Timeline for Why are there so many smooth functions?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 5, 2016 at 0:45 | answer | added | Margaret Friedland | timeline score: 6 | |
| Jan 5, 2012 at 19:27 | comment | added | Tom Goodwillie | No. A homeomorphism may be homotopic to a smooth map without being homotopic to any diffeomorphism. | |
| Jan 5, 2012 at 18:11 | comment | added | Anonymous | Excuse my ignorance but doesn't the original post make a false claim when stating "Also, (although I do take it on faith), any map of two manifolds is homotopic to a smooth one"? Doesn't the existence of exotic differentiable structure (e.g. spheres, R^4) contradict this? | |
| Jan 5, 2012 at 0:17 | comment | added | Liviu Nicolaescu | @Paul Siegel: There is a universe that includes besides analytic functions of the type $e^{-1/t}$ and this universe has the same rigidity as the real analytic universe. This was discovered by logicians (model theorists) who refer to objects in this universe as o-minimal. I recommend Tame Topology and O-minimal Structures, L. P. D. van den Dries, London Mathematical Society Lecture Note Series(No. 248) It's a far reaching generalization of real (semi)alpgebraic geometry. | |
| Jan 4, 2012 at 22:14 | answer | added | Matthias Ludewig | timeline score: 5 | |
| Jan 4, 2012 at 19:43 | comment | added | Vít Tuček | By the way, does anybody know of a reference on the statement about the homotopy of smooth map between manifolds to a continuous map? | |
| Jan 4, 2012 at 16:58 | answer | added | Liviu Nicolaescu | timeline score: 27 | |
| Aug 6, 2011 at 11:54 | vote | accept | Piotr Pstrągowski | ||
| Aug 6, 2011 at 5:46 | comment | added | Paul Siegel | I've always wondered if real analysis is what complex analysis would be if our universe of functions included functions with essential singularities in addition to holomorphic functions. Maybe there are deep insights to be gained by carefully studying the proof of big Picard. | |
| Aug 6, 2011 at 5:42 | comment | added | Paul Siegel | Here is a related observation that I have always found particularly mysterious. The key tool for passing back and forth between the continuous world and the smooth world is the existence of bump functions. But to construct a bump function you don't actually need all that many smooth functions - in fact, you really just need a single smooth function $f(t)$ with the property that $f^{(n)}(0) = 0$ for every $n$. The quintessential example of such a function is $f(t) = e^{-1/t}$, and in complex analysis this happens to be the quintessential example of a function with an essential singularity. | |
| Aug 6, 2011 at 4:46 | answer | added | Phil Isett | timeline score: 21 | |
| Aug 6, 2011 at 3:33 | answer | added | Terry Tao | timeline score: 71 | |
| Aug 6, 2011 at 0:15 | answer | added | Gjergji Zaimi | timeline score: 5 | |
| Aug 5, 2011 at 23:46 | answer | added | David Roberts♦ | timeline score: 29 | |
| Aug 5, 2011 at 23:18 | comment | added | Douglas Zare | In terms of cardinality, there aren't many smooth functions among all functions. Smooth functions are continuous, continuous functions are determined by their values on a dense set, and manifolds are second-countable so they have a dense countable set. Thus the cardinality of the smooth functions on a manifold is the same as the cardinality of constant functions. | |
| Aug 5, 2011 at 23:03 | history | asked | Piotr Pstrągowski | CC BY-SA 3.0 |