Timeline for Does this formula have a rigorous meaning, or is it merely formal?
Current License: CC BY-SA 4.0
28 events
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| Mar 15 at 22:33 | comment | added | Hans Lundmark | @TimothyChow: Yes, that's the one! Andreas moved to another university long ago, so the link I gave probably hasn't worked for the last 10 years at least... (And the book is great, by the way!) | |
| Mar 15 at 18:12 | comment | added | Timothy Chow | @HansLundmark The link in your comment is broken. I believe that book you're referring to is now published: Geometric Multivector Analysis: From Grassmann to Dirac, by Andreas Rosén. | |
| Apr 5, 2020 at 23:18 | history | edited | Adam P. Goucher | CC BY-SA 4.0 |
Punctuation
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| Feb 4, 2011 at 8:50 | comment | added | Hans Lundmark | @Dick: I think the new name was introduced for marketing reasons, and is mainly used by the followers of David Hestenes. | |
| Feb 4, 2011 at 5:03 | comment | added | Dick Palais | @Hans:You said: "GA is really just another name for Clifford algebras" Yes, so I eventually figured out. When I finally started reading about GA, it quickly looked familiar, and finally Clifford algebras got mentioned and I realized why. Why this renaming of a standard, well-known, and well-studied structure? They were a popular topic of study back in the 60s because of their use in the Index Theorem, and I even recall writing a section explaining them in the IAS Seminar on the Atiyah Singer Index Theorem volume. The TOC of your colleagues book looks great ! | |
| Feb 4, 2011 at 1:56 | history | edited | Andrey Rekalo |
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| Feb 3, 2011 at 12:54 | answer | added | Andrey Rekalo | timeline score: 17 | |
| Feb 3, 2011 at 11:00 | comment | added | Hans Lundmark | ...and by the way, I wish I could refer you to the book that a colleague of mine is writing, but unfortunately it is not finished yet: mai.liu.se/~anaxe/GMA.html | |
| Feb 3, 2011 at 10:57 | comment | added | Hans Lundmark | @Dick: GA is really just another name for Clifford algebras, and there are determinants everywhere if you do coordinate calculations in a Clifford algebra. For example, the highest graded part of the Clifford product of two (homogeneous) multivectors is the exterior product of those multivectors, and exterior product is related to determinants in a way that you're probably familiar with. | |
| Feb 3, 2011 at 7:19 | vote | accept | Dick Palais | ||
| Feb 3, 2011 at 0:04 | answer | added | Patrick I-Z | timeline score: 5 | |
| Feb 2, 2011 at 23:30 | answer | added | Deane Yang | timeline score: 15 | |
| Feb 2, 2011 at 21:31 | answer | added | Denis Serre | timeline score: 5 | |
| Feb 2, 2011 at 20:59 | answer | added | Greg Kuperberg | timeline score: 32 | |
| Feb 2, 2011 at 19:33 | comment | added | Dick Palais | @Ryan Budney: "...It's tricky answering your questions because I figure you've seen everything already." On the contrary, it is absolutely amazing to me (and more than a little humbling) how much I have learned from answers and comments to the questions that I and others have asked here. | |
| Feb 2, 2011 at 19:26 | comment | added | Ryan Budney | It's tricky answering your questions because I figure you've seen everything already. But I don't like making those kinds of assumptions about people, so here we are. :) | |
| Feb 2, 2011 at 19:21 | comment | added | Dick Palais | @Steve Huntsman: Thanks, Steve, that certainly looks promising. I had heard of GA before, but it always seemed too formal for my taste and I never looked into it carefully---perhaps this is a good opportunity to look at it more carefully. But, I didn't see any mention in the Wikipedia article about a relation between GA and determinants, although there clearly must be one. Do you know where I can look for that? | |
| Feb 2, 2011 at 19:08 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Feb 2, 2011 at 19:05 | comment | added | Dick Palais | @Ryan: Well, I think I understand the Hodge star isomorphism reasonably well (at least, I have written a number of articles purporting :-) to explain it to others in various places, e.g., Seminar on the Atiyah-Singer Index Theorem), and yes, I was hoping for an acceptable definition of det that would make everything kosher, but I don't want to dissuade you from answering via Hodge theory. | |
| Feb 2, 2011 at 19:04 | comment | added | Steve Huntsman | en.wikipedia.org/wiki/Geometric_algebra | |
| Feb 2, 2011 at 18:58 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Feb 2, 2011 at 18:57 | comment | added | Ryan Budney | Would you be happy with a nice formulaic interpretation of a "Hodge star" isomorphism $\mathbb R^3 \wedge \mathbb R^3 \to \mathbb R^3$ ? Or do you really want determinants defined at some enhanced level of generality? | |
| Feb 2, 2011 at 18:52 | comment | added | Mariano Suárez-Álvarez | A silly way out is to view your vector space as a bimodule over the field, and then you can compute the determinant without any guilt :) | |
| Feb 2, 2011 at 18:47 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Feb 2, 2011 at 18:47 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Feb 2, 2011 at 18:46 | comment | added | Ryan Budney | There's a natural way to side-step your question in that the cross product is dual (hodge dual + vector space isomorphic to its dual via an inner product) to the wedge product of forms. And your formula is essentially an expression of that duality. In the same way you can define the "cross product" of $n-1$ vectors in $\mathbb R^n$, etc. | |
| Feb 2, 2011 at 18:46 | history | edited | François G. Dorais | CC BY-SA 2.5 |
fixed quotes
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| Feb 2, 2011 at 18:42 | history | asked | Dick Palais | CC BY-SA 2.5 |