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This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers references which show this bridge from functional analysis into algebraic geometry.


I am not sure if its related but what are the good references for ``functional analysis on manifolds"? Like how do we characterize the function space based on the domain manifold properties or for specific manifolds like say spheres. (the related things I see are courses like, http://www.math.uiuc.edu/~palbin/Math524.Spring2012/LectureNotesMay1.pdf or http://www.math.harvard.edu/~canzani/math253.html but these seem more about understanding specific differential operators on manifolds rather than the space of functions on a manifold)

Like is there any meaning to wondering, "What is the Hilbert space of functions on a sphere?"

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    $\begingroup$ 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. $\endgroup$ – M.G. Aug 1 '16 at 16:58
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    $\begingroup$ 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. $\endgroup$ – Nate Eldredge Aug 1 '16 at 17:03
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    $\begingroup$ The interface was, I believe, Alexander Grothendieck ;-) $\endgroup$ – José Figueroa-O'Farrill Aug 1 '16 at 17:26
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    $\begingroup$ @Jake Why on earth does it come to mind for you? What about Connes's NC(D)G is algebro-geometric? $\endgroup$ – Yemon Choi Aug 2 '16 at 3:25
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    $\begingroup$ 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". $\endgroup$ – Avi Steiner Oct 22 '17 at 4:17
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I do not know if this is what you are looking for but I heard people speaking of Banach Algebraic Geometry. I can also note the fascinating work of Semyon Alesker (using algebra-geometric tools on say space of convex bodies).

On a high level there is certain similarities between the ideas in two fields. But, I personally prefer more concrete connections better. The real algebraic objects coming out of optimization problems, or convex objects with a lot symmetry coming out of Quantum Information Theory seems to be forcing an interface between algebraic geometry and convex geometric analysis ( local theory of Banach spaces if you like). There is also new developments in random real geometry which seems to require a blend of func analysis with AG.

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As an example in which algebraic geometry and functional analysis (mildly) interact: in the book Several complex variables with connections to algebraic geometry and Lie groups by J.L.Taylor, if I remember correctly, there is a chapter on GAGA in which sheaves of Fréchet spaces are considered.

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