Talk:Singularity theory

Merge?

Shouldn't this page be merged with Mathematical singularity ? --Piotr Konieczny aka Prokonsul Piotrus 21:36, 12 Jul 2004 (UTC)

Well, no. A rational function, for example, has singularities; but they have nothing to do with singular points in the sense of singularity

Charles Matthews 06:18, 13 Jul 2004 (UTC)

All right. Perhaps there should be some clarification then, or at least a see also link for laics like me - I honestly thought both pages are talking about the same thing. Oh, and singularity page should be updated with links to one of those pages - I have no idea how to phrase the links though. --Piotr Konieczny aka Prokonsul Piotrus 20:48, 13 Jul 2004 (UTC)

The introduction is overly "talky" in this section. The chit chat about dropping strings should be moved to another section called something like "informal discussion" and the introduction should be expanded a little bit. I am not expert enough in this topic to make these changes myself.

Also, this page in general is poorly organized. I am well versed in Analysis and know a little Differential Geometry but couldn't get the basic essense of the theory from this page. 169.229.250.214 (talk) 20:48, 29 February 2008 (UTC) A. Kaiser, UC Berkeley 169.229.250.214 (talk) 20:48, 29 February 2008 (UTC)

Yeah, I think this page needs the attention of an expert. It seems to touch on some subjects and use the vocabulary without really communicating any content. —Preceding unsigned comment added by 131.96.4.208 (talk) 15:59, 6 October 2009 (UTC)

Agree. There are too few sources. I think a merge into Singular point of a curve, or Singular point of an algebraic variety or mathematical singularity would improve the article. Mange01 (talk) 09:45, 29 December 2010 (UTC)

There are regularly international worshops on Singularity Theory. This page makes sence. I am an expert on the field. I intend to take care of it. Give me a couple of months.Coffeebrake60 (talk) 12:56, 2 February 2014 (UTC)

Some general comments

To merge with singularity of varieties is, I think, a mistake. The idea of singularity theory is, in some sense, the contrary namely that a generic family of varieties will contain singular elements. This is made clear in the sequel of the article but the examples which are given are very complicated (gravitational collapse!). For a beginner, the first thing is to give a precise to generic singularities which appear in a family. For instance, one could for instance consider plane cubics, the nodal elliptic curve and the cuspidal define manifold of codimension 1 and 2 respectively. So it is not that people are interested in singularities for themselves but in generic one parameter family, we cannot avoid them. So one seek for all kind of relation of this singularity with geometry, toplogy and so on. I some sense singularity theory is the mathematical theory of phase transitions. In this perspective Whitney's theorem (pleat and cups) should be added. The important discovery is the interaction between the local singularities and the global structure. The most striking case for a beginner is Brieskorn's theorem which states that the Milnor number is the rank of the Brieskorn lattice. I think this fundamental result could be added. I did not make any change to the page for the moment. MauricioGaray (talk) 08:58, 11 December 2014 (UTC)MauricioGaray

Do not unify; this is a real topic worthy of an article

Singularity theory (also catastrophe theory) are much distinct from the general concept of a mathematical singularity. They are a much more particular field of study of when, precisely, such abstractions as algebraic varieties or manifolds under unstable limit flow, really do yield singularities, and which forms they can really then take. They try to build a theory of what those singularities might seem like, both statically an sich, and also under dynamical considerations/on the margin/under perturbation.

Even if I don't understand such theory myself too well, I can with confidence argue that conflating such rather well-developed, more particular theories with the highest level intuitive idea of a singularity would be a grave mistake. Because these theories try to explain the *genesis* of a singularity, and quantify/qualify its true nature. They don't leave it at the popular idea of a singularity being something big and mystical, as in say a black hole, but instead are bona fide, entire *theories* of which of the many kinds of singularities in mathematics are like, and how they can come about.

Then, at the same time, I can understand why the question came up. The current article on singularity theory is to my eye really badly written. It does show some understanding of the field in many points. But then it also seems like all of those points were added as singular examples, by different people. The article seems to hang together pretty much by a thread, if even that, and certainly doesn't give you the kind of idea of the mathematical relevance of the field I just jotted out above.

So, I'd argue that the article shouldn't be unified with anything else. Instead, we should e.g. via the Wikipedia Math Project find an expert in the field who'd be willing and capable to clean this mess up. Make this article into a truly encyclopedic one, so that my inner mathematician would feel comfortable using it as a source in discussion. Decoy (talk) 01:06, 21 May 2016 (UTC)

Yes, unify

The first request for this to be unified with Mathematical singularity received the answer from the original author of the article,

Well, no. A rational function, for example, has singularities; 
but they have nothing to do with singular points in the sense of singularity

Since then, someone has fixed up the mathematical singularity article with a new definition of what a singularity is, the definition is that a singularity is where something becomes undefined. That is not perfect, but it seems OK, and that article says that for example at a cusp point of a curve the tangent space becomes undefined. This definition has the slight weakness that it also describes points where rational functions become indeterminate, however come to think about it, that may be why the word seems to be used in two different contexts. That is, the function y/x becomes indeterminate at (0,0) in the plane even though the plane is smooth, and would sometimes be called a singularity of the function itself. Some people would call the point (0,0) a singular point of the vector field x\partial/\partial x + y \partial/\partial y, just because the vector field becomes zero there while the coefficients have no common divisor.

Now, I'm not saying that this definition of mathematical singularity is the best one to consider. The issue is that (depsite the opinions of the eminent V Arnold, or his admiration of Thom), there really shouldn't be one subject known as 'singularity theory,' just as there is not really a subject known as 'collective animal behaviour.' That is a really good example of how having a Wikipedia article about such-and-such theory serves to focus on a term or point of view without any underlying substance.

I could be wrong: a really nice Wikipedia article about a 'theory' is the article Morse theory, for instance. In that case, a particular person had a particular theory which started at a particular time, had well-defined boundaries.

The Wikipedia String theory article is nice, because it does not pretend to be about anything but a 'theoretical framework'.

If someone had the time/energy/ability, it seems clear that the mathematical parts of this article should be (perhaps abbreviated and) added to the mathematical singularities article, and those parts that don't talk about mathematical singularities, so for instance which talk about black holes, should be moved into articles about black holes. Or, if black holes are an instance of a mathematical singularity (are they?) then they can be referenced there.

It is really not good to have a free-floating article about a 'theory' which is just a catch-all without any boundary for what it's about.

Actually, here is my real criticism: that in some places the article says "was getting a lot of attention" and "distaste" and other notions of what is or is not fashionable. So that does seem to be tracing the history of a 'theory' as it is formulated various ways, sometimes rejected, sometimes accepted. On the other hand, the actual definitions relate to straightforward notions in mathematics. So the article reads as if we were to write an article about equilateral triangles saying

 Theory of equilateral. A triangle is said to be equilateral if all 
 three of its sides have equal length. Dionysius of Crete dispaired 
 of the importance of equilateral. It became fashionable again in 50 BC.
 The universe is an equilateral triangle. The notion of equilateral 
 depends on a notion of length. The length of the day is 24 hours.
 

Createangelos (talk) 13:49, 31 May 2016 (UTC)

Absolutely not. I think many people don't quite realise how big a field of mathematics Singularity Theory is. It is its own unique discipline with research groups [1] and conferences [2], [3],[4]. The field of Singularity theory is specifically about just one of the definitions in Mathematical singularity, the algebraic geometry version rather than the discontinuities of real and complex analysis. Yes the article needs a lot of work, but there is enough of a unique identity of the topic. --Salix alba (talk): 15:30, 31 May 2016 (UTC)

What Salix Alba says. My interaction with this stuff was in classical mechanics, which starts with simple examples "which way will a pencil balanced on it's tip fall?" and moves on to complex examples (formation of turbulence) and goes through the ideas of Arnold et al. on the stability and dissipation of singularities. 67.198.37.16 (talk) 05:53, 12 June 2023 (UTC)