Timeline for What is the interface between functional analysis and algebraic geometry?
Current License: CC BY-SA 4.0
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| Aug 25, 2023 at 13:54 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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| Feb 9, 2021 at 23:51 | answer | added | Paul Siegel | timeline score: 10 | |
| Feb 9, 2021 at 21:47 | answer | added | Gabe K | timeline score: 6 | |
| Jul 19, 2018 at 0:04 | comment | added | Yemon Choi | Since this question has been bumped to the top again, can I just reiterate my view that it is too broad; it almost seems to be asking "where are places in one area where I might see some parts of the other area?" | |
| Jul 18, 2018 at 22:34 | review | Close votes | |||
| Jul 25, 2018 at 3:02 | |||||
| Jul 18, 2018 at 21:32 | comment | added | R. van Dobben de Bruyn | There are some conceptual links between the two. For example, the Gelfand–Naimark theorem provides almost an anti-equivalence between locally compact Hausdorff spaces and commutative $C^*$-algebras. In algebraic geometry, on the other hand, spaces are in some sense defined in terms of the functions on them, turning this principle around. In both cases, people have been looking for non-commutative analogues. This is still an active area of research (well, two somewhat unrelated active areas of research, as @YemonChoi's last comment shows). | |
| Jul 18, 2018 at 20:53 | answer | added | Qfwfq | timeline score: 5 | |
| Jul 18, 2018 at 20:46 | comment | added | Qfwfq | Well, as for Hilber spaces tout court on manifolds, i.e. $L^2(M,\mu)$, there isn't much to say: if the measure is the one induced by a Riemannian metric (which seems to be the case you might be interested in) then, up to a null measure set, $(M,\mu)$ is equivalent to an open $U\subseteq\mathbb{R}^n$ with the corresponding Riemannian volume form $\mu'$ transported on it, hence $L^2(M,\mu)=L^2(U,\mu')$. | |
| Jul 18, 2018 at 20:13 | answer | added | alpx | timeline score: 4 | |
| Oct 22, 2017 at 4:17 | comment | added | Avi Steiner | Take a look at the theory of algebraic analysis. Things like Sato's hyperfunctions, microlocal analysis, singular support, etc. A good (albeit terrifying) reference is Kashiwara and Schapira's "Sheaves on Manifolds". | |
| Aug 2, 2016 at 3:25 | comment | added | Yemon Choi | @Jake Why on earth does it come to mind for you? What about Connes's NC(D)G is algebro-geometric? | |
| Aug 2, 2016 at 1:49 | comment | added | JJJ | Noncommutative Geometry is what immediately comes to mind for me. | |
| Aug 1, 2016 at 23:04 | review | Close votes | |||
| Aug 2, 2016 at 11:31 | |||||
| Aug 1, 2016 at 22:49 | comment | added | Nate Eldredge | For instance, see Part 2, Section 48.3 of these notes. sbseminar.wordpress.com/2011/02/22/sobolev-spaces-on-manifolds also looks useful. | |
| Aug 1, 2016 at 22:49 | comment | added | Yemon Choi | @gradstudent Thank you for your response. It still seems to me that you are hoping for a connection where there is no reason to expect one, but since I am not an algebraic geometer perhaps I am overlooking something. Note that the categories in question look rather different! | |
| Aug 1, 2016 at 22:37 | comment | added | gradstudent | @NateEldredge Do you have a reference which talks of these Sobolev spaces and Hilbert spaces on the sphere? That looks like a starting point! | |
| Aug 1, 2016 at 22:36 | comment | added | gradstudent | @YemonChoi I am looking for a connection to "algebraic geometry". The "spectral" theory, as say done in the two references I gave, seem to be more of a connection to differential geometry. If the same you are looking at is an algebraic variety then does that description lead to an understanding of the function space on that variety? | |
| Aug 1, 2016 at 20:21 | comment | added | Yemon Choi | To repeat myself: why do you say algebraic geometry rather than differential geometry? Which is the one that you are hoping to be connected to functional analysis? | |
| Aug 1, 2016 at 17:31 | comment | added | M.G. | Another direction, where so called $L^2$-methods enter, is the study of (algebraic) vector bundles and positivity. There is whole manuscript by Demailly dedicated to this and literally called "Analytic Methods in Algebraic Geometry". | |
| Aug 1, 2016 at 17:26 | comment | added | José Figueroa-O'Farrill | The interface was, I believe, Alexander Grothendieck ;-) | |
| Aug 1, 2016 at 17:07 | comment | added | M.G. | I think, for the second question (i.e. functional analysis on manifolds), the right "buzzwords" are Geometric Analysis, Global Analysis, Geometric Measure Theory and similar. If you search for these, you'll find plenty of references. | |
| Aug 1, 2016 at 17:03 | comment | added | Nate Eldredge | You might want to consider asking your two questions in separate posts. At least the second part might be considered "too elementary" by some (though it seems okay to me), so math.stackexchange.com is another option. | |
| Aug 1, 2016 at 17:03 | comment | added | Yemon Choi | Differential geometry would seem more natural than algebraic geometry in this context. How much algebraic geometry are you familiar with? | |
| Aug 1, 2016 at 16:58 | comment | added | M.G. | One important connection between functional analysis and algebraic geometry is Hodge theory, for which you need to understand the Laplacian (elliptic) as well as some related operators. A concise reference are the books of C. Voisin. | |
| Aug 1, 2016 at 16:57 | comment | added | Nate Eldredge | There are pretty natural answers to your second paragraph. A Riemannian manifold $(M,g)$ has a canonical volume form $\mathrm{Vol}$ which induces a Borel measure. Then $(M, \mathrm{Vol})$ is a measure space and so the most natural Hilbert space of functions on $M$ is $L^2(M, \mathrm{Vol})$. We can also consider Sobolev spaces; for instance, the $H^1$ norm is defined by $\|f\|_{H^1(M)} = \int_M (|f|^2 + g(\nabla f, \nabla f))\,d\mathrm{Vol}$. Is that what you are looking for? I don't see how it helps build a bridge to algebraic geometry, but then again, I know little of the latter. | |
| Aug 1, 2016 at 16:30 | history | edited | gradstudent | CC BY-SA 3.0 |
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| Aug 1, 2016 at 16:22 | history | asked | gradstudent | CC BY-SA 3.0 |