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::Here is as precise a definition of Perelman entropy as exists for a non-geometer: Ricci flow on two manifolds which are basically the same (ie, diffeomorphic) can still be different. Perelman discovered a similar flow on manifolds which depends on not only the geometry of the manifold, but some other function (it can be just about any function). While we can create many different flows of this sort by altering that other function, there is a diffeomorphism of the manifold for which any flow you can think of will be identical to the Ricci flow. Perelman further modified this flow to be scale invariant, and called this discovery "Perelman entropy."<ref>Actually, he didn't name it. It is common for mathematicians to leave their discoveries unnamed, and for the next paper on the topic to refer to that discovery by the name of the discoverer.</ref>::{{reflist}}