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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 …

Answer to (2) is negative. For a counterexample, consider a $2$-sphere and blow up the radius. The isoperimetric ratio $C_H$ remains the same, but the Cheeger constant $h$ shrinks, essentially becau …

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 …

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 …

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 …

Not a survey per se, but a good article on the theorem is the one below which gives a quick, clear sketch of the second part of Schoen and Yau's proof of the positive mass theorem. It might be helpful …

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 …

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 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 …

Yes, these pictures have now been made rigorous. Another paper which you might be interested in on this topic is Simon, M. (2000). A class of Riemannian manifolds that pinch when evolved by Ricci flo …

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, …

In general relativity one has the Schwarzchild metric for a non-rotating black hole $g_{SC} = -\phi^2 \: dt^2 + \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $ and from this one has the spacelike Schwarzc …

I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system i …

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 …

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 …