Epstein Secrets

l structure. "Some areas of mathematics are very rich in structure," says Peter Shalen, a topologist a

tics, only that it's still unproved. Late last year, after Colin Rourke, an English mathematician, became the latest victim of what's known as the Poincare conjecture, Dave Gabai, of Caltech, explained the obsession. "For a mathematician," said Gabai, who has a reputation for cracking seemingly unsolva

Discover: July 1987 What happens when hubris meets nemesis By Gary Taubes I have committed the sin of falsely proving Poincare's conjecture. But that was another country; and besides, until now no one has known about it. -- John Stallings Conjectures are what mathematician

for the last two centuries. Almost all the other rich countries have followed a different path. In the

for the motions of the bodies. Even in a three body system the motions of the bodies become chaotic and unpredictable at a detailed level. In 1890 Poincare demonstrated that it is actually impossible to solve the equations for a three body system in a simple field system, so even a system as simple as t

and accurate enough measurement today, the future course of the universe could be predicted. For mathematicians, that dictum was dashed in 1899 by Poincare's proof of the existence of chaos. PoincarE showed that not only was it impossible to derive a formula which could predict 165 EFTA00625293 the

ron was by Gerstein and Mandelbrot (1964). Though rich with possibilities, it has been only very recentl

1864 Steriade, M., McCarley, R.W. (1990) Brainstem Control of Wakefulness and Sleep. Plenum. N.Y. 289 HOUSE_OVERSIGHT_013789 Stillwell, J. (1985) Poincare’s Papers on Fuchsian Groups. Springer-Verlag, N.Y. pp 1-50 Stockbridge,L.L., French,A.S. (1989) Characterization of a calcium-activated potassium ch

ormly divergent (and convergent), “hyperbolic” chaotic systems, such as brain systems, behave in Euclidean space, an intuitive similarity about which Poincare experienced his famous vacation bus trip epiphany (Stillwell, 1985). It should also be noted that encoding neural spike trains in one dimension for s

table. Rigid things can more easily fracture. The rich, altered states of consciousness that emerge whil

context of mathematical epistemology, the instantaneous images of a geometer contrast with the labored sequential logic of the mathematical analyst. Poincare’ claimed that inclinations toward one or the other of these two cognitive styles and their associated mathematical tools arise from different kinds o

stical measures, Hr, Hu, AC and srs? The yield is rich and unexpected. We find an enjoining of values of

blished a model of it through internal reconstructions of sequential sensory experiences that_accompanied our exploratory movements. A world. It was Poincare’s habit to topologize the dynamics of motion in mathematical problems that lacked analytic solutions. In this way, simple algebraic operations repla