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Last term I started a weekly seminar like thing. The basic idea was this -- At the beginning of the term each person will choose a topic that they don't know about and isn't usually covered in the standard undergraduate math curriculum. They'll work on it throughout the term and present it at some point during the term. Overall it was a success as most people involved got exposed to mathematics that they wouldn't have otherwise.

Last term I suggested topics to each person one by one based on their preferences and experiences. For example some of the talks we had were on: "How to have fair elections - Cryptography and Voting theory", "Quandles - The algebra of knots", "Hypercomplex Numbers", "Basic Elliptic Curves", "Ultrafilters and Non-standard analysis".

However this term I am struggling to come up with topics for the 20 or so people that have signed up to speak. So I'd like to ask if you have any suggestions for topics that are interesting and accessible to 2nd, 3rd year undergraduates but that are also not covered in a standard undergraduate curriculum.

PS So far for this term I have "P-adic Analysis" (the book by Katok is great), "Bernoulli Numbers and their Application in Number Theory", "The mathematics of Bitcoins", and "Penrose Tilings". I also understand that any book in the Student Mathematical Library will work.

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    I can write several. One could be **fuzzy logic**. – 2017-01-13
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    Sure, please add all the topics you can think of in an answer. – 2017-01-14
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    I also help run this and for further reference I'd like to just add that we did find many topics to speak on -- here is a link to the suggested topics page on our site: http://uwseminars.com/potential-topics And of course, we are in the process of adding the many wonderful topics listed here! So if you have any other topics please keep adding them. – 2017-03-08

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You might try pilfering a few topics from summer schools (particularly ones your students don't have access to, for whatever reason). For example, the LMS Summer School page has links to what was talked about in previous years, and for some you even get free lecture notes.

Among the topics listed there, I would recommend (mostly out of my own interests):

  • Continued fractions and hyperbolic geometry, taking a brief look at the geometry of continued fractions via the Farey tessellation (see also Francis Bonahon's page for some very pretty pictures; he has published a textbook about this kind of thing).
  • Quivers and Platonic solids, linking quivers to the Platonic solids via Gabriel's theorem.

In particular, a set of notes for each of these topics can be found on the LMS pages linked above, which include references to other useful sources.

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Going a bit broader, here are a few super cool fields of mathematics that have important connections to other fields but are often overlooked. An introduction to them, and then proving a cool theorem might be a good assignment.

Chang and Keisler's book on Model Theory, which is a branch of mathematical logic that has a lot of applications in philosophy and linguistics. It also has an exciting intersection with combinatorics. One of my favorite proofs ever uses the Model Theory to prove the Ax–Grothendieck Theorem.

Reverse Mathematics (two essays: introduction and advanced) is a branch of mathematics that answers the question: why does it make more sense to say "Use the Bolzano-Weierstrass Theorem to show that if $f : [0, 1] \to \mathbb{R}$ is continuous, then f is uniformly continuous" then it does to say "use the four color theorem to prove that there are infinitely many primes"

Measure Theoretic Probability Theory is the proper way to do probability on infinite sets. It will introduce you to the wonderful and weird world of $0-1$ Laws too.

For more stand-alone topics... fractal dimension, Unique Games Conjecture, Minkowski's Theorem and it's many applications to Number Theory, a second shout out to using Model Theory to prove the Ax–Grothendieck Theorem, the connection between category theory and functional programming.

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    The question does not allow jokes as answer because they are interesting but not useful as certainly **Reverse Mathematics** is. Also **Measure Theoretic Probability Theory** is commonly included in undergraduate study (it depends on the degree program). **Unique Games Conjecture, Minkowski's Theorem** are results, not topics (aren't answers either) – 2017-02-15
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    @ErnestoIglesias I don't understand what you're saying about jokes. This answer isn't a joke. – 2017-02-15
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    Sorry then, I don't want to hurt any feeling, I've just think **Reverse Math** can't be an answer because is fancy rather than useful. It's a matter of opinion anyway. People thinks the same for non-Euclidean geometry and thing have changed since then. – 2017-02-17
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The following references are free available (and the first paragraph in spanish but) I will write the topics in english (I hope that my translation is right for the specification of this topics is right). I write this literature and you can search such key words to find the literature or ideas that is required for your class.

In the hope page of professor Chamizo, from Universidad Autónoma de Madrid Apuntes de Modelización II, there are topics as gyroscopic movement (pages 19-22), soap bubbles (25-30), and so other topics for which I no longer wrote the page, for example heat transfer, or the Radon transform and tomography, the JPEG format.

In his lecture notes : Chamizo, Geometría IV (tensores, formas, curvatura, relatividad y todo eso), you can find a section dedicated for example to de Rham cohomology, the Schwarzschild metric and black holes or Einstein field equations.

Additionally you can think in topics (at least I think that are beatiful) as what's an invariant in physic or mathematics?, Paul Dirac and the beauty of the equations (I am saying the study of the symmetries of some equation related with Paul Dirac), what's the Church–Turing thesis? (or a different work of Alan Turing, for example with respect the how he did the cryptanalysis of the Enigma), Euler equations in fluid dynamics (Córdoba, Fontelos and Rodrigo, Las matemáticas de los fluidos: torbellinos, gotas y olas. La GACETA de la Real Sociedad Matemática Española Vol. 8, No. 3 (2005)), flamenco and mathematics (Díaz-Báñez, Sobre problemas de matemáticas en el estudio del cante flamenco, La Gaceta de la RSME, Vol. 16 No. 3, (2013), ), Interpolation (spaces, interpolation of operators) and PDE (see Lunardi, How to use interpolation in PDE's, Summer School on Harmonic Analysis and PDE's, Helsinki, August 2003.) Orbifolds (here I haven't find a free access, but as previous this has the more high quality: Montesinos Amilibia, Orbifolds in the Alhambra. Memorias de la Real Academia de Ciencias Exactas, Fisicas y Naturales de Madrid. Serie de ciencias exactas, 23 . p. 44. ISSN 0211-1721).

Mathematics and oceanography (see Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Springer, (2006) see www.springer.com; I presume that one can find information about mathematics and..., for example mathematics and the exploration of Mars. A proof or detailed examples of the Cauchy–Kowalevski theorem. Or mathematics and quantum mechanics (I found this Heathcote, Undounded operators and the incompleteness of quantum mechanics Philosophy of Science 57 (3):523-534 (1990)). See Ratlif, Linear Algebra and Robot Modeling ,also as a second example of topics that you can find in papers, from Institut für Parallele und Verteilte Systeme (Universität Stuttgart). Also you can find literatue about, for example, applications of fractals in the real life and the mathematics beyond this ideas (fractal antenna or fractal geometry in medicine...).

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    Maybe some topics (like the Schwarzschild metric and black holes or Einstein field equations) aren't the best by their difficulty but it is an idea – 2017-01-14
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Conway's On Numbers and Games. Definitely.

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I think it might be a good idea to look at past IMO and Putnam papers

Some of the questions in those papers when looked into deeply have enough scope of discussion for a whole lecture

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  • Integer-point enumeration in polyhedra. This is a subject with a low entrance threshold, but which is still at the frontiers of current research. It also blends ideas from combinatorics, number theory, and geometry in a natural way. The book Computing the Continuous Discretely is great resource for this subject.
  • Numerical analysis. This is an extremely useful subject that doesn't get its due recognition in the curriculum nowadays.
  • Clifford algebras. This is a useful tool which is gaining popularity among physicists as a somewhat unifying language for geometric ideas.
  • Matroids. These are a large and growing field of combinatorics, which are easily accessible.
  • Differential topology. The reason this is a viable subject is because of the beautiful book by Milnor, Topology from the Differential Viewpoint.