and minimal
immersions of surfaces" by Nadirashvili, "Extremal metric for the first
eigenvalue on a Klein Bottle" by Jakobson, Nadirashvili and Polterovich
and "How large can the first eigenvalue be on a surface of genus two?" by
Jakobson, Levitin, Nadirashvili, Nigam and Polterovich. The authors find
among metrics of fixed area one that maximizes the fundamental frequency
of the torus and Klein Bottle.

Whether the 3D incompressible Euler or Navier-Stokes equations
can develop a finite time singularity from smooth initial data has been
an outstanding open problem. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equations.
We show that the local geometric properties of vortex filaments can lead
to dynamic depletion of vortex stretching, thus avoid finite time blowup
of the 3D Euler equations. Further, we perform large scale computations of
the 3D Euler equations to re-examine the two slightly perturbed anti-parallel
vortex tubes which is considered as one of the most attractive candidates
for finite time blowup of the 3D Euler equations. We found that there is
tremendous dynamic depletion of vortex stretching and the maximum vorticity
does not grow faster than double exponential in time. Finally, we present
a new class of solutions for the 3D Euler and Navier-Stokes equations,
which exhibit very interesting dynamic growth property. By exploiting
the special nonlinear structure of the equations, we can prove the global
regularity of this class of solutions.