# A Sharp Li-Yau gradient bound on Compact Manifolds

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We present a recent result showing that a sharp Li-Yau gradient bound for positive solutions of the heat equation holds for all compact manifolds. However, no sharp Li-Yau bound holds for all noncompact manifolds. This answers an open question by a number of people.

# Finite Element Complexes

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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

# Positive Scalar Curvature and the Einstein Constraint Equations

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I’ll describe some recent results on the geometry and topology of manifolds with positive scalar curvature (PSC), and the related subject of the constraint equations of general relativity. I’ll cover a number of results which generalize various classics : Schoen-Yau’ 1979 topological obstructions to PSC, Schoen-Yau’s 1979 Positive Mass Theorem, and the Density Theorem of Corvino-Schoen 2005. In dimensions less than 8, these results settle the positive mass with arbitrary ends conjecture made in Schoen-Yau 1988, and the positive mass theorem for initial data sets with boundary (the case without boundary was resolved by Eichmair-Huang-Lee-Schoen 2015, Huang-Lee 2017, Eichmair 2013). Time permitting, I’ll discuss the proof of one of these in more detail. This is based on four papers, two with R.Unger and S.T.Yau, and two with D.Lee and R.Unger.

# Recent and ongoing work on minimal doublings and related topics

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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.

# A nonlinear spectrum on closed manifolds

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Abstract: The p-widths of a closed Riemannian manifold are a nonlinear

analogue of the spectrum of its Laplace--Beltrami operator, which was

defined by Gromov in the 1980s and corresponds to areas of a certain

min-max sequence of hypersurfaces. By a recent theorem of

Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like

the eigenvalues do. However, even though eigenvalues are explicitly

computable for many manifolds, there had previously not been any >=

2-dimensional manifold for which all the p-widths are known. In recent

joint work with Otis Chodosh, we found all p-widths on the round

2-sphere and thus the previously unknown Liokumovich--Marques--Neves

Weyl law constant in dimension 2.

# Hamilton-Ivey estimates for gradient Ricci solitons

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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.

# Geometric Flows of G2-Structures

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There are several natural flows of G2-Structures that may be useful for resolving important problems in the theory of G2-Holonomy Manifolds. Here we review what is known and discuss some open problems for two such flows: The Laplacian flow and the Laplacian co-flow. We will then present some new results for each of these flows.

# Degeneration of 7-dimensional minimal hypersurfaces with bounded index

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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.

# Rigidity and instability of SU(n) symmetric spaces

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Focusing on the Lie group SU(n) and associated symmetric spaces,

I investigate two related topics. The main result is that the bi-invariant

Einstein metric on SU(2n+1) is isolated in the moduli space of Einstein

metrics, even though it admits infinitesimal deformations. This gives a

non-Kaehler, non-product example of this phenomenon adding to the famous

example of Koiso from the eighties. I also explore the relationship between

the question of rigidity and instability (under the Ricci flow) of Einstein

metrics, and present results in this direction for complex Grassmannians.