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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 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.

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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

73 Answers 73

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".
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"Can be used to illustrate why we need rigorous proofs of things that are "obvious"." I don't see how this is an example for this - an illustration of need of rigor would be a situation where "obvious intuition" turns out to be wrong. Even if you accept the isoperimetric inequality without proof, as "obvious", nothing bad happens. –  Marcin Kotowski Apr 10 '11 at 9:54

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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Having come across this question by searching within the mathematics-education tag, I will try to answer it from the perspective of someone in the field of Mathematics Education.

Theorem: $n^2 - n$ is even for all natural numbers $n$.

It is quite possible that very good undergraduates (I am imagining freshmen) will laugh at seeing such a "theorem" written on the board; it is almost certain that professional mathematicians will scoff. Nevertheless, this is a talk that I have given in the past to graduate students in Math Education who wish to teach secondary school mathematics in the future. Under some reasonable interpretation of the parameters given in this question, I should think these two groups alike enough to outline the talk here.

After writing the theorem on the board, I then write down a collection of headers, each of which is intended as suggesting a method of proof. Once the headers are written out, I give the students three minutes to prove the theorem using one method that they are sure they can carry out, and to attempt a proof using another method they are less sure of. Below I will write the headers, followed parenthetically by the sort of remark I might say aloud as I write them down, and then a brief indication of the proof.

Cases: (Probably you don't need more than two) The cases I am thinking of are even and odd; check what happens when $n = 2k$ and then check what happens when $n = 2k+1$.

High School Algebra: (Factoring) Write $n^2 - n = (n-1)n$ as the product of consecutive integers, hence once of them must be even; so the product is even.

Number Theory: (This might not mean so much to you all as freshmen; we'll return to it later!)

Arithmetic: (I'm thinking of adding up a certain arithmetic sequence) Consider the sum of the first $n-1$ natural numbers; this gives some natural number $k = (n-1)n/2$. Multiplying both sides by $2$, we find that $n^2 - n = (n-1)n = 2k$ is even.

Geometry: (How would you represent $n^2$ with a geometrical picture?) Consider an $n \times n$ array of squares; remove the $n$ squares along the diagonal. The number of squares remaining is $n^2 - n$ and one sees symmetrically that they have been split into two groups of equal size. Hence the total is even.

n = 4

Combinatorics: (I'm thinking of forming two person committees...) The number of two person committees in a group of $n$ people is some integer $k = (n-1)n/2$. Cf. Arithmetic.

Mathematical Induction: (For students familiar with induction, you might give this a shot) The base case is clear; suppose $k^2 - k$ is even and note $(k+1)^2 - (k+1) = k^2 + k = (k^2 - k) + 2k$ is the sum of two even numbers, and hence even.

The point of the above is to demonstrate that even a seemingly simple statement can be proved in a number of different ways. Such a demonstration, more than any particular theorem, is likely to be useful for all students (as specified by the OP). I usually have students discuss their answers and then use the theorem we've proved to talk about something else that ought to be useful for everyone: generalization.

The proofs above made frequent use of the following fact: $(n-1)n = n^2 - n$.

How would you generalize the following statements?

Statement A: If $n \in \mathbb{N}$, then $2$ divides $(n-1)n$.

Statement B: If $n \in \mathbb{N}$, then $2$ divides $n^2 - n$.

The former statement suggests (in my mind) that $k$ divides $k$ consecutive numbers; the latter statement suggests (in my mind) that $k$ divides $n^k - n$.

Consider when $k = 3$.

Then the statements become:

Statement A: If $n \in \mathbb{N}$, then $3$ divides $(n-1)n(n+1)$.

Statement B: If $n \in \mathbb{N}$, then $3$ divides $n^3 - n$.

Not only are these statements true, they coincide: $(n-1)n(n+1) = n^3 - n$.

This overlap breaks down for $n>3$, though, and we find that only A is true for $n=4$. (Perhaps a good point at which to mention how a single counterexample can disprove a for all statement.)

From here, the talk suggests that A is a good segue into modular arithmetic, while B practically begs us to find the $k$ for which it holds. Of course, we can answer this question using Number Theory (as mentioned early on!) and, more precisely, by appealing to Fermat's Little Theorem.

I believe the talk outlined above, with its messages about the possibility of finding multiple proofs and the interesting directions in which a simple proposition can be generalized, is a practical and doable thirty minute talk for first-year students in mathematics. I have done nothing close to applying Groebner bases or making use of ultraproducts, but I have tried to heed the OP's request to be realistic.

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Nice lecture :) –  Anton Petrunin Dec 19 '13 at 3:56

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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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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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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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

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

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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Quite unbelievable that I haven't seen that answer in the previous ones.

Cantor's Theorem & Cantor's Diagonal.

Both of these are quite short, and one can squeeze them into a 30 minutes discussion including the definition of "cardinality".

I find them useful, even if not directly applicable, the shock that infinite objects (and generally, mathematical objects) need not match our finite intuition is probably one of the most important things that new mathematicians should learn. When you know that you don't know what to do, you work with the definitions slowly and carefully and eventually you develop the intuition that allows you to run freely in the field.

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Completeness theorem for first order logic.

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  • 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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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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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
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

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

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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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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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

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.
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I know that playing “elementarier-than-thou” isn't really much fun, but how can you possibly conceive of this as a lecture with no prerequisites? For example, it seems doubtful that one could convince students (usefully) that a ring is a geometric object if they didn't first have the idea that a ring was an algebraic object …. –  L Spice Apr 8 '11 at 18:09

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.

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

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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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Stone's representation theorem.

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I suppose I just find it no more implausible than taking 30 minutes to introduce metric spaces and partition of unity, and to convince students who've never encountered even those definitions of the significance of what you're talking about. I second the sentiment of Willie's and Yemon's comments (to the original question): from the dismissive response you're giving to many answers just for involving a concept like, say, ultraproduct, I confess that it is not at all clear to me what you're after for these 30 minute talks. I'll try one more answer :-) –  Ed Dean Apr 3 '11 at 21:27

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.
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