The Gauss-Bonnet theorem and the Cohn-Vossen inequality show that the only complete surface with positive curvature is either the sphere, RP2, or the plane. In higher dimension, the curvature tensor is far more complicated. There are several commonly used partial components of the curvature tensor that had been studied for the whole century. The simplest one is the scalar curvature which is the average of all curvatures at one point. It appeared in the Hilbert action for general relativity. This was studied extensively in connection with general relativity.

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

We construct a cohomology theory of transitive digraphs and use it to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

We propose a polynomial time computable graph invariant, which comes out directly from a modified version of discrete Greens function on a graph. It is our hope that it can help to construct fast algorithms for the graph isomorphism testing. We explain the physics behind this discrete Greens function, which is a very basic idea applied to graphs.

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime (Wang and Yau in Phys Rev Lett 102 (2): 021101, 2009; Commun Math Phys 288 (3): 919942, 2009), the anti-de Sitter spacetime (Chen et al. in Commun Anal Geom, 2016. arXiv: 1603.02975), or the Schwarzschild spacetime (Chen et al. in Adv Theor Math Phys 22 (1): 123, 2018). In each case, the reference spacetime admits a conformal KillingYano 2-form which facilitates the application of the Minkowski formula in Wang et al.(J Differ Geom 105 (2): 249290, 2017) to estimate the quasi-local energy. As a consequence of the positive mass theorems in Liu and Yau (J Am Math Soc 19 (1): 181204, 2006) and Shi and Tam (Class Quantum Gravity 24 (9): 23572366, 2007) and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.