it's a conspiracy.

it's been said that the last mathematical universalist was J. Henri Poincaré, and he died sometime early last century. since then, mathematicians have demonstrated the tendency to specialise in their own fields and then subfields and then (sub)²fields ..

but occasionally, mathematics appears as a conspiracy. there are too many coincidences ..

.. and i don't mean this upcoming film called "the number 23" with jim carrey (see the trailer @ apple.com), which looks really atrocious.

more precisely, i mean how concepts are related.

for example, earlier this summer i was browsing through c. villani's book, topics on optimal transportation because i had heard in a lecture, years ago @ Carnegie Mellon University, of a way to perceive the Wasser$tein space of measures on a metric (length?) space as having some sort of Riem@nnian structure.

the analogy is initially formal, though in the case of euclidean space there is a way of discussing this rigorously, through a complicated construction of gradient flows.

so i decided that this is really a formal thing, and not much can be done. of course, i was wrong.

lott and villani have recently worked on optimal transport and Ricci curvature (in the aleksandrov sense), and lott has even made sense of a Riemannian connection and curvature of the Wasserstein space, in the case of a compact smooth manifold.

as i may have said once, i'm perfectly willing to be both happy and wrong. the days when i am right are the pessimistic days, when the worst-case scenario actually happens.

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the other case at hand involves the notion of 'concentration of measure' which i learned from hearing lectures a year or two ago in a class @ um. the ideas were wholly nonintuitive yet intriguing, and it was really quite something!

but i thought that it would simply be an idle pursuit, that i'd never actually see it used in my field of interest. again, i was wrong.

more, lott-villani have just recently related notions from the 'concentration of measure' phenomenon to local/global Poincaré inequalities on length spaces -- the latter being the 'bread and butter' to metric analysts like myself.

amazing. if i didn't have to research and write a thesis, i'd love to explore these notions and see what can be said. but alas .. if i want to finish in five years .. \: