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

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 2

Something that I found very interesting and very useful is Singular value decomposition. It shows that every operator is "almost diagnosable", and is skipped in a lot of basic linear algebra courses I have seen.

I has many application, for example - solving sum of least squares of example. You can give a 30 minute talk on this in various levels as well.

There are prettier theorems (Stokes, Uniformization, and many more) but I think with the 3 constraints (interesting, useful, little background) this is a good topic.

Answer 3

Every matrix can be represented by the linear combinations of four orthogonal matrix

Answer 4

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 5

Yoneda Lemma. :D

Answer 6

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.