## Elementary+Short+Useful

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?

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 2011 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 2011 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 2011 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 2011 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 2011 at 17:10

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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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 2011 at 9:54
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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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Robinston-Schensted-Knuth algorithm

This is a map between permutations to pairs of standard tableaux. So it immediately gives various wonderful facts. It is elementary, short and useful.

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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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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 2011 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 2011 at 15:08
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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 2011 at 11:38
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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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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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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 2011 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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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 : math.lsa.umich.edu/~wfulton/CurveBook.pdf – aglearner Nov 9 2011 at 22:39

[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.][http://en.wikipedia.org/wiki/Taylor's_theorem]

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Completeness theorem for first order logic.

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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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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 go technical work". – Anton Petrunin Apr 8 2011 at 16:37
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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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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 2011 at 4:26
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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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• 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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• The famous Heine - Borel theorem which says that a closed a bounded subset of $\mathbb{R}^{n}$ is compact.
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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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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 2011 at 21:27
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