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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.

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How many years of undergraduate education do those students have? What can we assume that they know? (It is a big difference between one who's in the second half of her third year and one that just started two months ago.) – Willie Wong Apr 3 '11 at 18:28
I find it hard to square the "no prerequisites" condition with the "partitions of unity" example. Or are we talking about ideal undergraduate students, who like ideal gases are only an approximation to the reality? – Yemon Choi Apr 3 '11 at 20:34
In my opinion, the "try to be realistic" injunction (which I approve of in all pedagogical questions; note that a lot of experienced teachers do see some of the more ridiculously ambitious pedagogical suggestions promulgated in some answers here and have a good laugh at the naivete of the authors) is hard to square with the vagueness of the question. The term "very good undergraduate" alone is a currency whose value will rise and fall according to where you go. It is tempting to close the question as "too localized" for this reason, but I'll think about it a bit more... – Pete L. Clark Apr 3 '11 at 23:03
I too find the partitions of unity example unrealistic. I do think this and some of the examples below could be made to work if one wasn't obliged to give a proof, but perhaps only an intuitive idea, and then explain why it was useful -- sort of like a colloquium talk for undergraduates. – Todd Trimble Apr 3 '11 at 23:10
Indeed, Anton, you can do all sort of things in 30 minutes... but unless the students already somewhat familiar about the subject you are talking about, it is rather unusual that you can introduce three new objects, two concepts, and a theorem to anyone and as a result get them to understand the significance of anything. – Mariano Suárez-Alvarez Apr 4 '11 at 17:10

78 Answers 78

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

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Hmm, "using ultraproducts"... – Anton Petrunin Apr 3 '11 at 20:54
I have a lot of reservations about this answer, which will be more or less valid depending upon how you interpret the parameters of the question (which I also think is rather vague). First of all the OP said "100% useful". Now I happen to know and like this exact result enough to have made it the climax of a short course I taught last summer. Nevertheless I have not yet used any form of the Compactness Theorem for anything in my own work (I am an arithmetic geometer), and I think probably the majority of working mathematicians would say the same thing.... – Pete L. Clark Apr 3 '11 at 22:53
Second, the course I taught consisted of eight two-hour lectures to math graduate students (who were "very good" according to at least one reasonable interpretation of the term). It was not assumed that they had any previous exposure to mathematical logic of any kind, nor any previous exposure to ultrafilters. (And in fact none of them did have any prior experience with these things.) I mentioned the Compactness Theorem in either the second or third lecture, at the time without proof. The proof came in the last lecture, after I introduced ultrafilters from scratch... – Pete L. Clark Apr 3 '11 at 22:55
And you want to do all of this in half an hour, for undergraduates? I suppose I could compile a nonempty set of undergraduates (Qiaochu Yuan, Akhil Mathew, Zev Chonoles,...) for which this might have a chance of flying, but as a general suggestion this comes off as being much more likely to blow up in one's face. – Pete L. Clark Apr 3 '11 at 22:58
I think the compactness theorem is useful even if you don't apply it in your work. I think it is the best way to understand what the difference between first order sentences and others is. – Michael Greinecker Apr 4 '11 at 6:15

Sperner's lemma (Two-dimensional case)

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I don't know about 100% useful, but since I have seen a striking use for it (in the proof of Monsky's theorem about cutting a rectangle into congruent triangles) I won't object on that account. I note that the OP's suggestions seem more reasonable than some of the answers... – Pete L. Clark Apr 3 '11 at 23:10
Pete, can you give me a reference on that? – Mariano Suárez-Alvarez Apr 4 '11 at 17:53
@Mariano: sure, how about this? (I was going to give you the link to Monsky's original article, but to my surprise he does not explicitly use Sperner's Lemma. But that's the way it was presented in a talk given by Aaron Abrams in the graduate student seminar at UGA a few years ago. By the way, this was maybe the best talk I have seen in my five years in Georgia...) – Pete L. Clark Apr 4 '11 at 19:29
In my first comment, please replace "congruent" with "equal area". (Oops!) – Pete L. Clark Apr 5 '11 at 14:09

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 "elemetary" 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.

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As with other answers, I have to downvote here, because it just does not fit to the question; unless you give a specific motivation and application, why this would be interesting and is not just a part of elementary linear algebra. – Martin Brandenburg Apr 12 '11 at 6:26

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.
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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.

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I don't think this is needed to prove Pascal. How are you using it? (NB: the case when one of $P$ and $Q$ is linear is trivial.) – darij grinberg Nov 9 '11 at 19:17
Dear Darij, well that is not me who is using it ... This is Fulton, Miles Reid, Seilverman, and many many others (basically any algebraic geometer who wrote a book on curves)... you can check page 62 here, for example : – aglearner Nov 9 '11 at 22:39
My point is that when one of the polynomials $P$ and $Q$ factors into linear factors, you cannot talk of a real application of Bezout - it's a triviality. – darij grinberg Dec 3 '11 at 17:01
Darij, sure, I understand your point. The proof of Pascal using Bezout is non-trivial. It is applied to the conic and to one specific cubic in the pencil generated by two reduced cubics consisting to two triples of lines of the hexagon. – aglearner Jan 16 '12 at 0:30
Lecture 1 gives a half page proof of slightly weaker statement – aglearner May 9 '12 at 16:41

The Archimedes proof that the uniform distribution on the sphere projects on the uniform distribution on a diameter.

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I don't understand what the statement is supposed to be, nor am I confident that Archimedes is someone who proved it. Could you link to a reference, please? – Todd Trimble Mar 27 '14 at 17:05
@ToddTrimble If you consider a sphere to have a uniform mass distribution (constant surface density), and consider the vertical diameter, then if you project all mass horizontally onto this diameter, this diameter (segment) acquires a uniform mass distribution (line density). This is related to the fact that the surface area of a "spherical zone" is proportional to its height, $A=2\pi Rh$. This was certainly known to Archimedes and is associated with him. It is related to the Lambert cylindrical equal-area projection ("Archimedes projection"). – Jeppe Stig Nielsen Sep 25 '15 at 20:43
@JeppeStigNielsen Thank you! That was very helpful. – Todd Trimble Sep 25 '15 at 22:02

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.

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Transfinite recursion is wonderful, but 30 minutes are not enough for it. (To test their understanding, ask them to show that it fails for non-well-orders.) – Goldstern Jun 6 '11 at 16:23

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...

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I think convolutions and approximations to the identity is a great idea, if it can be achieved. – Todd Trimble Apr 5 '11 at 11:39
(If done right, I think it could.) – Todd Trimble Apr 5 '11 at 11:40

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 appaears 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, writeen 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.

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Well, there are by now many proofs, lending themselves to different directions and generalizations, and such might make for an interesting 30-minute lecture to undergraduates. – Todd Trimble Dec 19 '13 at 23:15
@ToddTrimble Agreed; I think What would you do? ought to encompass more than just the stated theorem... – Benjamin Dickman Dec 19 '13 at 23:44
In fact, there is some controversy as to whether the "traditional" even-odd reductio ad absurdum proof was the first one. Many sources assert that the original proof extended to irrationality of $\sqrt{d}$ for $d<17, d\ne 1,4,9,16,$ which would be consistent with not using elementary divisibility properties of primes. Also, some authors believe that a geometric proof involving the diagonal and the side of a square (the one that is equivalent to the non-termination of the continued fraction expansion of $\sqrt{2}-1$) was invented concurrently with or earlier than the even-odd argument. – Victor Protsak Dec 20 '13 at 1:46
@BenjaminDickman I agree with you; perhaps smyrlis would like to add more details. – Todd Trimble Dec 20 '13 at 1:56
@VictorProtsak I've also heard it said that the proof of irrationality of $\sqrt{5}$, based on the geometry of the pentagon, may well have preceded that of $\sqrt{2}$. – Todd Trimble Dec 20 '13 at 1:57

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

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

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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.

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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.

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Yes, that and Galois connections, which are closely related. – Todd Trimble Feb 2 at 23:10

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

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This is certainly a high point in a first course on representation theory, but why is it a worthy stand-alone topic? Will it be useful to a student who otherwise knows no representation theory? (Or will it persuade a student to study representation theory?) – Pete L. Clark Apr 4 '11 at 14:46
When I was an undergraduate, I was persuaded to read Serres book when an older student told me about that result. – Johannes Ebert Apr 4 '11 at 15:08
Once one knows a bit of representation theory, one is certainly set up to appreciate this as a surprising and exciting result; but, for a typical undergraduate audience, I would think one would have first to define a representation—which, itself, if done and motivated well, should take a big chunk of the time. – L Spice Apr 8 '11 at 16:48

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) $.

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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!)

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I also considered the Yoneda lemma, but I think it's a tricky case. To me the Yoneda lemma is just about the deepest "triviality" (if that isn't too self-contradictory!) in all of mathematics, but I think its profound significance takes quite some time to sink in, and it's not so easy to get that across in 30 minutes (I don't think). – Todd Trimble Apr 5 '11 at 11:38
@Todd: Well, it might be worth a try... (I'm in a better mood today, I guess.) – Pete L. Clark Apr 5 '11 at 14:13
To explain the Yoneda Lemma to undergraduates, you need to introduce the concept of a category, that of a functor, and that of a natural transformation (unless that is taught in an undergraduate course, but if it is, then the Yoneda Lemma is probably taught in that course, too). Then you can start working on the lemma. I don't see how this can reasonably done within 30 minutes, in particular because just giving definitions does not given the students any intuition. – Niemi Sep 9 '13 at 8:55
  • The famous Heine - Borel theorem which says that a closed a bounded subset of $\mathbb{R}^{n}$ is compact.
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[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.]['s_theorem]

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You should definitely give the integral form of the remainder. This form somehow seems less well known despite being at least seven hundred thirty times as useful. – Phil Isett Dec 4 '11 at 4:22

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.

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Completeness theorem for first order logic.

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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

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Fundamental Theorem of Finitely Generated Abelian Groups.

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Jordan normal form.

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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.

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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.

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Tarski's fixed point theorem.

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In what sense will this be 100% useful? – Douglas Zare Jan 10 at 6:04

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.

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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.

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

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