Since 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.

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.)

Measure theory: Monotone convergence theorem.

Commutative ring theory: Nullstellensatz.

Crap. I should have gotten that last one…

Differential geometry: the Stokes theorem

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?)

Apologies — an anti-Cantorian (Weierstraussian?) gremlin crept into my post: I should have said:

Set Theory: Cantorâ€™s theorem that the real numbers are uncountable.

Algebraic Topology: Van Kampen’s Theorem.

Functional Analysis: Hanh- Banach’s Theorem.

Numerical Analysis (PDE): Lax equivalence Theorem.

What is the Lax equivalence theorem?

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…

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)

Theoretical Physics: Noether’s Theorem (symmetries give rise to conservation laws)

I just found a different proof of Lax E.Thm., by using Fourier methods:

http://www.acm.caltech.edu/~acm210/2005/WINTER/week3.pdf

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