Fundamental theorems [closed]

Question

I am certainly sure that any one who has read Gil Kalai's witty community wiki has benefited a lot. Here I follow a similar track in asking this question. So let's compose a list of fundamental theorems in mathematics which may not even have the tag "fundamental" but have serious wight in the respective branch of math.

I will start with the elementary and very popular ones.(Please add a description if the theorem is fundamental but still not so well-known)

Thanks for all your effort.

1. FTA: The Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem): ->Any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers.

2. FTA: The Fundamental theorem of Algebra: -> The field of complex numbers is algebraically closed

3. FTC: The fundamental theorem of calculus: -> Has two parts and specifies the relationship between the two central operations of calculus: differentiation and integration.

4. FTLP: The fundamental theorem of linear programming: -> In a weak formulation, states that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners.

Answer 1

In his book Topics in Geometric Group Theory, Pierre de la Harpe calls the following result the Fundamental Observation of Geometric Group Theory (though he also calls it a theorem!). It is also often called the Svarc--Milnor Lemma. Roughly speaking, it asserts that the coarse geometry of a group is captured by any suitably nice action of that group by isometries on a metric space.

Theorem. Let $X$ be a metric space that is geodesic and proper, let $\Gamma$ be a group and let $\Gamma$ act properly discontinuously and cocompactly by isometries on $X$. Then $\Gamma$ is finitely generated, and furthermore for any $x_0\in X$ the map $\Gamma\to X$ given by

$\gamma\mapsto\gamma x_0$

is a quasi-isometry.

Remarks.

1. $\Gamma$ is endowed with the word metric (with respect to some choice of finite generating set).

2. A map of metric space $f:Y\to X$ is a quasi-isometric embedding if there are constants $\lambda\geq 1$, $\mu\geq 0$ such that

   $\lambda d_Y(y_1,y_2)+\mu\geq d_X(f(y_1),f(y_2))\geq \frac{1}{\lambda} d_Y(y_1,y_2)-\mu$

   for all $y_1,y_2\in Y$. It is a quasi-isometry if, furthermore, for every $x\in X$ there is $y\in Y$ such that $d(x,f(y))\leq \mu$.

Answer 2

I used to joke that The Fundamental Theorem of Combinatorics is interchange of summation.

Answer 3

To add to the Gowers examples: the fundamental theorem on finitely-generated abelian groups. It seems at least a mildly interesting linguistic point. German discriminates between Hauptsatz and Fundamentalsatz, i.e. main theorem and fundamental theorem (if satz is not quite "theorem"). That distinction seems less clear in the English usage. The German Wikipedia admits the Fundamental Theorems of Algebra, Analysis and Arithmetic, but others in pure mathematics aren't obvious. I would myself think of Galois theory (the perfect duality of subfields and subgroups) and projective geometry (collineations semi-coordinatised) as having "fundamental theorems".

Answer 4

The Fundamental Theorem of Asset Pricing (FTAP) in mathematical finance also comes in two parts. The first part says, more or less, that a market is arbitrage-free if and only if there is an equivalent martingale measure for the discounted price process. The second part says that the market is complete (all European options can be hedged) if and only if the equivalent martingale measure is unique.

(In some models, you may need an appropriate definition of "arbitrage-free", such as the notion of "no free lunch with vanishing risk", and you may replace "equivalent martingale measure" with "equivalent local martingale measure". But the idea is the same.)

Answer 5

Wikipedia says the Fundamental Theorem of Riemannian Geometry is the unique existence of the Levi-Civita connection. I've never heard it called that myself, so this is maybe an anti-answer.