Vela wrote:So who is a math head here? I stumbled across a presentation of hyperspheres and the poincare conjecture midweek, and it sounds absolutely fascinating and completely beyond my grasp. Has anyone got any book recommendations that cover it? I've seen one by an author donal o'shea but there's nothing between that and the proceedings of a meeting where the proof was presented.
acemuzzy wrote:There's a maths thread, fwiw. Not immediately sure of book recommendations.
SpaceGazelle wrote:Dat Einstein guy was a clever cookie wasn't he?
WorKid wrote:Nice link. Ta Gaz.
Vela wrote:So who is a math head here? I stumbled across a presentation of hyperspheres and the poincare conjecture midweek, and it sounds absolutely fascinating and completely beyond my grasp. Has anyone got any book recommendations that cover it? I've seen one by an author donal o'shea but there's nothing between that and the proceedings of a meeting where the proof was presented.There's a maths thread, fwiw. Not immediately sure of book recommendations.Rightyo. I've finished reading and mulling over the Donal O'Shea book on the Poincare Conjecture and how it was proven by Grigory Perelman. A very interesting book because it delves into the history of the subject, and the author takes care to describe and define concepts such as manifolds versus shapes, and n-dimensional surfaces. The idea of a closed, simply connected manifold in 3 dimensions is basically what the conjecture was about - namely whether or not the 3-sphere is the only one. Turns out it is. The book closes with an argument that it is quite possible the universe (the total universe, not just the observable one) is a 3-sphere. But the theory also holds that it is possible for higher dimension geometries in smaller spaces too. It suggests that a 3-sphere, if that's what the universe is, would have measurable and slightly positive curvature. In this scenario, if the universe was small enough you could theoretically see distant supergalactic clusters in one direction, and then again in other directions - assuming the light had time enough to travel that way. But if the universe was extremely large, you might not be able to see that far and to top it off, the curvature you might measure might be extreeeeeeemly slight and appear flat to the limit of measurement. Lawrence Krauss claims that we have measured, to about 5 decimal places, the curvature of the universe to be roughly zero (i.e. flat). The 3-sphere if we are in one must then be extremely large, or we are not in one. If you take Krauss argument about a zero total energy universe being self-creating, it does present a problem for his philosophy if they measure any amount of global (glomal) curvature, doesn't it? It's also interesting to see how much of the Poincare conjecture/proof is informed by (or informed) the understanding of space-time. It is solved with the aid of Ricci flow, it forecasts an ever-expanding universe resulting in de Sitter space at infinite time, and it confuses the hell out of me. Hopefully someone here is familiar with the subject and can clarify anything I've typed.Dat Einstein guy was a clever cookie wasn't he?
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