Talk:Floer homology

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This article may be too technical for a general audience. Please help improve this article by providing more context and better explanations of technical details to make it more accessible, without removing technical details.

I'm a college student with a strong background in math and physics and I can't make head or tail of this. Could someone please explain what the heck this article is about? —Keenan Pepper 23:54, 19 May 2006 (UTC)

Shouldn't you place the {technical} tag in the main article page, rather than the talk page? --Byakuren 04:56, 9 July 2006 (UTC)

How about I read before I blurt stuff out??? Never mind... --Byakuren 05:00, 9 July 2006 (UTC)

Yeah, a kinda wish I were allowed to put it on the article itself; then maybe people would pay attention. Do you have any idea what this is? Maybe you can explain it to me. —Keenan Pepper 05:27, 9 July 2006 (UTC)

Sorry, dude, this is way out of my field. I'm just an engineer, and I've taken up until graduate-level PDE's, and I have no idea what this article's about... good luck! --Byakuren 01:09, 10 July 2006 (UTC)

I rewrote the introduction in an attempt to help out. The first paragraph will hopefully give you some idea of what parts of mathematics Floer homology belongs to. It's definitely a technical topic, but it's also very rich mathematically. One explanation for both its technical nature and its current relevance is that it pulls together tools from many areas of mathematics: topology, geometry, differential equations, analysis and mathematical physics. Feedback is welcome.  : David M. Austin (talk) 15:23, 21 December 2007 (UTC) : 15:23, 21 December 2007 (UTC)

Contents 1 Prerequesites for understanding this article 2 Technical tag 3 Please notice 4 References 5 Homologies 6 Loop space or free loop space?

[edit] Prerequesites for understanding this article

I can't possibly understand this article now, but that doesn't mean I don't have hope of understanding it some day. Here's the bulletin for the math department of my university: undergraduate and graduate. So far I've taken calc III, differential equations, and complex analysis. Which courses should I take in order to understand Floer homology? —Keenan Pepper 05:40, 9 July 2006 (UTC)

I don't work in this subject exactly, but I think you need courses in differential topology (manifolds, forms, vector bundles), differential geometry (connections), functional analysis (Sobolev spaces), more differential equations. Then you need to read chunks of a book like McDuff's and Salamon's Introduction to Symplectic Topology to learn some symplectic stuff. It would also be nice to know a little more complex analysis, along the lines of Chapter 0 of Griffiths and Harris, Principles of Algebraic Geometry.

If you want to use Wikipedia to understand the math behind this article, you might pursue the link Morse homology. If you want to understand the physics, you might pursue topological quantum field theory. Both of these links appear in the intro, but I do agree with posters above that this article needs more motivation, physical explanation, etc. Joshua Davis 14:47, 10 July 2006 (UTC)

Random side question: how many people in the world really understand this? Is it closer to a thousand or a million? —Keenan Pepper 05:44, 9 July 2006 (UTC)

Maybe just hundreds. But it is intimately connected to quantum cohomology, string theory, algebraic geometry, etc., which have broader appeal (thousands). Keep in mind that all of this is work in progress; no one knows yet which parts of the subject will have lasting value in math or physics. As such, the article could be considered too technical and premature for an encyclopedia, but I think it's valuable in naming a number of different subtopics and explaining how they relate to each other. Joshua Davis 14:47, 10 July 2006 (UTC)

[edit] Technical tag

I'm removing it because I don't think the person who added the tag realized the technical level of the topic. It's really not possible to make this accessible to someone who doesn't already know something like morse homology. Additionally, the number of people who can improve this article is very small, so the tag is kind of pointless. I think people working on this page now realize they should work on accessibility some more, so mission accomplished as far as the tag's concerned. --C S (Talk) 02:10, 11 September 2006 (UTC)

If it's really impossible to explain even the basics to an average person with a college education, it should be deleted. Wikipedia is an encyclopedia for general use, not a textbook of wizard math. —Keenan Pepper 13:01, 11 September 2006 (UTC)

General use to whom? One of wikipedia's strengths is its breadth. We have Britannica for the people who don't like things they don't understand. Perhaps we could add a section called Basics that gives required reading for understanding, if the links don't appear in an article like this. Orthografer 15:51, 11 September 2006 (UTC)

I agree; the subject deserves coverage in Wikipedia. On the other hand, this article needs improvement.

• The second paragraph of the intro is already tough reading even for a mathematician.
• Have these invariants ever been used to solve any problem in math? The Arnold conjecture is lost in the shuffle. Tell us what the point is in the intro, and perhaps in a "math applications" section as well.
• The intro mentions TQFT; how about a physics section that explains the theoretical physics relevance?
• I think someone with a college education can get an intuitive idea of Morse homology (the subject, not the article as it stands). I would like to see one of these Floer theories explained in more detail, based on its inspiration in Morse homology, so that someone who actually understands Morse theory could get an intuitive idea of Floer homology. Then, perhaps, a college-educated person could at least glimpse Floer homology. Joshua Davis 21:43, 11 September 2006 (UTC)

These are great suggestions! I vote for Heegaard-Floer homology as a good candidate for something we can draw some nice pictures and compute some examples. --C S (Talk) 15:35, 19 September 2006 (UTC)

I'm glad you like them. (Perhaps I should mention that I don't know enough about this subject to implement them...!) Heegaard-Floer seems reasonable, but I think the "math applications" and "physics applications" are more important, and easier to write? Joshua Davis 03:06, 20 September 2006 (UTC)

Keenan, I think that would generally be considered as an extreme deletionist stance. Wikipedia is populated by a number of technical articles (not even in mathematics) that are difficult for even people like me (I see myself as having had a decent college education) to get even more than the foggiest notion of what they are about. For example, NK_cells, found just after a few semi-random clicks of the mouse. I believe that this phenomenon is considered by many to be a strength of Wikipedia. I don't think it really hurts to have very technical articles...how does absence of this information further Wikipedia's mission of giving "every single person on the planet...free access to the sum of all human knowledge"? --C S (Talk) 15:32, 19 September 2006 (UTC)

My point wasn't that it should be deleted, my point was that if it's worth keeping, then there must be some way to explain the basics. You said "it's really not possible to make this accessible to someone who doesn't already know something like morse homology" but I think that's a ridiculous claim. Take your example, Natural killer cell. From the first paragraph of that article I know that they're cells of the immune system, and they kill cells that don't look friendly. The rest of it is mostly gibberish, but at least I know the basics. This article is gibberish from the very first sentence, but I don't think it has to be. I want the intro to tell me what kind of mathematical object it is, what it's useful for, and things like that. —Keenan Pepper 03:10, 21 September 2006 (UTC)

Well, I certainly can't see from what you said why my statement is ridiculous. You'll notice that besides you, everyone else here with a somewhat extensive math background thinks this page cannot be made that accessible. Joshua probably is more optimistic than me, but even he appears to think NK cell (which looks like gibberish to me) has a better future as an accessible article. You admit that that you only understand the very beginning of NK cell; presumably you understand that very basic part because you have a vague idea (or better) of what an immune system is and what a cell and microbe are.

So what did you get from NK cell? That it's a cell of the immune system that kills stuff that "doesn't look friendly"? Well, I mean you already knew there were such things, right? But you didn't actually learn more than a name. If you're happy with that very small nugget of info, I really don't see why you are picking on this article.

The beginning of this article explains that

Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. Some of these theories are due directly to Andreas Floer, while others are derived or inspired by his work. They are all modelled upon Morse homology on finite-dimensional manifolds...

You say this is gibberish, here presumably because you don't know what homology is or what a finite dimensional manifold is. But just as the NK cell article is a bad place to explain what an immune system is, this is a bad place to explain what Morse homology, or even what homology is. A person that has a vague notion of these topics would get just as much, if not more, I suspect, than what you got from NK cell.

Let me put it another way. I think Joshua has a point insofar as a person that knows what Morse homology is should be able to get a better idea of what Floer homology is from the intro, but that is not the case right now. So that certainly can be improved, along the lines of what he has suggested.

I understood your point; I just don't think it's possible to explain the basics in a manner you would be satisfied with. My point was (and is) that even if the article never reaches a point where you are happy with it, deletion serves no purpose and is detrimental. Perhaps I am mistaken about what you would be satisfied with; as a test case, what are your thoughts on the example paragraph by Joshua (right below this response)? That of course is not a lead paragraph, but I imagine Joshua would like to implement similar changes in the lead.--C S (Talk) 06:41, 21 September 2006 (UTC)

I support Keenan's restoration of the technical tag here. I would also support removal of the tag when the introduction and sections deal with motivation/applications in math and physics. Some soft, intuitive language would also be nice, along the lines of

Lagrangian submanifolds are distinguished subspaces in a symplectic manifold that reveal much of the manifold's properties. The Lagrangian intersection Floer homology essentially counts the number of disks, satisfying a natural energy-minimization property, that connect chosen Lagrangian submanifolds. The Floer homology organizes the results of this counting process into a ring, an algebraic structure that reflects the geometric structure of the Lagrangian submanifolds and hence of the symplectic manifold.

(Apologies if it's wrong; I told you, I don't know this stuff.) However, for the majority of the article we cannot hope to be even as accessible as Natural killer cell. The average reader knows a lot more about cell (biology) than manifold. Joshua Davis 04:36, 21 September 2006 (UTC)

If people want to add the tag, that's fine. I'm not going to stop anyone; however, there are already a lot of articles with technical tags, and quite a few that shouldn't be. If you think adding the tag will do something positive, go ahead! I would be happier myself if people only stuck the tags on articles that have reasonably good chances of being accessible to the mathetmatically literate non-mathematician. There, at least, we have a shot. --C S (Talk) 06:41, 21 September 2006 (UTC)

C S brought up an important point above: "...don't know what homology is or what a finite dimensional manifold is. But just as the NK cell article is a bad place to explain what an immune system is, this is a bad place to explain what Morse homology, or even what homology is." I largely agree; those topics have their own articles, and Wikipedia is a reference work, not a textbook.

On the other hand, I think it's quite reasonable to have a brief, intuitive overview that reviews the prerequisites for understanding the article. Then someone coming across this article gets a vague notion of what's going on, which she can carry in her head as she reads the prerequisite articles. Otherwise it is very hard to learn math from Wikipedia --- it's a giant recursion of isolated definitions. Our articles should not be disjoint; they should overlap and patch together, to help the reader move around.

Wikipedia: Manual of Style (mathematics) doesn't specifically address this, but it does say that continuity should not just be "the preimage of every open set is open" but also "can be drawn without lifting the pencil". Joshua Davis 13:15, 21 September 2006 (UTC)

[edit] Please notice

Please notice: There is a German version of this article now [1], made by user Claude J [2]. It is a complete and perfect translation of the. text. There may be open questions, e.g. relative to the term “differential”. Let me recommend it to your attention. Erkabo 09:32, 8 March 2007 (UTC)

[edit] References

Dear All, I came across this article for the first time today. It's well-written, by and large, but to my taste at least there are too many vague references. It seems to me that an article on a topic like this (which is still very much under construction) should be especially careful to give unamibguous references. (The reference to a "forthcoming book" of Mrowka and Kronheimer is a notable case in point --- this surely violates WP:ATT.) Artie P.S. 09:09, 23 April 2007 (UTC)

Yes, at the very least we should be able to give one citation in each section. Each type of Floer homology has some kind of "foundational paper" in which it was invented. These shouldn't be too hard to track down. (This is on my to-do list after I finish my dissertation and graduate. The eventual goal is to see each of these become a reasonably fleshed-out article in its own right.) VectorPosse 16:47, 23 April 2007 (UTC)

[edit] Homologies

I've never before seen the plural of the word homology (or cohomology, for that matter!). In fact, I thought that it had an attributive function (like school board), with the noun (initially, it was homology group') dropped, in which case plural would be impossible. But it stayed in the article for so long that I am just wondering if there is anything that I am missing here? Arcfrk 03:13, 4 May 2007 (UTC)

Sometimes, homology is kind of like a collective noun. So it is okay to refer to "the homology" of a chain complex, meaning the collection of algebraic structures associated to a chain complex. I don't know how correct it is, but one might refer to "a homology", meaning a homology theory. And as there are many flavors of homology, one might refer to a bunch of them as "homologies". If it bugs you, though, "homology" in this sense can be replaced by "homology theory" and "homologies" by "homology theories". VectorPosse 07:08, 4 May 2007 (UTC)

A couple of Google hits: [3] [4] VectorPosse 07:13, 4 May 2007 (UTC)

Merriam-Webster has this [5] and this [6]. I am no mathematician and not even a native speaker of English, but right from the first time I heard this word - which was more than 30 ears ago - it has always been used as a noun. And I think using a term as e.g. 'Floer homology' implies that there are other homologies, too. --Rolf of Erkabo 07:36, 4 May 2007 (UTC)

Thanks you for the links! Yes, in Russian you can use the plural гомологии for homology groups, where the second (plural) noun is implied and omitted, and a similar use is common in French, but Russian grammar is very different from English grammar (in fact, it's the inflected plural forms, rather than the singular form гомология, that are presently used in Russian). Thus I wouldn't put too much faith into the translation of Arkhangelskii's book. Of the two examples in Merriam-Webster, the first one defines homology in other contexts, where definitely plural form is acceptable (including a distinct mathematical use for a certain type of transformations in projective geometry); while the second one, during exclusively with the mathematical notion, does not mention a possibility for plural cohomologies. This seems to bear out my suspiction that homologies (in the topological sense) is also incorrect. Usually, The Oxford English Dictionary is the authoritative source. But on this subject it's remarkably archaic: the word cohomology is not even in the dictionary, and the only mathematical meaning of homology is from projective geometry (this may be the only example that I am aware of where a word is referenced in MW but not in OED!). After some head scratching, I came up with the idea to put in homologies* in MR Search (Math Sci Net), and discovered two interesting phenomenae:

• There are a couple of dozen papers (in English) where homologies appears in the title not in the projective geometry sense, with relatively large portion of them actually about Floer theory; and
• Virtually all authors of these papers do not appear to be native English speakers! The only exception that I was able to spot is an old paper of Sullivan, with the title

A foliation of geodesics is characterized by having no "tangent homologies".

Well, I do not know if any of this "research" is of slightest importance, I certainly hadn't expected this to turn out to be such a convoluted question! On the other hand, grammatically correct or not, Floer homologies is supported by the literature. Arcfrk 04:05, 5 May 2007 (UTC)

...and I would add "knot homologies" due to Rasmussen's article. (And he is not only a native speaker of English, but also a very trustworthy source of correct terminology.) VectorPosse 04:43, 5 May 2007 (UTC)

[edit] Loop space or free loop space?

Several times this article mentions loop spaces. For example "For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the loop space of M". I think the correct statement is that the Floer homology is the homology of the free loop space of M. This probably applies to the other instances of "loop space" in the article, though I'm not an expert so I can't be sure. Would someone who is sure fix it? (As for the article being too technical for a general audience, sure, but at least I found it useful....)

Response: It's the free loop space, and I've changed the article accordingly. I believe that Abbondadolo-Schwarz's paper also contains a way of getting the homology of the based loop space out of Lagrangian intersection Floer theory.