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## A transcript of my qualifying exam

I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript.

Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

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## New page on reading recommendations

I’ve added a new page of reading recommendations, mostly for undergraduates, to the top. The emphasis is intended to be on well-written and accessible books. Comments and suggestions welcome.

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## The Summer Program on Applied Rationality and Cognition

This summer I will be teaching at a newish high school summer math program, the Summer Program on Applied Rationality and Cognition (SPARC). We’ll be covering a wide range of topics, including probability, Bayesian statistics, and cognitive science, with the general theme of learning how to make rational decisions (both practically and theoretically). Many interesting people are involved, and I’m excited to see how the program will go.

I think SPARC will be an extremely valuable experience for talented high school students. If you are (resp. know of) such a student, I strongly encourage you to apply (resp. forward this information to them so that they can apply)! Questions about the program not addressed in the FAQ should be directed to contact@sparc2013.org.

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## Update

I’ve uploaded notes for the classes I’m taking this semester again. This semester I’m taking the following:

• C*-algebras (Rieffel): An introduction to C*-algebras from the noncommutative geometry point of view. Should be quite interesting.
• Discrete Mathematics for the Life Sciences (Pachter): An introduction to computational genomics. I’m hoping to learn something about what kind of mathematics get used in biology.
• Algebraic Geometry (Nadler): Algebraic geometry from the point of view of categories of (quasi)coherent sheaves, their derived categories, etc. Should also be quite interesting.

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## Notes

I’ve uploaded current notes for the classes I’m taking. I make no claim that these notes are complete or correct, but they may be useful to somebody. The notes for Dylan Thurston’s class are particularly fun to take; I’ve been drawing the pictures in Paper and I’m generally very happy with the way they’re turning out.

Edit: It would probably be a good idea for me to briefly describe these classes.

• Homological Algebra (Wodzicki): An introduction to homological algebra aimed towards triangulated categories. Taking these notes is a good exercise in live-TeXing commutative diagrams.
• Curves on Surfaces (Thurston): An introduction to various interesting structures related to curves on surfaces. There are cluster algebra structures involved related to Teichmüller space, the Jones polynomial, and 3- and 4-manifold invariants, but the actual curves on the actual surfaces remain very visualizable. Taking these notes is a good exercise in drawing pictures like this (a curve on a thrice-punctured disc being acted on by an element of the mapping class group, which in this case is the braid group $B_3$):

• Quantum Field Theory (Reshetikhin): An introduction to the mathematics of quantum field theory. The course website has more details. Taking this class is a strong incentive for me to learn differential, Riemannian, and symplectic geometry.

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## Update

I put up a post over at the StackOverflow blog describing a little of what I’ve been up to this summer.

Curiously enough, the Zipf distribution which shows up in that post is the same as the zeta distribution that shows up when trying to motivate the definition of the Riemann zeta function. I’m sure there is a conceptual explanation of this connection somewhere, probably coming from statistical mechanics, but I don’t know it. I suppose the approximate scale invariance of the zeta distribution is relevant to its appearance in many real-life statistics, as described in Terence Tao’s blog post on the subject here.

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## “MathUnderflow” now in public beta

The new StackExchange math site has finally gone out of private beta. I would like this site to succeed, and I had forgotten how much fun it is to actually be able to answer questions, so I hope everything works out. Come on over!

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## Nationals 2010

MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but

• I happen to be a member of the Rubik’s Cube club here, and
• Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.

Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?

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## Update and bleg

Sorry for the lack of updates! I’ll try to get back to commutative algebra at some point, but real life has been intervening a lot lately. In the meantime, I’d like to shamelessly ask that my readers help me out on some unanswered questions of mine on Math Overflow:

Any responses are appreciated!

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## GILA IV: The unreasonable effectiveness of generating functions in the combinatorial sciences

We’ve got everything we need to prove the Polya enumeration theorem. To state the theorem, however, requires the language of generating functions, so I thought I’d take the time to establish some of the important ideas. It isn’t possible to do justice to the subject in one post, so I’ll start with some references.

Many people recommend Wilf’s generatingfunctionology, but the terminology is non-standard and somewhat problematic. Nevertheless, it has valuable insight and examples.

I cannot recommend Flajolet and Sedgewick’s Analytic Combinatorics highly enough. It is readable, includes a wide variety of examples as well as very general techniques, and places a great deal of emphasis on asymptotics, computation, and practical applications.

If you can do the usual computations but want to learn some theory, Bergeron, Labelle, and Laroux’s Combinatorial Species and Tree-like Structures is a fascinating introduction to the theory of species that requires fairly little background, although a fair amount of patience. It also contains my favorite proof of Cayley’s formula.

Doubilet, Rota, and Stanley’s On the idea of generating function is part of a fascinating program for understanding generating functions with posets as the fundamental concept. I may have more to say about this perspective once I learn more about it.

While it is by no means comprehensive, this post over at Topological Musings is a good introduction to the basic ideas of species theory.

And a shameless plug: the article I wrote for the Worldwide Online Olympiad Training program about generating functions is available here. I tried to include a wide variety of examples and exercises taken from the AMC exams while focusing on techniques appropriate for high-school problem solvers. There are at least a few minor errors, for which I apologize. You might also be interested in this previous post of mine.

In any case, this post will attempt to be a relatively self-contained discussion of the concepts necessary for understanding the PET.

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