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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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    http://en.wikipedia.org/wiki/Fair_division 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.2011-10-12
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    http://mathoverflow.net/questions/47214/how-to-present-mathematics-to-non-mathematicians might be useful.2011-10-12
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    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.2011-10-12
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    Related, and maybe useful: [Suggestions for a public talk about art and mathematics](http://math.stackexchange.com/questions/19400/suggestions-for-topics-in-a-public-talk-about-art-and-mathematics).2011-10-12
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    @AmiteshDatta Perhaps since the degree that you are enrolled in emphasizes "research", why not give a talk on the projects you are doing now? It would be a real challenge explaining them to an audience like that! For example, you can say you are doing a project on algebraic geometry. Maybe you can link that to Fermat's Last Theorem?2011-10-12
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    @simplicity Thank you very much for your suggestion! I think this is a very good idea.2011-10-12
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    @Asaf Thank you very much for the link!2011-10-12
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    @Ragib Thank you very much for your suggestion! Indeed, I also think that graph theory is a very good idea for the topic of the talk.2011-10-12
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    @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 ...2011-10-12
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    @Benjamin For example, at the moment I am thinking about fibered products (e.g., separated and proper morphisms) and normal schemes in algebraic geometry, constructable sets (e.g., Chevalley's theorem) and associated primes in commutative algebra, and Fourier analysis on the torus (e.g., pointwise convergence of Fourier series and Bochner-Riesz summability) and singular integral theory (e.g., Calderon-Zygmund decomposition) in harmonic analysis ...2011-10-12
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    @Benjamin ... If someone does not care about a scheme, a derived functor, or a Banach spaces *at the very least* (and these are non-trivial concepts even for mathematics student), then there is not much I can talk about. However, there are beautiful topics in "classical algebraic geometry" which I can talk about; I will consider doing this. For example, I could talk about moduli spaces. Thank you very much for your suggestion!2011-10-12
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    @AmiteshDatta I understand the difficulty in explaining concepts like that to students, but here is an example of what you can do. Students know what a vector space is now at least. You can explain that in algebra there are many objects with less structure than a vector space like a group, or more structure like say a field (then again what more or less mean is not well defined) to people when say explaining what a ring is. Problem is, mentioning anything even a little more than what students has done in these courses, like say algebra in the next course will inevitable disinterest, which2011-10-13
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    @AmiteshDatta .... is sad considering how I remember my eyes lighting up just standing outside the seminar room listening to people talk about "connections" and "convariant derivatives" and "fibre bundles"..... Even last semester my supervisor was trying to explain to me something on differential equations and how it depended on some topological space being seperable (he mentioned it in context of the Riesz Representation Theorem). Even though I understood maybe $10^{-6}$ percent of it, still in my mind I try to imagine the day I could do and understand these things!!2011-10-13
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    @Benjamin Yes, this is indeed a possibility. However, in my opinion, there does not seem to be a purpose in me explaining the definition of a group in 15 minutes. One first needs to be interested in studying such things or else the relevant definitions will appear to be unnecessary technicalities. I certainly know and appreciate that group theory and ring theory are very interesting branches of mathematics. However, most results in these areas are interesting primarily because they are results purely about groups or rings; not because they are relevant to physics or chemistry ...2011-10-13
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    @Benjamin ... Of course, there is a chance that a mathematics student in the talk will find these topics interesting. However, I think that I was the only one last year to give a talk on mathematics, which suggests to me that the audience consists primarily of non-mathematics students.2011-10-13

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