Question

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You need to present a theorem which will be 100% useful for them.

What would you do?

One theorem per answer please. Try to be realistic.

For example: 30 min is more than enough to introduce metric spaces, prove existence of partition of unity, and explain how it can be used later.

P.S. Many of you criticized the vague formulation of the question. I agree. I was trying to make it short --- I do not read the questions if they are longer than half a page. Still I think it is a good approximation to what I really wanted to ask. Here is an other formulation of the same question, but it might be even more vague.

Before I liked jewellery-type theorems; those I can put in my pocket and look at it when I want to. Now I like tool-type theorems those which can be used to dig a hole or build a wall. It turnes out that there are jewellery-type and tool-type theorems at the same time. I know few and I want to know more.

Answer 1

The well-ordering theorem and an application (that uses transfinite recursion, after well-ordering a set). Many interesting sets and examples can be built that way. Or maybe Axiom of Choice/Zorn's lemma (show one from the other) and then show the well-ordering theorem.

Answer 2

Heisenberg's uncertainty principle.

• Everyone should be exposed to quantum mechanics.
• Appears frequently in analysis and probability (not to mention physics).
• Showcases some of the highlights of Fourier theory.

Answer 3

The isoperimetric inequality.

• Ubiquitous in geometry.
• Among the easier examples of variational problems.
• Can be used to illustrate why we need rigorous proofs of things that are "obvioius".

Answer 4

Uniform convergence of the averages of the partial sums of the Fourier series, for any continuous function $f$ on $[0, 2 \pi]$ with $f(0)=f(2\pi)$:

$$ \sigma_N(f, \theta) = \sum_{n = -N}^N \left(1-\frac{|n|}{N+1} \right) \widehat{f}(n)e^{in \theta} \to f(\theta) $$

And the Weierstrauss Polynomial Approximation Theorem: the polynomials are uniformly dense in $C[a,b]$. This is a corollary of the Fourier series result, or it can be proved similarly. Finally, if time permits, the Stone-Weierstrauss Theorem.

Of course, it would be nice to talk about approximations to the Dirac Delta, convolutions, fundamental solutions to PDEs, e.g. the Heat Equation, etc. etc. but I suppose only a REALLY good class could absorb all this in half an hour...

Answer 5

Hilbert projection theorem

Answer 6

Robinston-Schensted-Knuth algorithm

This is a map between permutations to pairs of standard tableaux. So it immediately gives various wonderful facts. It is elementary, short and useful.

Answer 7

Let $G$ be a finite group and $V_i$, $i=1,...,r$ be the irreducible representations, $d_i:=dim(V_i)$. Then $|G|=\sum_i d_{i}^{2}$.

Answer 8

At the risk of incurring the wrath of some here, I would propose the Yoneda Lemma, along with the minimum of necessary category theory. Like it or not, category theory is hugely useful to algebraists, and early exposure can be very helpful. (It was to me!)

Answer 9

Gödel's incompleteness theorems

A non-technical overview could be done in a fairly short amount of time, thus allowing for some discussion of its various implications, particularly regarding possible roles of mathematics.

Answer 10

I've always been thrilled by the fact that the coefficients of a (monic) polynomial are obtained by taking the elementary symmetric functions in (minus) the roots of that polynomial:

$$\prod_{i=1}^n (X+\alpha_i) = \sum_{k=0}^n (\sum_{i_1 < \cdots < i_k} \alpha_{i_1}\cdots \alpha_{i_k})X^{n-k}$$ A lot is built on this, I think. I'd like to explain the connection to automorphisms and fixed fields and how the roots of a polynomial are permuted by an automorphism that fixes the coefficient field of that polynomial. Then maybe mention the beginnings of Galois theory.

Answer 11

The Gelfand-Naimark theorem: every commutative C* algebra is $C_0(X)$ for some locally compact Hausdorff space $X$.

• The spectral theorem is a corollary.
• The theorem introduces students to the idea that a ring is a geometric object
• Certain constructions in topology, e.g. the Stone-Cech compactification, become more transparent.

Answer 12

My first choice was taken, Picard iteration using Fixed point principles. I'll try not to have a repeat. I have been teaching a history of math class this semester so this sort of thing has been on my mind recently.

I would definitely consider different choices depending on how advanced the students I expected were.

Pre-Calculus but talented: Archimedes method for finding $\pi$. Calculus: Fermat method for finding the integral of $x^n$ Differential Equations: Picard iterations/fixed point principles more advanced. The Brachistichrone.

Another topic that I like, specifically for analysis is to take some of the different definitions of continuity and show that they are equivalent.

Answer 13

Pursuant to Johannes's answer, I would like to give a talk entitled “How to factor $x_0^4 + x_1^4 + x_2^4 + x_3^4 - 2x_0^2 x_1^2 - 2x_0^2 x_2^2 - 2x_0^2 x_3^2 - 2x_1^2 x_2^2 - 2x_1^2 x_3^2 - 2x_2^2 x_3^2 - 8x_0 x_1 x_2 x_3$”.

Answer 14

I would tell them "What is real maths". To achieve this use Lakatos way about Euler's formula ( $ V - E + F = 2 $ ).
It is a set of successive reformulations (more and more precise) each followed by a counter example justifying the next reformulation.

Reference is : I. Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery

Answer 15

Fundamental Theorem of Finitely Generated Abelian Groups.

Answer 16

Jordan normal form.

Answer 17

I would introduce Bezout's Theorem (there is an article on wiki). It will be hard to prove this statement in the full generality, but the proof of the weaker statement:

The system of two polynomials $P(x,y)$ and $Q(x,y)$ without common factors of degrees $m$ and $n$ correspondingly has at most $mn$ solutions.

takes one page at most and uses only the fact that polynomials of two variables have a unique factorisation in irreducible polynomial. (for example, you can check page 244 in an appendix of the book "Rational Points on Elliptic curves" of Silverman and Tate).

The well-known beautiful (or, say, elementary) application of this theorem is Pascal's theorem.

Answer 18

[I would introduce Taylor's theorem and point out that it has many applications for instance in physics but also in differential geometry. On the one hand very elementary proofs can be given, but on the other hand, for practical computations with "nice" functions it is always helpful to have that theorem in full generality at the ready. For instance in Riemannian Geometry, one uses Taylor expansion in combination with Jacobi fields to expand the metric tensor locally. This does show that locally, we can find coordinates s.t. the metric behaves like the standard Euclidean metric, but there have to be some corrections such as one term involving the Riemannian curvature tensor.][http://en.wikipedia.org/wiki/Taylor's_theorem]

Answer 19

Completeness theorem for first order logic.

Answer 20

Maybe a stretch, but...

Finiteness of the class number via Minkowski's theorem.

• Everyone should at least have a rough idea what the class number is.
• Minkowski's theorem has other amusing and useful applications (e.g. well-definedness of the signature?)
• One of the first (of many) interesting theorems involving the geometry of lattices.

Answer 21

The Martingale stochastic process

Answer 22

I can just imagine what would have happened if I was introduced to Kepler's Conjecture and Thomas Hales' approach earlier ...

Answer 23

Schur's Lemma. After which one can as an application, classify the simple modules for cyclic groups.

Answer 24

Sperner's Theorem on antichains in subset lattice and the Sunflower Lemma. Two great theorems in combo which require little to no theory to introduce and have extremely beautiful proofs.

Answer 25

Integration by Parts It's a powerful analytical tool and it can be used for reduction of order on complex functions.

Answer 26

• I would go for Cayley's theorem which asserts that every group is isomorphic to a subgroup of $S_{n}$ for some $n$.

One, can even look into this following post:

• http://math.stackexchange.com/questions/10029/importance-of-cayleys-theorem

Answer 27

• The famous Heine - Borel theorem which says that a closed a bounded subset of $\mathbb{R}^{n}$ is compact.

Answer 28

The Arithmetic Mean-Geometric Mean Inequality.

Answer 29

Moore closures, their relation to collections of Moore-closed sets and a characterization for closure under finitary operations.

One can then discuss why Moore-closed sets form a complete lattice and a lot more, if one feels so inclined.

This is certainly something students will encounter over, and over, and over again in different guises. Moore-closures are certainly among the most useful trivialities I know.

Answer 30

Stone's representation theorem.