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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
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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
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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
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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
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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

Stone's representation theorem.

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I doubt that you can explain the formulation in 30 min. If you can, then how? – Anton Petrunin Apr 3 '11 at 20:52
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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

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
    
+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. – Kevin Ventullo Apr 8 '11 at 19:12
    
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 – Yemon Choi Apr 10 '11 at 9:18
    
@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. – Paul Siegel Apr 29 '11 at 2:10

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

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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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And it can also win you the national medal of science: mathoverflow.net/questions/53122/mathematical-urban-legends/… :-) – Willie Wong Apr 13 '11 at 18:18

The Arithmetic Mean-Geometric Mean Inequality.

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This needs more self-contained elaboration. – Douglas Zare Jan 10 at 6:05

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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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". – Anton Petrunin Apr 8 '11 at 16:37
    
@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 :) – pageman Apr 9 '11 at 14:10

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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This is a different theorem by the same person... – Gil Kalai Apr 12 '11 at 4:26
    
[deleted earlier comment as it was based on a misreading] – Yemon Choi Apr 14 '11 at 2:53
  • 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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Consider some metric spaces, then Hausdorff distance and Gromov-hausdorff convergence

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

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

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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 useds without any proof).

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

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Example of use, maybe? – Michal Kotowski Apr 10 '11 at 9:49

In a metric space context, using natural metric, construct reals as equivalence classes of Cauchy sequences of rationals.

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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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Singular value decomposition has already been suggested. mathoverflow.net/questions/60457/elementaryshortuseful/… So has Stokes mathoverflow.net/questions/60457/elementaryshortuseful/… – Willie Wong Apr 8 '11 at 18:50

Yoneda Lemma. :D

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Someone else has already suggested this. mathoverflow.net/questions/60457/elementaryshortuseful/… – Willie Wong Apr 7 '11 at 14:54

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