The Almgren-Pitts min-max theory is a Morse theoretical
type variational theory aiming at constructing unstable minimal
surfaces in a closed Riemannian manifold. In this talk, we will
survey recent progress along this direction. First, we will discuss
the understanding of the geometry of the classical Almgren-Pitts
min-max minimal surface with a focus on the Morse index problem.
Second, we will give an application of our results to quantitative
topology and metric geometry. Next, we will introduce the study of
the Morse indices for more general min-max minimal surfaces arising
from multi-parameter min-max constructions. Finally, we will
introduce a new min-max theory in the Gaussian probability space and
its application to the entropy conjecture in mean curvature flow.

We prove interior H ̈older estimates for the spatial gradient of vis- cosity solutions to the parabolic homogeneous p-Laplacian equation

ut = |∇u|2−pdiv(|∇u|p−2∇u),

where 1 < p < ∞. This equation arises from tug-of-war-like stochastic games with noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre. 

Do you have fond memories of your favorite college professor? What made their teaching memorable? What do you want your students to remember about your own way of teaching?  In this talk, we will share inspirations for good teaching in mathematics.

In the numerical simulation for fluid-structure problems, many different possible combinations of coordinate systems can be used for fluid and structure problems.   While Eulerian coordinate is almost always used for fluid problems,  either Eulerian or Lagrangian coordinate can be used for structure problems.   In this talk, I will discuss some mathematical and numerical issues for these different choices of coordinate systems.  In particular, I will report some convergence analysis for eXtended Finite Element Methods (XFEM) for Eulerian-Eulerian coordinates and the study of the well-posedness and preconditioning for finite element discretization based Arbitrary Lagrangian-Elerian (ALE) coordinates.  I will also present some numerical results for some practical applications such as hydroelectric power generator and abdominal aortic aneurysm.

Just as we study varieties by utilizing vector bundles over them, we
often study symplectic manifolds by utilizing holomorphic curves.
While holomorphic curves are by far the most useful tool in
symplectic geometry, the analytical details can often be a
bottleneck. In this talk, we'll talk about how the most computable
cases of holomorphic curve theory may conjecturally be recovered by
purely topological (i.e., non-analytical) means---namely, through the
algebraic structure inherent in cobordisms. As an example theorem, we
will show that if two exact closed Lagrangians submanifolds are
related by an exact Lagrangian cobordism, then their Floer theories
are identical in a very strong sense.