All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number theorem, and the Poincaré conjecture. I am planning to give a talk on the history of "depth" in mathematics, and for
that reason I would like to have a longer list of examples and, if possible, some thoughts about what makes them deep.
Most examples, I expect, will be from after 1800, but I am also interested in examples before that date.
When it comes to the meaning of "depth," I am interested in both specific and general explanations. In specific cases, one might point to the introduction of unexpected methods, such as analysis in Dirichlet's theorem, or differential geometry in the Poincaré conjecture, which are not implicit in the statement of the theorem.
In most cases, it is probably not provable that these methods are necessary (e.g. there are "elementary" proofs of Dirichlet's theorem), but in some cases it is provable, by general theorems of logic. Both types of explanation are welcome.
Update. I am a little surprised that nobody mentioned reverse
mathematics, which seems to offer a precise sense in which certain
theorems are "equally deep." For example, on pages 36–37 of
Simpson's Subsystems of Second Order Arithmetic there is a list
of 14 theorems, including the Brouwer fixed point theorem and
Riemann integrability of continuous functions, which are equally
deep in a precise sense. Admittedly, these are not the deepest
theorems around, but they're not shallow either. Later in the book
one finds other results of equal, but greater, depth. How do MO members
view such results?
Best Answer
There are many possible meanings of word "deep" one can detect in the common speech. I'll list three good and three bad but I do not pretend the list is anywhere near complete.
1) Very difficult (Fermat-Wiles, Carleson, Szemeredi, etc.) These theorems usually stand as testing tools for our methods and we can measure the development of the field by how easily they can be derived from the "general theory". Their "depth" in this sense deteriorates with time albeit slowly.
2) Ubiquitous (Dirichlet principle, maximum principles of all kinds). They may be easy to prove but form the very basis of all our mathematical thinking. This depth can only grow with time.
3) Influential (Transcedence of $e$, Furstenberg's multiple recurrence.) This meaning relates not as much to the statement as to the proof. Some new connection is discerned, some new technical tool becomes available, etc.
1') With an ugly proof (4 color, Kepler's conjecture). They usually reflect our poor understanding of the matter
2') Standard black boxes used without understanding ("By a deep theorem of..." something trivial and requiring no such heavy tool follows.) They are used to produce junk papers on a conveyor belt and create high citation records.
3') Hot (I'll abstain from giving an example here to avoid pointless discussions). They reflect the current fashions and self-promotion.