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As an algebraist i frequently receive questions from my friends-mathematicians and non-mathematicians about applications of my topic "in real life". I study algebraic groups in the stream of researches of N. Vavilov, A. Bak, B. Sury, A. Stepanov and so on. And actually, I have no explicit applications of all this stuff in my head. Well, crystallography condition on root systems makes sence to look up for them in physics. Once, i saw a paper about describing the universe with $E_6/E_7/E_8$, but it was insane... Anybody knows any applications which could be explained to non-algebraist (well, let's not consider non-mathematicians now)?

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This should be Community Wiki in my opinion. –  Igor Rivin Oct 14 '11 at 10:59
Does the Standard Model of particle physics (and various GUT extensions) qualify? –  S. Carnahan Oct 14 '11 at 11:06
And what applications does portrait painting have to real life... it gets used on currency? Seriously, I think you should address to the non-mathematician friends not your specific research interest but its broader context: mathematics of higher-dimensional spaces. That's not just relevant to physics (4 dimension for relativity) but anywhere that linear algebra is used (solving systems of linear equations in gazillion variables, where the solution techniques are often supported by methods with a geometric meaning, such as contraction mappings or projections). –  KConrad Oct 14 '11 at 21:03

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Like many of the people here, I have a different view of real life than most of the population of the planet. Lattices like $SL(n, Z)$ and $Sp(2g, Z)$ come up frequently in topology (often via the study of the mapping class group, see, eg, Farb-Margalit's book, available online), and in algebraic geometry (including Deligne's proof of the Weil conjecture, see the considerable oevre of Nick Katz). Also in the study of singularities of differential equations (all three of the above are related). This has much to do with number theory (eg, recent work on Sato-Tate, though I will not embarrass myself further by talking about things I have no idea of). Physicists (in the weak sense, that is, string theorists, who are primarily mathematicians in my view) have used much of the above in the study of Calabi-Yau manifolds.

I believe that the applications to real real life (such as building bridges and constructing iPhones), you might be, sadly, out of luck.

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Oh, bad luck, it was such a great hope, that my work helps to construct iPhones :) Anyway, thank you for this overview, although it doesn't really solve the problem of describing it to non-professional... well, may be it's impossible. –  iRyoka Oct 14 '11 at 11:30

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