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I am to give a mathematics talk in a couple of weeks and I am writing to request suggestions for possible topics of the talk. The relevant information is as follows:

(1) The audience for the talk consists of first and second year students in science; some of the students might be first or second year students in mathematics. In particular, I cannot expect that any member of the audience is particularly knowledgeable in mathematics; I think at a minimum I can assume a knowledge of the elements of calculus. However, I would prefer the talk to not be directly based on calculus.

(2) The audience consists of very intelligent students. Nevertheless, I probably should not expect the audience to have to do a great deal of thinking while listening to the talk.

(3) The talk is to be 15 minutes in length. In particular, I probably have to focus on one theme during the talk.

I would like to talk about a mathematics topic that fits the following description:

(a) The talk appeals to an intelligent person who is not particularly knowledgeable in mathematics but is not too trivial that it does not appeal to a mathematics student.

(b) The talk illustrates a beautiful piece of mathematics.

(c) The talk is solely based on mathematics.

Thank you very much in advance for all suggestions for the topic of the talk! I will certainly acknowledge you in the talk if I use your suggestion for the topic of the talk.

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2 is a good topic to talk about. Cutting the cake algorithm is intuitively obvious and is deep in the sense it gives fair division. I think your best bet is to talk about some sort of game theory stuff. – simplicity Oct 12 '11 at 12:19
2… might be useful. – Asaf Karagila Oct 12 '11 at 12:29
First, show them the Seven Bridges of Königsberg problem, ask them to think about it to themselves for a little bit. Almost definitely not trivial to anyone. Introduce Graphs, the notion of Eulerian graphs, it's relevance to the Chinese Postman Problem, and Euler's solution to the Seven Bridges of Königsberg problem. Graphs are things easy to introduces and visualize, and the solution is very intuitive, it will give the audience a "ahhh!" moment. – Ragib Zaman Oct 12 '11 at 13:00
Related, and maybe useful: Suggestions for a public talk about art and mathematics. – Mike Spivey Oct 12 '11 at 16:24
@Benjamin Thank you very much for your suggestion! I briefly considered the idea of discussing either commutative algebra, algebraic geometry or harmonic analysis in the talk. However, the problem is that these topics are all very advanced and it is very difficult to explain them to a first or second year non-mathematics student! The problem is that there is so much technical machinary that one needs to introduce before one gets to beautiful applications that it is not possible to talk about anything particularly deep or interesting in the framework of a 15 minutes talk ... – Amitesh Datta Oct 12 '11 at 23:12

I think that the periodicity of Fibonacci numbers modulo $m$ is a topic that can be introduced to your desired audience and about which non-trivial, appealing things can be said within 15 minutes (you might want to mention some of the more advanced, more appealing things at the end without proof).

Specifically, I imagine you could introduce the Fibonacci numbers (but I think they have seeped into popular knowledge to a significant extent), modular arithmetic, observe the periodicity in a few example cases, prove that the period is always even, and prove that the period modulo $m$ is less than or equal to $m^2-1$ (which is just the pigeonhole principle). You could then mention without proof that in fact the period modulo $m$ is less than or equal to $6m$, and various observations about the period modulo a prime number.

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+1 Thank you very much for this interesting suggestion! – Amitesh Datta Oct 12 '11 at 12:33
@Amitesh Datta: $15$ minutes is very short. Tempting to pack a lot in and rush. But useless, almost nobody will follow. Less is more! If you have a web site, put details, links there for people who want more. – André Nicolas Oct 12 '11 at 13:51
Maybe problem about $4$-colouring plane so that points distance $1$ apart have different colours. Show need at least $4$ (nice picture, not too hard to understand). Mention (maybe sketch) that $7$ enough. Mention not known what minimum number of colours is. – André Nicolas Oct 12 '11 at 13:59
I agree with @André that KISS is very much applicable here, especially since you're talking to nonspecialists; that there is a greater risk of boring your audience to tears if you pack in too much for them to follow. – J. M. Oct 12 '11 at 16:21
I agree that a topic that gets a head start being more relatable (like fair division, as was suggested above, or coloring problems, like Andre suggested) would probably be better given the time restriction. I am just very partial to this topic and wanted to mention it :) – Zev Chonoles Oct 12 '11 at 16:31

If I am right, your talk is going to be for our conference in a few weeks. I can tell you a little bit about the first year audience: Most of them are not well acquainted with mathematics and few actually will understand its true power - including me. Perhaps those in second year who have done the rigorous analysis course will be able to appreciate more of what you are going to talk about.

Most of the students, at least those in first year are now doing the standard calculus/linear algebra course offered. I think at this point they have not seen how analysis and linear algebra come together beautifully and perhaps your talk can be based on showing how different fields of mathematics come together (at the moment most students - at least in my year - think that analysis and linear algebra are not related at all).

For me I would suggest a talk on non-Euclidean geometry. Several reasons are:

$\textbf{(1)}$ I think students would be fascinated about the fact that the sum of the angles in a triangle being 180 degrees holds only for Euclidean geometry. Besides, how are so called "curved" surfaces not to impress anybody? :D

$\textbf{(2)}$ You can show a lot of pretty pictures by M. C. Escher (he has a work called "circle limit" which is the Poincaré model of hyperbolic geometry, you can explain why the "devils" go smaller towards the edge) that can fascinate people - it is way easier for people at this level to imagine "beauty" like this than to explain say what Maschke's Theorem is. The latter would require having to explain a lot of technical language first.

$\textbf{(3)}$ Students of physics I understand have learned at least special relativity. You can mention Minkowski spacetime and how this is related to the hyperboloid model of hyperbolic geometry. In particular maybe you can link this to why the inner product used in class has signature $(1,1,1-1)$.

$\textbf{(4)}$ I think one can present a topic on this while expecting minimal prerequisites from the audience. You can have a look at Needham's book as mentioned above - there is a chapter in there on hyperbolic geometry. Proofs were neat and elegant, many of them using so called symmetry arguments like proving why every automorphism of the unit disk to itself has the form $e^{i\theta}$ times some linear fractional transformation which I don't remember.

I hope this helps!

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Some practical thing is always good in short-only expositions. Also if it is some problem of non-zero relevance solved with primitive/homebrewed means I would have liked it as a young student.

One idea which comes to mind were farming or even architecture in ancient orient where they wanted to assure rectangular angles for the house/temple/garden/farm. Then this leads to the pythagorean triangles and the rope-with-knots-solution for the workers (a rope with 3+4+5 knots, for instance, which he could manufacture at any place he happend to be and any size required).

For your talk to get then the shift to deeper problems of science/math one could introduce the question, what happens to the number of knots on the hypothenuse, if the architect wanted a square - to come out with the observation, that generalizations of a problem sometimes lead to the introduction of new concepts (non-rational numbers). Which are sometimes even anticipated in social/scientific disagreement (do irrational numbers exist? When -historically- was this generally understood?) Do we see acceptance-problems even today which express some intrinsic difficulty for our pre-scientific daily-thinking? And what's the role of a rigorous framework for the science (here: the subject is math/geometry/architecture)?

That raw idea came after I've just read some nice books about early cultures - I didn't yet think really deep about formulating the math for this current answer, though. But perhaps it is a triggering idea anyway? At least it seems to be not too much for a 15-min-talk...

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One possibility is to give a discussion of scaling laws in engineering, physics, biology, etc. Google "square cube law" for some ideas. You can talk about the fallacy of the strength of fleas, biological issues (breathing, walking, etc.) for insects the size of dinosaurs in all those 1950s "B grade" movies, problems in designing huge skyscrapers, what wind tunnels can tell you about airplanes and what they can't, fractals, why small hot objects cool faster than large hot objects, etc. Depending on the background of your audience, you can work plenty of mathematics into this topic, such as relative growth rates of functions or how the scaling features of the area of a square can be used to show the same for the interior of any "nice" planar figure (fill the region with lots of tiny squares). Also, with this topic you can readily incorporate pictures and graphs into your talk.

(5 minutes later) I just noticed your last requirement (pasted below), which I missed earlier. I was thinking of a talk involving mathematics (but not purely mathematics) for first and second year science majors.

(c) The talk is solely based on mathematics.

(next day) Here's something that is more "pure math" based. The URL below takes you to a sci.math thread in which a method is given for obtaining, in a relatively simple way, quotients of integers whose decimal expansions contain arbitrarily long (when the length is specified in advance) initial segments of the Fibonacci sequence. For example,


$$=\;\;\; 0.\;\;\;00001\;\;\;00001\;\;\;00002\;\;\;00003\;\;\;00005\;\;\;00008\;\;\;00013\;\;\;00021$$



One way to see how this works makes use of generating functions, and other ideas are explored in this thread. Also, in a later post I give the details for obtaining quotients of integers whose decimal expansions contain arbitrarily long initial segments (when the length is specified in advance) of a general 2nd order positive-integer-linear difference equation: $x_{n+2} = r \cdot x_{n+1} + s\cdot x_{n},$ where $a,$ $b,$ $r,$ $s$ are positive integers and $x_{0} = a,$ $\;x_{1} = b.$ Some relevant published expository journal papers are also cited in this thread.

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Hmn, how about complexity theory? Does that count as pure Math? Maybe you can take something practical like finding patterns in a DNA sequence, and explain an algorithm for finding these patterns, and take them step-by-step on finding the complexity of the algorithm.

Else, how about something simple like going through finding the closed-form generating function for the Fibonacci numbers? It's something quick, interesting (well, personally I find it pretty neat), and Fib. numbers is something everyone would have heard of. The Math is not advanced, at the same time it's not trivial or boring.

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