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I would say Penrose is a mathematical physicist and I don't think he can be considered (at least not primarily) to be a pure mathematician. For example, his argument for the Penrose inequality is a …

I have noticed that in the literature on causality in general relativity one sees apparent counterexamples to the cosmic censorship hypothesis (somehow you have models for gravitational collapse which …

It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow. A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of …

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in pa …

A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which ar …

I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, …

Generally speaking, you could say they are a special type of solution to the field equations of gauge theories. More specifically, an instanton is a classical solution in a classical Euclidean field …

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the …

It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves). I …

I think third-order and fourth-order derivatives in particular are very common in both pure and applied mathematics. For example, I once worked with some equations modelling the local film thickness …

I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $R$, I can find a manifold such that the Ricci tensor has all zero components except for one …

Having thought about this more and discussed it with others, the answer seems to be that there are likely no counterexamples to the Penrose inequality, even if one allows for unphysical violations. Fo …

The Ricci flow deforms a Riemannian metric. I was wondering if there was something very similar which deforms a pseudo-Riemannian metric or if not, is there reason why such a geometric flow cannot ex …

I am reading a 2007 article of Bressler et al. on deformation quantization of gerbes. In the article, the authors state that a gerbe on a manifold is defined using certain two-cocycles $c_{ijk}$ b …

My personal suggestion is 'Differential Geometry, Gauge Theories, and Gravity' by M. Gockeler and T. Schucker. However, it assumes a fairly high degree of mathematical sophistication (it's one of the …