Publication of the week: Dr Stuart Hall

Hall, S., “Perelman’s entropy for some families of canonical metrics”, Experimental Mathematics 23.3 (2014), 277-284. DOI:10.1080/10586458.2014.898220.

In the early 1980s a mathematician called Richard Hamilton showed that one could treat the curvature of abstract mathematical spaces (things called manifolds) a bit like heat distributed about the space.  He showed how you could ‘cool down’ the curvature via a partial differential equation called the Ricci flow. The steady state solutions are called Ricci solitons and have solutions of Einstein’s equations as an important subset.  In 2003 Grigori Perelman found an entropy-like quantity for this flow which (as entropy always increases under the flow) allowed mathematicians to order the geometries that manifolds admit.

In this paper the entropy is calculated for manifolds that admit families of Ricci solitons.  This gives a vague idea as to how the geometry of these spaces is evolving under the Ricci flow.  One interesting consequence of the results found here is that one family of solutions to the Einstein equations called warped-product metrics appear to have higher entropy than ordinary products.  This is a counter-intuitive result and suggests interesting further directions to pursue.

The journal Experimental Mathematics was founded in 1992 by David Epstein of Warwick University.  It is devoted to publishing mathematics that supports hypotheses and conjectures rather than necessarily proofs of big results.  This paper can be viewed as a few experiments with the Ricci flow and Perelman’s entropy that suggest further areas of study.

An arxiv preprint of the paper is available at http://arxiv.org/abs/1402.5625.

Stuart Hall is a research lecturer in mathematics based in the Department of Applied Computing at Buckingham.