Timeline for Definition of area
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 19, 2015 at 17:21 | comment | added | Daniel Litt | @roysmith: Yes, I think the computation of $K(Poly)$ is basically Euclid's theory of equal content from Elements. | |
| Nov 19, 2015 at 17:06 | comment | added | roy smith | Isn't this essentially Euclid's approach? | |
| Nov 19, 2015 at 15:21 | comment | added | Daniel Litt | @JeffStrom: This is actually well-studied. The $3$-dimensional version gives $\mathbb{R}\oplus (\mathbb{R}\otimes \mathbb{R}/\mathbb{Z})$, where the first factor is ordinary volume and the second factor is the Dehn invariant. IIRC we don't know a complete description of the $4$-dimensional version. But lots of people (e.g. Bloch, Goncharov, and others) have worked on this sort of thing--the hyperbolic version has relations to periods and other things; also some people (e.g. Inna Zakharevich) have studied "higher $K$-groups" in this setting. | |
| Nov 19, 2015 at 14:15 | comment | added | Jeff Strom | What is the higher-dimensional analog? That is, you could argue from this approach that the natural target of the area function is $K(Poly(2))$, and the fact that this turns out to be isomorphic to $\mathbb{R}$ is just a happy accident. So---for example---what is $K(Poly(3))$, the `natural target' of volume? | |
| Jan 26, 2013 at 19:50 | comment | added | Daniel Litt | Ah, fair enough. | |
| Jan 26, 2013 at 19:45 | comment | added | Anton Petrunin | @Daniel, maybe 40 is bit too much, but try to imagine that you are writing this for students, I do not think one can make it in 10 pages... | |
| Jan 26, 2013 at 19:02 | comment | added | Daniel Litt | Ah, I haven't seen the book. What could possibly take them so long? | |
| Jan 26, 2013 at 18:59 | comment | added | Anton Petrunin | This is the same as in Millman and Parker --- with proofs it takes 40 pages, not sexy at all. | |
| Jan 26, 2013 at 18:53 | comment | added | Daniel Litt | I don't think so, Mariano---I define a topology on K(Poly) in the 3rd paragraph, which agrees with the usual topology on R. | |
| Jan 26, 2013 at 18:48 | comment | added | Mariano Suárez-Álvarez | For your limit procedure to work as expected, one need a bit of extra structure (for example, to trace the action of dilatations on the k-group) for otherwise, as the isomorphism $K(poly)\cong R$ is only an isomorphism of abelian groups and R has many automorphisms, the limits you are taking might do weird things. | |
| Jan 26, 2013 at 18:40 | history | answered | Daniel Litt | CC BY-SA 3.0 |