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!

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