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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!

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closed as off topic by Kevin H. Lin, Daniel Moskovich, Denis Serre, Harry Gindi, S. Carnahan Dec 12 '10 at 16:01

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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
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Community wiki? –  Todd Trimble Dec 11 '10 at 23:05
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Community wiki would be appropriate, together with the "soft-question" tag and the "big-list" tag. See the FAQ and the community wiki discussion on meta. –  Ryan Budney Dec 11 '10 at 23:40
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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 0:20
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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 2:20

17 Answers 17

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.

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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).

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at least, it used to be a sub-discipline in the late 60's and early 70's... –  sleepless in beantown Dec 12 '10 at 7:02

Knots. Quandles and Racks.

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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 6:24
    
@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 6:57

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.

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

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Hopf fibration, Icosahedron, Henon map, Hilbert (space filling) curve, Conic sections

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Godel's constructible universe L.

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Do you mean this in the sense that every mathematician studies objects in L or in the sense that the study of L is an important branch of logic? –  Qiaochu Yuan Dec 12 '10 at 5:56
    
The latter${}{}$ –  Amit Kumar Gupta Dec 12 '10 at 9:24

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

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And when the inevitable question arises on MO "Which mathematics books have the shortest title ?", I'll recycle this answer and quote Serge Lang's $SL_2 \mathbb R$. –  Georges Elencwajg Dec 12 '10 at 1:45
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A=B is a shorter title. –  Felipe Voloch Dec 12 '10 at 2:12
    
Sure, Felipe. I am only enjoying the fantasy of lazily answering two MO questions with just four typographical characters... –  Georges Elencwajg Dec 12 '10 at 3:02
    
+1 to Felipe, and Georges' original question, as I agree with the adjective "evil" most whole-heartedly. –  Adam Hughes Dec 12 '10 at 6:36

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.

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$\pi{}{}{}{}$ ${}{}{}{}$

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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 4:10
    
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 6:31

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

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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 0:15

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

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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 0:22
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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 6:37

$C[0,1]$. 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.

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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 2:06
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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 3:35
    
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 6:35
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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

I think the set of ultrafilters on the set of natural numbers would be an answer. BTW, what anyone can tell me about this set?

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The set of ultrafilters on the set of natural numbers is essentially the same thing as my #3. –  David Feldman Dec 11 '10 at 23:29

The braid group.

The Monster group.

The Steenrod algebra.

The representation ring of the symmetric group.

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+1 for Steenrod and representation ring of the symnmetric group. I'd give +1 for each if I could. –  Adam Hughes Dec 12 '10 at 6:29

$E_8$

separable Hilbert space

maybe, Thompson's group $F$

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