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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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    $\begingroup$ 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? $\endgroup$
    – Yemon Choi
    Commented Apr 3, 2011 at 20:34
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    $\begingroup$ 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... $\endgroup$
    – Pete L. Clark
    Commented Apr 3, 2011 at 23:03
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    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented Apr 3, 2011 at 23:10
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    $\begingroup$ Anton: you can prove it, but why do you think your audience will absorb either the details or the significance? You seem to be assuming a fair bit of prerequisite knowledge, hence my confusion as to what examples you are really after. I could probably, if I rack my brains, define a ring, a module, a projective module and then do Schanuel's lemma in the alloted time, but would anybody who didn't know it really follow? $\endgroup$
    – Yemon Choi
    Commented Apr 4, 2011 at 7:34
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    $\begingroup$ 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. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 4, 2011 at 17:10

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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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    $\begingroup$ 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?) $\endgroup$
    – Pete L. Clark
    Commented Apr 4, 2011 at 14:46
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    $\begingroup$ When I was an undergraduate, I was persuaded to read Serres book when an older student told me about that result. $\endgroup$
    – Johannes Ebert
    Commented Apr 4, 2011 at 15:08
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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Commented Apr 8, 2011 at 16:48
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Completeness theorem for first order logic.

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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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    $\begingroup$ 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 …. $\endgroup$
    – LSpice
    Commented Apr 8, 2011 at 18:09
  • $\begingroup$ +1. I don't study C*-algebras, but this is one of the prettiest theorems I know. This is definitely a "jewelry-type" theorem. On the other hand, the non-commutative analogue (GNS construction) lies at the foundation of the theory of operator algebras; I think most functional analysts would view this as a "tool-type" theorem. $\endgroup$
    – Kevin Ventullo
    Commented Apr 8, 2011 at 19:12
  • $\begingroup$ Personally, I view it as a theorem telling me that (locally) compact Hausdorff spaces can be wild and savage beasts. Though as Paul says, it is the result which allows one to construct continuous functional calculus for normal elements in C*-algebras, and that is most definitely a useful "tool-type" theorem $\endgroup$
    – Yemon Choi
    Commented Apr 10, 2011 at 9:18
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    $\begingroup$ @L Spice - Perhaps this one is a stretch, especially for a 30 minute talk. But I could imagine using this result as motivation for the abstract definition of a ring. One could start out by defining C_0(X) as just a set of functions and then start listing its extra structure. Then one can pose the question: how much structure do we need to pile on before we have enough information to recover X? I've never actually tried giving a talk like this, but it doesn't seem totally inconceivable. $\endgroup$
    – Paul Siegel
    Commented Apr 29, 2011 at 2:10
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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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Fundamental Theorem of Finitely Generated Abelian Groups.

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Jordan normal form.

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Diamond lemma.

(Introduce rewriting systems, prove the diamond lemma and give couple of applications.)

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Stone's representation theorem.

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  • $\begingroup$ I doubt that you can explain the formulation in 30 min. If you can, then how? $\endgroup$
    – Anton Petrunin
    Commented Apr 3, 2011 at 20:52
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    $\begingroup$ 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 :-) $\endgroup$
    – Ed Dean
    Commented Apr 3, 2011 at 21:27
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Integration by Parts It's a powerful analytical tool and it can be used for reduction of order on complex functions.

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An earlier answer by Andrey Rekalo suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: fractals.

The proof of BFPT is so easy that, without a "flashy" application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

$(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. this survey paper by Natoli.

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)

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

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    $\begingroup$ It is nice way to impress students, but I do not see anything useful, except a message "do not be afraid to do technical work". $\endgroup$
    – Anton Petrunin
    Commented Apr 8, 2011 at 16:37
  • $\begingroup$ @Anton thanks for the comment :) actually that's the point - the students are from the a Game Development and Design course and we will soon have an Alienware laboratory - I might as well come up with something that they can put all that computing power to good use :) $\endgroup$
    – pageman
    Commented Apr 9, 2011 at 14:10
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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 Brachistochrone.

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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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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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.

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    $\begingroup$ This is a different theorem by the same person... $\endgroup$
    – Gil Kalai
    Commented Apr 12, 2011 at 4:26
  • $\begingroup$ [deleted earlier comment as it was based on a misreading] $\endgroup$
    – Yemon Choi
    Commented Apr 14, 2011 at 2:53
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In a metric space context, using natural metric, construct reals as equivalence classes of Cauchy sequences of rationals.

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If it is an election year in your country, you can discuss and prove Arrow's theorem. I especially like Terry Tao's treatment of the theorem, here. The proof requires only (very) elementary set theory (knowing what the intersection of two sets is) and knowing about proofs by induction. Even though the theorem is easy to state, it is counter-intuitive, which is a rarity in mathematics.

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I would show the students the Cantor-Schroeder-Bernstein Theorem and tell them they can themselves contribute to the list of elementary by providing a solution to the following open problems:

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Perhaps slightly off with respect to usefulness but certainly accessible : The four color theorem. (I am a bit surprised that nobody mentioned it, if I am mistaken I should change my glasses). It has a fascinating history and the use of computers in its proof is still slightly controversial. This would allow to discuss the notion of 'proof' (the first proof was riddled with errors but was nevertheless more or less accepted since all errors could easily be corrected). It has interesting reformulations (for example in terms of cross-products) and can be generalized to graphs embedded in other surfaces. It can also be complemented for example by mentioning the (still unknown exact) value of the chromatic number of the Euclidean plane (minimal number of colors needed to color all points of $\mathbb R^2$ with no points of identical color at distance $1$) or generalizations to other surfaces. One could also end with a Theorem of Steinitz (a graph is the $1$-skeleton of a polytop if and only if it is planar and can not be disconnected by removing less than $3$ edges), mention the five platonic solids (and perhaps even Schlaefli's classification of regular polytops in higher dimensions).

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Consider some metric spaces, then Hausdorff distance and Gromov-Hausdorff convergence

Also, introduce categorical notation, it may be very useful.

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Riesz theorem or, more general, Lax-Milgram theorem.

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  • 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:

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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.

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Yoneda Lemma. :D

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