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Certain mathematical objects have a theory so rich that their study alone arguably constitutes a distinct (sub)discipline. My own list would begin with

1) the absolute Galois group of the rationals;

2) the Mandelbrot set;

3) the Stone-Cech compactification of the integers;

4) the three-dimensional Cremona group;

5) the Riemann $\zeta$ -function;

6) the hyperfinite type $II_1$ factor;

7) the set of rational prime numbers;

8) $SL_2({\mathbb R})$;

9) the 27 lines on a cubic surface;

etc.

I suppose one might add "the real line," "the Euclidean plane," "the axioms of ZFC," but I'm looking for objects that have emerged out of research and whose richness itself might carry an element of surprise, rather than objects purpose-built for their universal or foundational character.

I think a survey of such objects would make a lovely text for an undergraduate capstone course, so I'm asking for your favorite examples.

My question has a sociological underpinning - there actually exist communities of mathematicians who would recognize the objects I've listed as central to their focus. I'm not allergic to suggestions of objects that should enjoy that level of attention, but for whatever reason, don't yet.

In the same spirit, I recognize that all the objects mentioned belong to broad categories, and could thus abstractly could be deemed mere examples, and certainly then studied in a broader context. But de facto, these objects enjoy a distinctive critical level of attention in relative isolation. For example, each makes an appropriate subject for a monographic treatment. But please don't hesitate to make a suggestion because your favorite object doesn't have a monograph yet!

Denis Serre
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David Feldman
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    I don't agree with your premise. Take any of your examples and look at the main contributors to their discovery and development and you'll notice few if any of those researchers lived inside a field devoted to the study of these objects. Douady, Hubbard, Fatou and Julia would certainly object to being described as Mandelbrot set theorists. Perhaps you have a pretty light definition of a subdiscipline, but I would imagine any such object would have to strictly contain at least one member -- someone that primarily lives in that realm? I doubt any of your examples satisfy this condition. – Ryan Budney Dec 11 '10 at 21:54
  • @Ryan - >Perhaps you have a pretty light definition of a subdiscipline, but I would imagine any such object would have to strictly contain at least one member -- someone that primarily lives in that realm?

    I take yours as a semantic quibble. Can you find me a better word than sub-discipline? In any case, it's not important to me whether any particular mathematician "lives primarily in that realm." If the single object has a large literature and tends to feature in the titles of works that study it, that might suffice for me.

     – David Feldman Dec 11 '10 at 22:47
  • OK, let me try to fiure out what you want: you want objects which looked small and hand-tame when they were discovered/introduced, but turned out to be full of complexity and mystery when studied. So things like "the Mandelbrot set" is okay because it was originally defined in its entirety, but things like "the category of representations of the symmetric group" are not because first came the representations, and only later they were artificially collected into a category. Now, I think "the absolute Galois group of the rationals" is more an example of the latter kind, as it is just an ... – darij grinberg Dec 11 '10 at 23:40
  • ... artificial way to collect the properties of various Galois extensions of $\mathbb Q$. Also, what is the theory of the 27 lines on a cubic? – darij grinberg Dec 11 '10 at 23:40
  • @Darij I think you grok my intentions. I would rather have too many suggestions than too few though. I anticipated that "the one versus the many" issues would arise. I think only an expert can say when the gluing of many objects into one is artificial. It would surely spoil my question to accept all answers of the form "the category of...," but sometimes one really does study a category primarily as an object unto itself rather than merely as a collection of interesting objects. Likewise, someone suggested "the Steenrod algebra" but of course there's one for every prime. – David Feldman Dec 12 '10 at 00:10
  • ... and for every prime power. ;) But probably the one for $p=q=2$ is already mysterious enough. – darij grinberg Dec 12 '10 at 00:14
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    I think $\mathrm{U}_q\left(\mathfrak{sl}_2\right)$, the quantum deformation of $\mathfrak{sl}_2$, is an example. In contrast to $\mathrm{SL}_2\left(\mathbb R\right)$, the interesting things about $\mathrm{U}_q\left(\mathfrak{sl}_2\right)$ are algebraic and still interesting over $\mathbb C$. – darij grinberg Dec 12 '10 at 00:20
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    @Darij That deserves to be an answer!

    As for the 27 lines, this gives some idea of the richness of the story: http://en.wikipedia.org/wiki/Cubic_surface

     – David Feldman Dec 12 '10 at 00:29
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    @David, this thread appears to becoming a duplicate of: http://mathoverflow.net/questions/4994/fundamental-examples Do you have a specific distinction between this list and the other? – Ryan Budney Dec 12 '10 at 02:04
  • @Ryan Thank you for pointing me to that question - some of those examples are useful here too, but I see a distinction. Namely this: a counterexample may alter the direction of a discipline or subdiscipline or field or line of research ... without become a focus thereof.

    Here's my philosophy. Many students of mathematics come away with the view that mathematics is a tool rather than a science. Students of the physical sciences often get excited about the objects studied by those sciences: galaxies, black holes, viruses, dinosaurs, quasicrystals, DNA, etc.

     – David Feldman Dec 12 '10 at 02:19
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    cont.-- I wanted...and I'm getting...a list of objects about which I can say...here are some things (as opposed to methods or facts or constructions) that mathematicians get excited about. I don't mean to reduce the study of mathematics to these objects, but merely to emphasize an aspect of mathematics for which most undergraduates I encounter in my teaching never get a feeling. – David Feldman Dec 12 '10 at 02:20
  • David: thanks a lot for the link. And you reminded me of this: a generic (i. e., the vertices are independent variables in a rational function field) inscribed hexagon ("inscribed" can mean inscribed in a circle, or, more generally, inscribed in a conic). Think of Pascal's theorem, and all that comes after it: Steiner points, Kirkman points, Cayley lines, Plücker lines... Oh, and of course the generic triangle is a whole science in itself. – darij grinberg Dec 12 '10 at 10:17
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    Sorry, I think the question as stated is not sufficiently focused to match the purpose of this site. – S. Carnahan Dec 12 '10 at 16:03
  • Although I like the question, I have the feeling that this list could be made very very long. The aspect that seems interesting to me is, how objects that where initially exceptions or counter-examples became themselves archetypes of sub-disciplines or even new theories. Examples of this are the Peano's and von Koch's curves, born as counterexamples to properties linked to differentiability, now archetypes of fractal objects. – Pietro Majer Dec 12 '10 at 16:07

16 Answers16

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The moduli spaces of curves, $\overline{\mathcal{M}}_{g,n}$.

Vivek Shende
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8

The braid group.

The Monster group.

The Steenrod algebra.

The representation ring of the symmetric group.

Todd Trimble
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8

The homotopy groups of spheres, $\pi_k(S^n)$.

J.C. Ottem
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    I know that one profitably studies stable homotopy groups of spheres as components of one grand mathematical object. Is this true for unstable groups as well? – David Feldman Dec 12 '10 at 00:15
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$\pi{}{}{}{}$ ${}{}{}{}$

Gerry Myerson
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    At some time in the past (I haven't looked recently) Citizendium's article about $\pi$, had a section titled something like "Fields in which $\pi$ is used". It was blank. The section was apparently intended to contain something, but that was future work. (I haven't looked recently, so for all I know maybe it still looks like that.) So someone commented that it was a complete list of fields of mathematics in which $\pi$ is not used. – Michael Hardy Dec 12 '10 at 04:10
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    I don't know if $\pi$ should or even really does constitute a separate subject, though I agree there are a lot of different things which relate to it and a lot of odd places where it shows up, I'm not so sure that that is a unified enough study for either of the worlds "should" or "do", but I do agree it is very common. – Adam Hughes Dec 12 '10 at 06:31
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The Korteweg–de Vries equation. For almost 90 years it was seen as just another non-linear equation stemming from fluid dynamics. Everything has changed after people discovered the world of solitons.

Andrey Rekalo
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5

$E_8$

separable Hilbert space

maybe, Thompson's group $F$

Fedor Petrov
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The Fermat equation $x^n+y^n=z^n$ is a candidate I guess.

J.C. Ottem
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  • Associated to these equations one has Fermat curves, which receive a certain amount of attention. But the real excitement has been over the set of rational points on these curves, and that turns out not such a rich object (for $n>2$). I do have a Platonic versus formalist bias here - by object I think I generally don't mean "an equation," but rather perhaps "the set of solutions" that equation. And then, I'm looking for existential, not merely logical, richness.

    But perhaps you see this a different way?

     – David Feldman Dec 12 '10 at 00:22
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    +1 because I remember my early days in algebraic number theory when we were told how much of the machinery for modern algebraic number theory came out of a desire to solve Fermat's last theorem. – Adam Hughes Dec 12 '10 at 06:37
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Hopf fibration, Icosahedron, Henon map, Hilbert (space filling) curve, Conic sections

Dick Palais
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$SL_2\mathbb R$ and its evil universal covering.

Georges Elencwajg
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Conway's Game of Life in 2-dimensions, as my exemplar instance in the class of (what used to be my overly general answer of...) Automata: deterministic finite state machines and nondeterministic and probabilistic automata and the theory behind them leading to things like acceptors of regular languages and the concepts of simulation, computational equivalence and computability as in Turing machines and "Turing equivalent", and the concept of "power of computing", computational complexity and complexity classes, bisimulation (and the equivalent computing power of single-tape vs. multi-tape and other classes of Turing machines, and the equivalent computing power of systems which can simulate other systems).

sleepless in beantown
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The Erdos-Renyi random graph model $G(n,p)$ - a single, concrete model that more or less created the field of random graph theory and is still studied.

Marcin Kotowski
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  • And for a single well-studied graph: this model applied to a countably infinite vertex set gives essentially only a single graphs (with probability one), the Rado graph, also called "the random graph". – M. Winter Aug 30 '18 at 09:51
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$C[0,1]$. Since every separable metric space embeds isometrically into $C[0,1]$ and every separable Banach space embeds isometrically isomorphically into $C[0,1]$, the study of $C[0,1]$ includes the study of the geometry of separable metric spaces as well as 90% of Banach space theory.

Bill Johnson
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    Be reasonable! I also was tempted to post an answer starting with "Almost anything would really do (with reasonable interpretation). An analytic function/the unit disk; A convex set/the simplex in high dimension; A graph/$\mathbb Z^3$" and ending with "And that's definitely a community wiki type question, if it is a question at all, which I have strong doubts about. Even if you had asked for "a single proof of a single statement about a single object", you would be in almost equally bad shape", but then decided to find some criterion for a good answer here instead (not that I succeeded :(). – fedja Dec 12 '10 at 02:06
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    What @fedja said. I don’t think “all widgets embed into $X$” together with “widgets are a field of study” makes $X$ a field of study. This is the same sort of argument by which a few logicians, set theorists, category theorists, etc. (and, more often, students meeting these subjects for the first time) claim “logic (set theory, category theory) subsumes all other mathematics”. – Peter LeFanu Lumsdaine Dec 12 '10 at 03:35
  • I also agree, if you consider the Sobolev spaces and in particular the Hilbert-Sobolev spaces, they embed nicely into other spaces and by infinite dimensionality and separability the latter are isometrically isomorphic to $\ell^2$, but at the same time the ways to go between them isn't really easy to recover the structure of one from the other, especially trying to figure out $H^k=W^{k,2}$ from just knowledge of $\ell^2$. – Adam Hughes Dec 12 '10 at 06:35
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    But studying a Banach space involves understanding the structure of its subspaces. 'Course the corollary to the universality of $C[0,1]$ is that we will never understand the structure of $C[0,1]$ as a Banach space. – Bill Johnson Dec 12 '10 at 16:15
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Godel's constructible universe L.

Amit Kumar Gupta
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The free group factor(s) -- not just because of the infamous free group factor problem, but also because, IIRC, $VN(\mathbb F_2)$ and relatives appeared very early on when von Neumann et al. were laying out the theory and looking for examples to demonstrate its richness.

Yemon Choi
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The hyperbolic space.

Guillaume Brunerie
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Knots. Quandles and Racks.

sleepless in beantown
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    I think the OP was looking along the lines of a particular object (e.g. a particular knot) rather than a particular class of objects. – Qiaochu Yuan Dec 12 '10 at 06:24
  • @Qiaochu-Yuan, oops, you are indeed correct. I've overgeneralized and will retreat to a more specific position after my late dinner, and edit this into particular objects, say "the Conway Game of Life cellular automaton", and maybe the "unknot", though there's probably a better knot candiadte than the un-knot. – sleepless in beantown Dec 12 '10 at 06:57