Archive for the ‘Uncategorized’ Category

My MaBloWriMo 2015 run met an untimely end on the 18th, when LaTeX stopped working on WordPress for me; I could no longer see any of the LaTex I was writing. It’s still not working for me in Chrome, but it’s now working for me in another browser, so hopefully I’ll get some posts up soon.

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Decisions, decisions

Newcomb’s paradox is the name usually given to the following problem. You are playing a game against another player, often called Omega, who claims to be omniscient; in particular, Omega claims to be able to predict how you will play in the game. Assume that Omega has convinced you in some way that it is, if not omniscient, at least remarkably accurate: for example, perhaps it has accurately predicted your behavior many times in the past.

Omega places before you two opaque boxes. Box A, it informs you, contains $1,000. Box B, it informs you, contains either $1,000,000 or nothing. You must decide whether to take only Box B or to take both Box A and Box B, with the following caveat: Omega filled Box B with $1,000,000 if and only if it predicted that you would take only Box B.

What do you do?

(If you haven’t heard this problem before, please take a minute to decide on an option before continuing.)


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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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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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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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So: I’m happy that I’ve kept up MaBloWriMo for 13 days so far, but I’m running out of steam. I’ve gone through essentially all of the posts in my backlog that were relatively easy to write, and the things I’d like to write about at this point either

  • really should be done with diagrams (and it’s not easy to finish a blog post with diagrams in a day) or
  • might take more time than I allot for blogging in a day to work through the relevant concepts.

Sticking to one post a day at this point is likely to drive down quality, so I think I am going to stop doing it. It was a good goal for awhile in that it got me to write some posts that I’d wanted to write for a long time now, but unfortunately it is now doing the opposite of that.

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MaBloWriMo is upon us

Three years ago I thought it would be fun to write a blog post every day of November. I’m not sure why I didn’t do this in November 2010 or 2011 because I’m pretty sure I learned a lot from doing it in 2009, so I’d like to do it again. The posts will probably be shorter this time.

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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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UC Berkeley

I’ve made my graduate school decision. Real math to follow soon in subsequent posts, I promise.

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Estimating roots

In lieu of a real blog post, which will have to wait for at least another two weeks, let me offer an estimation exercise: bound, as best you can, the unique positive real root of the polynomial

\displaystyle x^{10000} + x^{100} - 1.

The intermediate value theorem shows that x \in (0, 1), which was the subject of a recent math.SE question that provided the inspiration for this question. I provide a stronger lower bound on x using elementary inequalities and entirely by hand in an answer to the linked question, although I don’t try to improve the upper bound.

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