Elementary + short + useful

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 jewelry-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 turns out that there are jewelry-type and tool-type theorems at the same time. I know a few and I want to know more.

Answer 1

Stokes' Theorem

Answer 2

The definition of the tensor product and existence/uniqueness/associativity properties.

I know, this is perhaps not a single theorem but in my eyes one of the most useful "elementary" concepts. Personally, I had two semesters of linear algebra without mentioning the tensor product. And from this I suffered for a long time during my further studies. Now it is my first homework/exercise for students in my lectures (e.g. diff geo).

If the student is really clever, one can even do something like the tensor algebra in these 30 min.

Answer 3

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 4

Sperner's lemma (Two-dimensional case)

Answer 5

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 Weierstrass 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-Weierstrass 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 6

Theorem. $\sqrt{2}$ is irrational.

This is an ancient theorem, about 2400 years old, and its modern proof is identical to the one appearing in Euclid's elements. A simple number theoretic proof, where you get the chance to use the abductio ad absurdum (or εἰς ἄτοπον ἀπαγωγή).

Note. As Victor Protsak noted, the number-theoretical proof is not the first one. The first one is believed to geometrical, using anthyphaeresis (ἀνθυφαίρεσις), i.e., proving geometricallly that the euclidean algorithm of dividing $1+\sqrt{2}$ by $1$ is periodic: \begin{align} 1+\sqrt{2}&=2\cdot 1 +v_1, \\ 1&=2\cdot v_1+v_2, \\ v_1&=2\cdot v_2+v_3, \\ \text{etc} \end{align} and thus $1+\sqrt{2}$ and $1$ are inconsummerable (ἀσὐμμετρα). It is noteworthy that, although the number theoretical proof appears Euclid's Elements, which were written c. 300 BC, the fact that there is a proof that the square roots of positive integers less than 19 is mentioned in Theaetetus of Plato, written c. 380 BC. Anthyphaeresis works for every $n$, but it can get extremely complicated, as $n$ gets larger. In fact, for $n=19$, in order to establish periodicity of Euclidean algorithm, 6 steps are required, and huge geometrical figures to observe it! A few years ago I supervised a Master's thesis on this proof, and I think it makes an extremely interesting lecture.

Answer 7

Compactness of First Order Logic (using ultraproducts, not as a corollary of completeness; they get Łoś's Theorem for ultraproducts as a freebie.)

Answer 8

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 9

Combinatorial Nullstellensatz. You may prove it and then choose your favorite applications for as many minutes as you have. I personally like to include applications to evaluation of coefficients, as explained in this MO answer, after that to additive combinatorics, like Cauchy--Davenport theorem, and to graph theory, like 3-choosability of a planar bipartite graph.

Answer 10

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 11

I am surprised that no one mentioned the Baire category theorem.

I am not sure if you would have enough time to show many applications in 30 minutes but it is almost certain that they will end up using it at some point. Here are some applications discussed on MO.

Answer 12

Existence of Nash equilibria. This is feasible in 30 minutes, and builds surprising connections between game theory, elementary probability, elementary geometry, and algebraic topology.

Answer 13

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 14

Series representations for functions and the fact that $\mathbb C$ is "rigid" in contrast to $\mathbb R$ when discussing differentiability and series developments.

This "explains" for example how pocket calculators compute trigonometric functions, logarithms and exponentials.

Answer 15

Hilbert projection theorem

Answer 16

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 17

The fact that Fourier-Motzkin elimination gives a sound and complete procedure for determining whether a system of linear inequalities has a solution (and for computing the existential projection of the solution-set onto a subset of the variables). The proof is intuitive, very difficult to forget, and provides a useful procedure for hand-computations in low dimensions. Farkas' Lemma follows as an obvious consequence.

The general idea of variable elimination occurs throughout mathematics and logic and is closely connected to tools used to automate mathematics - for instance, the Resolution proof system is used in SAT-solving, for algebraic geometry we use Gröbner bases, for polynomial inequalities we use Cylindrical Algebraic Decomposition. Fourier-Motzkin elimination is a perfect introduction to the general technique and is often practically useful. Farkas' Lemma also motivates convex duality, a cornerstone of mathematical optimization.

Answer 18

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

Answer 19

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.

Answer 20

My suggestion -- assuming they have not yet taken a class on complex analysis -- would be to talk about Eulers formula and De Moivre's formula, along with the complex representations of the most common trigonometric functions. Perhaps, if there is time left, power series and the Cauchy product could be touched upon.

This could help the students to understand better how some trigonometric identities can be derived, which is usually not explained in detail until a first course on complex analysis.

Each of the topics is simple enough to introduce in a very short amount of time, so there would probably be time left to show some cool applications.

Answer 21

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 22

Sanov's theorem of large deviations.

I don't have to prove anything, right? If they want a proof, they'll look it up in a book later.

Assume the students already know about the central limit theorem. Explain how the two theorems talk about limits in different direction: let $ S_n $ be the sum of $ n $ independent variables of identical distributions (real valued, with zero mean and finite variance), the central limit theorem gives a limit of the unscaled probability $ P(S_n/\sqrt{n} < c) $, this limit is strictly between 0 and 1; whereas large deviation theorems give the rate of decrease of a probability like $ P(S_n/n < c) $.

Answer 23

The Martingale stochastic process

Answer 24

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 25

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

Answer 26

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 27

The Arithmetic Mean-Geometric Mean Inequality.

Answer 28

Tarski's fixed point theorem.

Answer 29

The Laplace approximation to integrals! This scores well on all points: it is elementary (could be taught in calculus 1) and very useful. Within 30 minutes there should be time to some application, dependent on the audience, maybe the Stirling approximation to the factorial (which is often used without any proof).

Answer 30

Cauchy's integral theorem and Cauchy's integral formula.

It's really an example of a jewellery-type and tool-type theorem at the same time. It can be introduced and proved for students that even don't know about functions of complex variables in 20 minutes. And other 10 minutes can be spend to say how many applications and generalizations these results have in theory of functions and applied mathematics.