Timeline for High-dimensional geometry: Top-down Vs. Bottom-up
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| May 29, 2012 at 15:56 | comment | added | Ollie | For instance, the existence of phase transitions for large numbers of particles can be inferred from the behaviour of an infinite dimensional system. The partition function for the finite system is analytic, whereas for the infinite dimensional system it has singularities. The singularities correspond to phase transitions. | |
| May 29, 2012 at 15:52 | comment | added | Ollie | It's not really a mathematical answer, but the use of infinite dimensional techniques for high dimensional problems is quite common in physics (esp. the thermodynamic limit and continuum mechanics). My knowledge in this area is limited, but I'm sure there must be corresponding mathematical statements. | |
| May 29, 2012 at 12:34 | comment | added | Benoît Kloeckner | @Simon: recall that translation invariance characterizes Lebesgue measure up to a constant; so a translation invariant measure on an infinite dimensional Banach space should put zero measure on any ball (since it would otherwise put infinite measure on any ball doubled - that's any ball). | |
| May 29, 2012 at 9:31 | history | edited | Simon Lyons | CC BY-SA 3.0 |
deleted 3 characters in body
|
| May 29, 2012 at 9:30 | comment | added | Simon Lyons | That's interesting, but I'm not quite convinced - I can't see why translation invariance should fail from this example. | |
| May 28, 2012 at 22:33 | comment | added | Will Sawin | I think the answer is, strangely, exactly the same as the spiky ball thing. In high-dimensional spaces, almost all the measure of the unit ball is contained in the shell with $r>1-\epsilon$, thus convex bodies appear spiky and nonconvex. In infinite-dimensional spaces, you replace "almost all" with "all", and derive a contradiction - none of the measure is in the insides, so there is no measure anywhere. | |
| May 28, 2012 at 22:16 | history | asked | Simon Lyons | CC BY-SA 3.0 |