Answer by Maryam Ajorlou for What are some deep theorems, and why are they...
How about the compactness theorem which appears almost everywhere in Model Theory? It says a set of the first-order sentences has a model if and only if every finite subset of it has a model. There is...
View ArticleAnswer by Dianbin Bao for What are some deep theorems, and why are they...
I do not see any body mentioned Moonshine phenomena. The Fourier coefficient of a modular function gives information about the degree of irreducible representations of Monster group, if I write right....
View ArticleAnswer by Ivan Meir for What are some deep theorems, and why are they...
Niels Bohr : "It is the hallmark of any deep truth that its negation is also a deep truth"A good example would be the Riemann Hypothesis. Although in this case the opposite is not particularly...
View ArticleAnswer by Lennart Meier for What are some deep theorems, and why are they...
I want to mention a few examples from homotopy theory:1) Bott periodicity (1957): This states (in a form) that the homotopy groups of the infinite unitary group $U = colim \, U(n)$ are $2$-periodic and...
View ArticleAnswer by Steve Flammia for What are some deep theorems, and why are they...
Noether's theorem is a very deep result, even if the proof is rather simple. The relationship between symmetry and conservation laws is far from obvious, and it gives a tremendous amount of insight...
View ArticleAnswer by Iddo Tzameret for What are some deep theorems, and why are they...
I recall Tim Gowers (rough) definition of depth in mathematics, on Dick Lipton's blog: "I agree with those who have commented that depth and logical strength are different things. I would define depth...
View ArticleAnswer by darij grinberg for What are some deep theorems, and why are they...
Just a few elementary suggestions I don't have the time to elaborate on:The first deep result in linear algebra is the Jordan normal form. Arguably the difficulty of its proof is not out of proportion...
View ArticleAnswer by fedja for What are some deep theorems, and why are they considered...
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...
View ArticleAnswer by Mark Gomer for What are some deep theorems, and why are they...
I typically interpret "deep" as implying a contrast between how little work you have to put into a theorem (e.g. the complexity of the relevant axioms and preliminary definitions) and how much work you...
View ArticleAnswer by Noah Schweber for What are some deep theorems, and why are they...
In a comment on his answer (What are some deep theorems, and why are they considered deep?), njguliyev argues that "all complete classification theorems seem deep to me." This is generally something I...
View ArticleAnswer by Benjamin Steinberg for What are some deep theorems, and why are...
I believe that Gromov's theorem that groups of polynomial growth have a nilpotent subgroup of finite index is an extremely deep result. It meets almost all criteria.The proof involved new and original...
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Answer by Joseph O'Rourke for What are some deep theorems, and why are they...
I feel this may be a cut below some of the other amazingly deep theorems already mentioned, but I think a case can be made in favor of Steinitz's Theorem: the 1-skeletons (vertex-edge graphs) of convex...
View ArticleAnswer by Timothy Chow for What are some deep theorems, and why are they...
The way I understand the word "deep" is that a deep theorem should reveal some important and non-obvious mathematical fact, and its proof should not be straightforward, simple, or obvious.All these...
View ArticleAnswer by Piyush Grover for What are some deep theorems, and why are they...
Kolmogorov-Arnold-Moser (KAM) theorem in the study of perturbed Hamiltonian systems. This landmark theorem clarified a major point in the field, and answered in positive the question of existence of...
View ArticleAnswer by user26107 for What are some deep theorems, and why are they...
The spectral theorem for self-adjoint operators.ADDED: I am not a native English speaker and therefore most times I try to give shorter answers. But after seeing some comments I feel I have to add some...
View ArticleAnswer by Noah Schweber for What are some deep theorems, and why are they...
How about the unsolvability of the quintic? The idea of studying an object by looking at how it sits inside a larger, "complete" version of itself is surely one of the most important ideas in modern...
View ArticleAnswer by Vidit Nanda for What are some deep theorems, and why are they...
As a very concrete example of a deep theorem different from the ones you've already mentioned, I'd nominate the Atiyah-Singer Index theorem and its more general cousins for consideration in your talk....
View ArticleAnswer by Noah Schweber for What are some deep theorems, and why are they...
I can think of at least three different meanings I routinely ascribe to the word "deep" in mathematics: A theorem is deep if its proof requires some fundamental new way of thinking;A theorem is deep if...
View ArticleAnswer by Liviu Nicolaescu for What are some deep theorems, and why are they...
Here are some examples from around or before 1800 that come to my mind.The reciprocity theorem for quadratic residues.The description by Gauss of the arithmetic-geometric mean in terms of elliptic...
View ArticleWhat are some deep theorems, and why are they considered deep?
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...
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