A Hilbert complex is a sequence of Hilbert spaces connected by a sequence of closed densely defined linear operators satisfying the property: the composition of two consecutive maps is zero. The most well-known example is the de Rham complex involving grad, curl, and div operators. A finite element complex is a discretization of a Hilbert complex by replacing infinite dimensional Hilbert spaces by finite dimensional subspaces based on a mesh of the domain. Usually inside each element of the mesh, polynomial spaces are used and suitable degree of freedoms are proposed to glue them to form a conforming subspace. The finite element de Rham complexes are well understood and can be derived from the framework Finite Element Exterior Calculus (FEEC). 

In this talk, we will survey the construction of finite element complexes. We present finite element de Rham complex by a geometric decomposition approach. We then generalize the construction to smooth FE de Rham complexes and derive more complexes including the Hessian complex, the elasticity complex, and the divdiv complex in two dimensions by the Bernstein-Gelfand-Gelfand (BGG) construction. 

We also present a direct approach for constructing finite element tensors. We construct polynomial complexes and Koszul type complexes, which leads to decompositions of polynomial spaces. We then characterize trace operators using Green’s identity. We construct conforming finite elements for tensor functions with extra constraint.

The constructed finite element complexes will have application in the numerical simulation of the biharmonic equation, the linear elasticity, the general relativity, and in general PDEs in Riemannian geometry etc.

This is a joint work with Xuehai Huang from Shanghai University of Finance and Economics. 

The zoom Link to the meeting is

https://uci.zoom.us/j/98420908393

I will discuss the current status of understanding for the geometry and constructions of minimal surface doublings. In particular I will discuss in more detail results related to the Linearized Doubling (LD) approach (Kapouleas: JDG 2017, Kapouleas-McGrath: CPAM 2019, and Kapouleas-McGrath: Camb. J. Math. (to appear); arXiv:2001.04240v3) and some ongoing work.

One special feature for the Ricci flow in dimension 3 is the
Hamilton-Ivey estimate. The curvature pinching estimate provides a lot of
information about the ancient solution and plays a crucial role in the
singularity formation of the flow in dimension 3. We study the pinching
estimate on 3 dimensional expanding and 4 dimensional steady gradient Ricci
solitons. A sufficient condition for a 3-dimensional expanding soliton to
have positive curvature is established. This condition is satisfied by a
large class of conical expanders. As an application, we show that any
3-dimensional gradient Ricci expander C^2 asymptotic to certain cones is
rotationally symmetric. We also prove that the norm of the curvature tensor
is bounded by the scalar curvature on 4 dimensional non Ricci flat steady
soliton singularity model and derive a quantitative lower bound of the
curvature operator for 4-dimensional steady solitons with linear scalar
curvature decay and proper potential function. This talk is based on a joint
work with Zilu Ma and Yongjia Zhang.

A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitely-many diffeomorphism types.

Asymptotically conical Calabi-Yau manifolds are non-compact Ricci-flat Kähler manifolds that are modeled on a Ricci-flat Kähler cone at infinity. I will present a classification result for such manifolds. This is joint work with Hans-Joachim Hein (Fordham/Muenster).