“Fundamental” Theorems
June 30th, 2005 by michaelSince most posts don’t get many comments, I thought I would make one the required audience participation. The subject is “fundamental” theorems in the various subjects. What I am going for is hard to actually describe, but encapsulates a theorem being fundamental, its utility, its depth. It is the result in the subject that would hurt the most not to have, but does not have to be the putative “fundamental theorem of X”
For example, my votes for a few subjects:
Calculus: Mean Value Theorem.
Probability: Linearity of expected value.
Model Theory: The compactness theorem.
June 30th, 2005 at 3:59 pm
In Riemann Surfaces (my field years ago) the Uniformisation Theorem: Every simply connected Riemann surface is conformally equivalent to C, the unit disk (or equivalently the upper half plane), or the Riemann sphere.
The universal cover of any Riemann surface is, by definition, simply connected. So every Riemann surface can be obtained as a quotient as one of the above by the surfaces fundamental group. So this gives a nice concrete handle on any Riemann surface. As all non-singular algebraic curves over C give Riemann surfaces we have an interesting way to look at these objects too.
(Or if you don’t like that, the Uniformization Theorem is a consequence of the Riemann Mapping Theorem.)
July 1st, 2005 at 12:29 am
Measure theory: Monotone convergence theorem.
Commutative ring theory: Nullstellensatz.
July 1st, 2005 at 12:35 am
Crap. I should have gotten that last one…
July 1st, 2005 at 4:01 pm
Differential geometry: the Stokes theorem
July 2nd, 2005 at 4:45 am
Mathematical Statistics: The Central Limit Theorem
Logic: Godel’s Incompleteness Theorems
Set Theory: Cantor’s theorem that the rationals are uncountable
Number Theory: The non-rationality of the square-root of 2 (due to Pythagoras?)
July 2nd, 2005 at 4:47 am
Apologies — an anti-Cantorian (Weierstraussian?) gremlin crept into my post: I should have said:
Set Theory: Cantor’s theorem that the real numbers are uncountable.
July 2nd, 2005 at 6:08 am
Algebraic Topology: Van Kampen’s Theorem.
July 2nd, 2005 at 6:18 am
Functional Analysis: Hanh- Banach’s Theorem.
Numerical Analysis (PDE): Lax equivalence Theorem.
July 2nd, 2005 at 11:17 pm
What is the Lax equivalence theorem?
July 2nd, 2005 at 11:26 pm
Graph Theory: Menger’s Theorem.
Let G be a graph and A, B be vertices in G. Then the minimum number of vertices separating A from B in G is equal to the maximum number of A-B disjoint paths in G.
There are three proofs available in Diestel.
Also Number Theory (and my favorite): There are infinitely many primes…
July 3rd, 2005 at 8:25 am
Lax Equiv. Thm: a consistent finite difference scheme for a well posed initial value problem is convergent if and only if it is stable.
Is a very deep theorem, which follows from the Uniform boundedness principle.
By the way, Lax won the Abel Prize this year: http://www.abelprisen.no/en/prisvinnere/2005/documents/abelprize_2005_EN.pdf , (previously, Jean-Pierre Serre in 2003, and MF Atiyah and IM Singer in 2004)
July 3rd, 2005 at 8:35 am
Theoretical Physics: Noether’s Theorem (symmetries give rise to conservation laws)
July 3rd, 2005 at 8:43 am
I just found a different proof of Lax E.Thm., by using Fourier methods:
http://www.acm.caltech.edu/~acm210/2005/WINTER/week3.pdf
July 15th, 2005 at 11:44 am
[...] Last week, when Michael asked for a list of fundamental theorems in different branches of mathematics, “>Juan de Mairena suggested the Lax Equivalence Theorem as a candidate. Today on ArXiv I spotted a paper that makes the rather dramatic claim that the theorem is “wrong” — not that it is wrong in the strict mathematical sense, but that its conditions are not realistic for real-world problems. I’m not in a position to evaluate the claim (I never even heard of the result until Juan’s comment), but I thought it was interesting to see a paper on the subject so soon after we discussed it. [...]
March 3rd, 2007 at 5:05 am
Lebesgue dominated convergence theorem
August 15th, 2007 at 5:01 am
hmm.. well that last spam just bombed.
August 15th, 2007 at 7:54 am
I think the companies that write spamming software have rolled out new versions. The spamming strategies have changed in a way that lets more slip past Akismet.