A week ago I asked Perplexity AI:
“Is there a monograph, or a comprehensive review, discussing infinite-dimensional symmetric spaces and Kahler manifolds?”
Here is the answer:
Yes. There are a few substantial sources on infinite-dimensional symmetric spaces, and many standard Kähler references; however, there is (to my knowledge) no single monograph that treats both topics together in a unified, exhaustive way.
“Infinite-dimensional symmetric spaces
For Riemannian symmetric spaces in infinite dimension, good “monograph-level” treatments are essentially PhD theses and long articles:
A. Dvorsky, Infinite dimensional symmetric spaces (PhD thesis, Göttingen, 1998). This is a several-hundred-page work developing the general theory for spaces arising from loop groups, with classification results in that setting.
B. Duchesne, Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator, Ann. Inst. Fourier 65 (2015), 211–244, and the expanded Numdam version. This gives a systematic classification for simply connected Riemannian symmetric spaces (finite or infinite dimensional) under a sign condition on the curvature operator, using L∗-algebras.”
It is difficult to understand how AI operates. The title of the first reference is correct, the link is correct, but the author’s name , the place and the year, are all fabrications of AI. Similarly as: “This is a several-hundred-page work“. The real reference is: Bogdan Popescu, Infinite dimensional symmetric spaces, PhD Thesis Augsburg 2005. It is 61 pages long.
The second reference is alright, except that there is no “expanded” version. The Numdam version is the exactly the same as the AIF version, which seems to be exactly the same as the arxiv version.
Then Perplexity lists some lectures on Kahler manifolds in general, to conclude with more pertinent to my question:
“Kähler manifolds and infinite-dimensional targets
There is also work explicitly involving infinite-dimensional Kähler targets, for example Kähler immersions into infinite-dimensional complex projective space:
A. J. Di Scala and S. Hishi, Kähler manifolds and their relatives, Ann. Scuola Norm. Sup. Pisa (downloadable preprint). They study Kähler immersions and note, following Kobayashi, the classical result that certain manifolds admit Kähler immersions into the infinite-dimensional complex projective space.”
This time the authors and the title were correct, but Perplexity messed up the link it has provided. The correct link is here: https://www.numdam.org/item/ASNSP_2010_5_9_3_495_0.pdf
I asked Perplexity to redeem itself by creating a short story about George Cantor, the mathematician who opened the infinity Pandora box. Here is what Perplexity offered:
In a quiet German university town, a solitary mathematician declares war on infinity.
Georg Cantor believes he has seen what no one else dares to imagine: that there are many infinities, stacked in a strange, invisible hierarchy, some forever larger than others. His proofs are clean, his logic ruthless—and almost everyone hates what he has found.
To his colleagues, Cantor is a heretic who has broken the ancient commandment that infinity must remain indivisible. To his students, he is a distant figure haunted by letters that never bring the recognition he craves. To himself, he is alternately prophet and impostor, chosen and condemned, staring into an abyss that sometimes seems to stare back.
As the Berlin establishment shuts its doors and rumors about his sanity begin to circulate, Cantor retreats deeper into a world where sets and symbols feel more solid than people. Yet far away, in the undiscovered future of mathematics, a young David Hilbert is already learning the language that will one day give Cantor’s vision a name: a paradise from which, he will vow, no one shall expel them.
This is the untold story of a man who tried to measure the unmeasurable, lost almost everything in the attempt, and quietly changed what it means to think at all.
Not too bad. Another version of the above I put on my Substack blog,
I first saw you mention Kahler manifolds 20 years ago during the first year of the Cass forum. Zero idea what they were then. You also mentioned symplectic which I also was clueless about. By now I know those and Born reciprocity relate to phase space that Anna brought up last year based on a Sasa question and your 2,2 signature toy model.
You also brought up a complex number field and a spectral triple recently. Complex number field I think I only knew of when Tony said Feynman’s checkerboard was over it while Tony’s bigger checkerboard was over the reals. Never heard of spectral triple though Connes and his noncommunitive geometry I had seen before. Spectral triple via Wikipedia looks Fock space-like to me.
In 2018 you asked about branes and infinity and while your recent infinity conversation makes the geometric algebra a countable infinity, all the branes seemed to be an uncountable infinity. So it seems like while a single brane universe state is a countable infinity geometric algebra, all the brane states and their geometric algebra become uncountable. Cantor-like it’s like the countable integers vs. uncountable reals. Geometric algebra even has a nice 2^n power set kind of structure. The 2018 conversation also had the spacetime dimensions replicating infinitely so perhaps tensor product-like.
Spectral triple also seems Dirac operator-ish and that was part of a conversation about Pauli symphony vs. Riemann radio whatever that means but it sounds spectral.
As I clarified it in yesterday’s session the “geometric algebra” in the above is NOT some Clifford algebra. Emil Artin wrote a book “Geometric Algebra”, but Clifford algebras are only a small part in his book. I asked Perplexity about it, and here is part of the answer:
“From the publisher’s and historical descriptions of the book, Artin’s project is the following. He starts from an abstract plane geometry (points, lines, incidence axioms, etc.) and asks how to generate an algebraic structure (a field and associated linear spaces) from purely geometric axioms. The resulting algebra—essentially a field with vector spaces and bilinear forms—then serves as the common framework in which he treats:
affine and projective geometry
symplectic and orthogonal geometry
the general linear group and classical groups
Clifford algebras and spinorial norms.
All of these are presented as aspects of one unified algebraic structure underlying geometry.
In other words, “geometric algebra” in Artin’s sense means: the algebra (fields, vector spaces, bilinear forms, classical groups, Clifford algebras) that one canonically reconstructs from geometry and then uses to treat geometry in a unified way.”
Trying to get some more clues from C’s was unsuccessful. The answer seems to be “coded, I guess for some reason According to this answer geometric algebra is about “infield relations”. I ahve no idea what “infield” stands for here. Infield is a diamond-shaped region in baseball, the term also appears in the matyhematics of agriculture. Perplexity instantly associated it to Born-Infeld nonlinear electrodynamics, bacause Inflled is so similar to Infeld! Perhaps that is what the code is supposed to refer to?
We might look at this from a bright side, sort of a silver lining kinda thing.
Not knowing the exact context, this could be way off base, so FWIW.
According to the C’s, densities are directly related to consciousness, and regarding describing consciousness in math or scientific terms they also said a while ago that we did not even have adequate concepts to start with at the time. If there have been no significant changes in that respect, as there might very well have been, it is only natural that they would not want to take away from us humans the thrill and joy of learning, discovering and describing something important as consciousness on our own. “It’s fun to see how much you/we can access!”, remember? 🙂
Regarding the “infield relations” coded hint, the C’s said not so long ago that information arranged by truth becomes consciousness. If everything is sort of (made of) the Information Field, one possible way to interpret “infield relations” might be in the context of that previous remark.
To describe densities as sort of a states of consciousness and/or awareness, the framework in the form of geometric algebra, and they did also say that the geometry would get us there, while the algebra would enable us to ‘move’, paraphrasing, would naturally deal with “infield relations” as arrangements of information (of the Field) by the truth or, from math perspective, as possibly completely novel geometric algebra ‘rule set’, that is still to be discovered by us humans.
And learning surely is fun! 🙂
Thanks. In fact I misread the answer. It was
A: It is about relationships within a field.
There was no “infield”. Anyway infinite dimensions is a sufficient clue for me. Gets me going.
IRC, they said in previous session that infinity in question is countable, which seems to exclude the set of real numbers. Is the set of rational numbers, m/n where m and n are natural numbers or integers, as possible set to describe dimensions of fractals, considered to be countable or uncountable?
Answering my own question, so FWIW.
According to Wikipedia, the set of rational numbers is countable (infinite), the same as the set of algebraic numbers.
The Cs once said Clifford with enhancements and the enhancements could certainly be the main algebra with Clifford algebra as a mother algebra as you once mentioned; more a container than anything else. I’d vote for in-field relations; in particular symplectic and orthogonal including with a degenerate metric. Perplexity for me keeps trying to make the metric 8-dim (though when degenerate it’s for the spacetime part) likely related to Born trying to extend the Minkowski space metric. I’d like to keep it 4 with connections for mother algebra reasons but maybe the complex number field is an extension for more than just non-physical stuff to make Fock space/spectral triple stuff easier as Perplexity said to me.
Ark, why do you need these infinite-dimensional symmetric spaces? You have a finite-dimensional compactified Minkowski space, which is perfectly suitable for populating it with fermions (double loops of a compactified isotropic cone). And if you want to add gauge symmetries to the properties of fermions, then a compactified isotropic cone comes to the rescue here too, but this time for spaces with an inverse Lorentz metric. The point is that the group of unbroken symmetries of the Standard Model acts on this compact manifold through motions of 1-, 2-, and 3-dimensional submanifolds (cycles and inertial manifolds). The infinite dimensionality of fermions arises when we consider them as a dynamical system.
We need fields, not just points. Fields, as a rule, have infinite-dimensional geometry. Even if points do not exists or inaccessible, fields may still be there, as primary objects. For instance in noncommutative geometry the primary object is “the algebra of fields”. I think this is an important issue.
True, the fields are infinite-dimensional, but they exist in finite-dimensional spaces. And why describe the fields using the intrinsic geometry (metric) of the foliation or level surface induced by these fields? It’s enough that we use intrinsic geometry to describe gravity.
” but they exist in finite-dimensional spaces. ”
“Spaces” may be questionable. I am thinking about “noncommutative geometry”. See e.g.
https://mathoverflow.net/questions/462707/is-there-a-notion-of-point-in-noncommutative-geometry
Added new post on Substack:
“Later is not necessarily better”
https://arkadiuszjadczyk.substack.com/p/later-is-not-necessarily-better