Isoperimetric properties of the mean curvature flow

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self evident. The first is a genuine, 5-lines proof, for the isoperimetric inequality for $k$-cycles in $\mathbb{R}^n$, with a constant differing from the optimal constant by a factor of only $\sqrt{k}$, as opposed to a factor of $k^k$ produced by all of the other soft methods (like Michael-Simon's or Gromov's). The second is a 3-lines proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of a paper of Giga and Yama-uchi. We then turn to use the above mentioned relation to prove a bound on the parabolic Hausdorff measure of the space time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon Isoperimetric inequality. To prove it, we are lead to study the geometric measure theory of Euclidean rectifiable sets in parabolic space, and prove a co-area formula in that setting. This formula, the proof of which occupies most this paper, may be of independent interes

Religion, religious fervour, and universalist education

This paper is conceived from a secular perspective, and designed to address three elements identified in the call for papers: “Pluralistic tendencies”, their counterpart of “exclusivist attitudes”, and “creating an ethos of inter-religious harmony”. I choose to tackle these aspects by (a) exploring the meaning of religion, (b) addressing a specific attitude often corresponding to religion, namely religious fervour, and (c) assessing the validity and instrumentality of facilitating a universalist education as a tool to defuse “mistrust and hatred among various faith-communities”. The following paper is intended to serve only as a preliminary discussion guidance paper