Category Archives: math

Random thoughts from 2012 ARML

Last weekend was 2012 ARML. It was my fifth ARML, and third as coach of the SFBA teams. I remember my first trip to ARML, with just a single team. This year, ARML has become a 7-team, 4-day expedition. This was also one of my most entertaining trips to ARML. Watching SFBA troll the Friday evening talent show was amazing.

As for results, SFBA’s teams did amazingly well: A1 and A2 were 3rd and 7th nationally, the first time that any organization has had two teams in the top 7 at ARML. Next year, we’re aiming for 1st and 2nd on the national scoreboard.

Some random thoughts from ARML:

  • Tie-dye was an amazing shirt color. Pink was pretty cool too.
  • I should try to avoid losing my bed next time.
  • Flip, Flop, Fliegerspiel
  • Bif! Bif! Bif!
  • I am Chen!
  • Yee!
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Formatting and math style in LaTeX

LaTeX gives almost infinite control over formatting, and it is possible to spend hours making small adjustments to a document, forcing every character and every pixel to be in an optimal location.

A much easier solution is to trust LaTeX to format documents. Though it might not look exactly as you might envision, LaTeX’s standard document classes do a surprisingly good job at formatting. Of course, some manual adjustments might still be in order; you might want to change the default indent size (e.g. with \parindent=0in), force some space between paragraph breaks (e.g. with \parskip=5px), or change the margins (e.g. with the geometry package or using \usepackage{fullpage}). The general structure of a document, however, is best left up to LaTeX’s default styles.

LaTeX can fairly easily create a title heading. Here is an example:

\title{All of Mathematics}
\email{someone@somewhere.com}
\author{C.F. Gauss}
\date{\today}
\begin{abstract}
      This paper explains all of mathematics, in alphabetical order.
\end{abstract}
\maketitle

The final \maketitle command tells LaTeX to display the title, author, email, date, and abstract information in the style defined by your documentclass.

LaTeX can also handle section headings and section numbering. To start a new section, simply say \section{My new section}. Similarly, you can define new subsections and subsubsections with \subsection{My subsection} and \subsubsection{A subsubsection}. These sections (and subsections and subsubsections) are automatically numbered sequentially. (To disable automatic numbering in LaTeX, add an asterisk, i.e. \section*{Unnumbered section}.)

LaTeX is also often used with mathematical expressions, for which there are also some stylistic points to keep in mind. Firstly, it is a good idea to use displayed equations. These can be used for long or complicated expressions to make them easier to read. When surrounded by text, they also serve to break up long paragraphs into more manageable chunks. Even simple equations can be displayed to make them stand out, such as

\displaystyle  e^{i\pi}+1=0.

Something to keep in mind is that all punctuation following a displayed equation should be placed in math mode, immediately following the equation.

Make it a habit to use math mode for all mathematical symbols, regardless of length. For example, it looks ugly to say “Consider a function f with f'(x) = x+5“; instead, say “Consider a function f with f'(x) = x+5“. Doing this allows you to clearly differentiate between math symbols and text, and also eliminates confusion resulting from math mode letters looking different from text mode letters.

LaTeX also provides a number of math operators for easy formatting. For example, log x looks ugly, while \log x looks much better. It is easy to define new operators as well. For example, if you want to use “disc” to denote a discriminant, then you can use \def\disc{\operatorname{disc}} or \DeclareMathOperator{\disc}{disc} to define the \disc operator. This allows you to type \disc K to get \mathrm{disc}\, K.

You can also use expanding delimiters in large math expressions. For example, compare

\displaystyle  P(\frac1x) \qquad P\left(\frac1x\right).

The symbol on the right is achieved by writing $P\left(\frac{1}{x}\right)$. All other delimiters expand as well, via commands like \left[, \left\{, \left|.

For clarity, inline fractions in text can be typeset horizontally. That is, in a typical font size, the fraction 79/128 is easier to read than the fraction \frac{79}{128}. Of course, there are plenty of cases when you want to write fractions vertically, such as when they are part of a larger expression (e.g. \frac12 + \frac13 = \frac56).

Long equations can be broken up. It’s no good when an equation is too long and runs off the page, and it’s ugly to see an equation reach into the margin. Instead, break the equation up at binary operators and equals signs, such as + or =. Long calculations can be broken up at each equals sign, such as

\displaystyle  \begin{array}{rcl}  K_m(s) &=& \frac{1}{m+1} \frac{1}{1- e^{it}} \left( \frac{1 - e^{-i(m+1)s}}{1 - e^{-is}} - e^{is} \frac{1 - e^{i(m+1)s}}{1 - e^{is}}\right) \\ &=& \frac1{m+1} \frac1{e^{is/2} (e^{-is/2} - e^{is/2})} 	\left( 		\frac{1 - e^{-i(m+1)s}}{e^{-is/2} (e^{is/2} - e^{-is/2})} 		-\frac{e^{is} (1 - e^{i(m+1)s})}{e^{is/2} (e^{-is/2} - e^{is/2})} 	\right) \\ &=& \frac1{m+1} \frac1{e^{-is/2} - e^{is/2}} 	\left( 		\frac{1 - e^{-i(m+1)s}}{e^{is/2} - e^{-is/2}} 		-\frac{1 - e^{i(m+1)s}}{e^{-is/2} - e^{is/2}} 	\right) \\ &=& \frac1{m+1} 	\left( 		\frac{ e^{i(m+1)s} + e^{-i(m+1)s} - 2 }{(e^{is/2} - e^{-is/2})^2} 	\right) \\ &=& \frac1{m+1} 		\frac{ \sin^2 (\frac{m+1}2 s) }{\sin^2 \frac s2}. \end{array}

Similarly, it is a good idea to align lists of equations at equal signs, such as

\displaystyle  \begin{array}{rcl}  \sin\theta &=& y \\ \cos\theta &=& x \\ \tan\theta &=& \frac yx \end{array}

Such an effect can be achieved using a construction like

\begin{align*}
       \sin\theta &= y \\
       \cos\theta &= x \\
       \tan\theta &= \frac yx.
\end{align*}

Integrals need extra space. Compare:

\displaystyle  \int x^2 dx \qquad \int x^2 \, dx.

The second integral separates the dx from the integrand, making the equation more clear. This was typeset using $\int x^2 \, dx$, where the \, serves to add a bit of space.

Technical writing

It is important to remember that even though technical writing aims to communicate ideas precisely and unambiguously, such writing is still ultimately read by human beings and not computers. Readers of technical writing already have to struggle with challenging ideas, and it is up to the writers to make their writing as approachable and clear as possible.

Use words between mathematical symbols. For example, instead of “Consider {A_{mn}}, {n < m}”, use “Consider {A_{mn}} where {n < m}”. This serves to delimit mathematical expressions and avoid confusion.

Consider using words instead of symbols to express simple ideas. For example, instead of “Take {m, n \in \mathbb{Z}}”, consider using “Take integers {m} and {n}”. The goal is to avoid burying the reader in a morass of unnecessary symbols and notation.

Avoid unnecessary symbols and abbreviations. Instead of using {\therefore}, {\forall}, {\exists}, write them out in words! It is easier to read “for every {\epsilon > 0} there exists {\delta > 0} …” than “{\forall \epsilon > 0}, {\exists \delta > 0} …”

Write in complete sentences. Consider replacing each mathematical expression with some filler word, like “blah”. Is each sentence still complete? Still properly punctuated? Using mathematical symbols does not absolve the writer from the rules of grammar. If a sentence ends with a displayed math expression, use a period! For example, we can define

\displaystyle  |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0. \end{cases}

Explain with words. It is often tempting to write pages of formulas and calculations and leave the reader to fend for himself while working through those computations. It is much nicer to the reader, however, to include some running commentary explaining what is being done.

Avoid walls of text. Nobody likes working through a dense argument that is one massive paragraph. Use paragraph breaks to give the reader some space to reflect on what’s been said so far. Use displayed equations to make them more readable, more visible, and simply as a way to break a paragraph into more manageable pieces.

Don’t be afraid to be redundant. It often makes things clearer to say the same thing in both words and symbols. For example, which definition of an open set is easier to read: “Every point {x \in X} has some neighborhood {B_\delta(x)} of radius {\delta > 0} around {x} such that {B_\delta(x) \subset X}” or “For all {x \in X} there exists {\delta > 0} such that {B_\delta(x) \subset X}”? They carry the same content, but in the first, the reader can focus on the concepts presented by the text, while in the other, the reader needs to translate from mathematical symbols into a more humanly parsable form.

Each theorem, lemma, corollary, definition should be self-contained. When scanning through a paper, its confusing to see

Theorem {f(x)} is a continuous but nowhere differentiable function.

Much better is to see “Let {f(x)} be as defined above in Definition 1.3. Then {f(x)} is continuous but nowhere differentiable.” Alternatively,

Theorem Let {f(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x)} where {b} is a positive odd integer, {0 < a < 1}, and {ab > 1 + \frac32 \pi}. Then {f(x)} is continuous but nowhere differentiable.

Give motivation and talk about the big picture. It is easy to blithely jump from one lemma to the next, giving clear and rigorous proofs of each lemma but losing sight of the reasoning behind those jumps. Before a chain of arguments and ideas, it helps to spend a paragraph talking about the big picture: Where is this argument leading? What’s the goal? Is there some intuitive picture or motivating example? All of this might be extremely clear for the writer, especially if he worked all of the details out on his own; for the reader, however, finding motivation and intuition often requires gaining a deep understanding of the topics under discussion. It only takes a few words to make everything much clearer.

Use clear formatting. Use section headings and appropriately space and align equations. This helps to communicate the structure of a paper. This is the subject of my next post.

For more on technical writing, see Knuth’s notes on mathematical writing.

ARML

Sorry about not posting for the past few months. I’ve been very busy recently. With the start of the summer, I should be able to post much more frequently.

I have been a coach for the San Francisco Bay Area ARML team for the past two years. Every year, people ask me why I help with ARML, and the answer is always the same: because it’s fun.

I enjoyed my first trip to ARML in 2008, and ARML has gotten better every year. Unlike most contests, ARML is heavily team-based, and as such, it is a great way to meet interesting people. ARML tends to build a sense of community that is lacking from most math events. Over the past few years, I’ve met many students and coaches at ARML, and they are a major reason why I go back to ARML every year.

Another big part of coaching ARML are the practices that we run every spring. It’s always great to see the gathering of people with similar mathematical interests. I believe in teaching math as a way to learn and understand something more thoroughly, and I hope to do more teaching at ARML practices in future years. Doing math is often most enjoyable as a social activity, and I think that our practices fully demonstrate that feature.

Here’s a list of interesting things that happened at ARML, in no particular order:

  • Green sharpies
  • Easy button thieves
  • A disappearing sword
  • Brilliant power round solutions
  • Paper balls and paper airplanes
  • Math!

A problem from the SMT

The 13th annual Stanford Mathematics Tournament was held on February 19, 2011. My favorite problem on the contest was the last problem on the calculus subject test.

Theorem 1 Compute the integral

\displaystyle  \int_0^\pi \ln (1 - 2a \cos x + a^2) \, dx

for {a > 1}.

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Content-free mathematics

All of mathematics is content-free.

That’s a claim that would probably get me lynched in any self-respecting math department, and anyone who has taken a real math class would feel that it isn’t true. How could anything that causes so much pain and confusion be content-free?

In some sense, however, every statement in mathematics is trivial. After all, a proof is no more than a series of logical implications, and every such implication is logically trivial (i.e. correct). The difficulty in understanding a proof is simply a parsing problem. For a technically challenging proof, there may be many definitions and lemmas to parse, and the writing may not be very clear, but all of the logical steps are there. In mathematics, proving is hard, but once a proof has been written, checking the proof requires much less inspiration and genius. In fact, given a proof, it could be written in computer-checkable form, and we all (hopefully) believe that computers aren’t yet sentient.

This isn’t to say that reading and understanding proofs is entirely easy; if this were true, learning math would take no time at all and checking Perelman’s proof of the Poincare Conjecture would not have taken many months and years. Even though I can try to justify a theoretical claim that mathematics is trivial, it is nonetheless true that I struggle to understand things. My brain isn’t a computer; though it’s probably better as being creative, it’s certainly worse at parsing. An argument about the triviality of math because purely philosophical and is itself content-free.

However, there are some parts of math that I do believe to be content-free, not just because all proofs are logically correct, but because they have been generalized to the point of being eaten up by other broader fields. These domains of mathematics become special cases of general statements, and though they may maintain their clever tricks and elegant arguments, work in these fields can contribute little that is not trivially known.

Consider Euclidean geometry as an example. In the time of Euclid and the Greeks, geometry was considered the essence of mathematics; they even used it to prove facts about the primes. Now, however, Euclidean geometry as moved into the realm of the content-free. Every statement in pure Euclidean geometry can be restated in terms of basic building blocks like collinearity and concurrency, and these can be translated into systems of polynomial equations that can be solved by standard techniques (e.g. Gröbner basis). So the ability to solve problems in Euclidean geometry is a corollary of an understanding of systems of equations, and the work of Euclid can now be automated by a computer.

Of course, plenty of people still care about Euclidean geometry. Even if the mathematical statements themselves are no longer important for the development of mathematics as a whole, Euclidean geometry still provides a good place to learn how to think about math, and that is a skill that transcends mere problems or mere fields. In addition, Euclidean geometry is still a great source of problems for contests because of its abundance of elegant results. For me, an argument in pure Euclidean geometry still carries an aesthetic appeal that a system of equations will never be able to match.

Will every mathematical field end up as a subfield of a more general theory? As I see more math, I believe that the answer is yes: There is always a more general framework in which old ideas are simplified and trivialized. There’s no reason that anyone would care about such a general framework, however, unless it can serve to unify seemingly unrelated ideas. The pursuit of generality for generality’s sake feels to me as rather silly; adding needless abstraction to a problem does not make it any more important or more interesting. Instead, I believe in abstraction when mathematics is ready — and not before.

In the end, there will always be problems and puzzles for mathematicians to ponder, and that is what is important. Regardless of whether mathematics is actually content-free (and throughout this discussion, we never defined “content-free”, so this is an subjective opinion), mathematics will always be interesting and worth studying.

Dirichlet’s Theorem

We will prove Dirichlet’s Theorem on Primes in Arithmetic Progressions.

Euclid proved that there are infinitely many primes. This, however, says nothing about the distribution of those primes. Dirichlet proved a stronger statement: there are infinitely many primes in every arithmetic progression; in this sense, the primes are “uniformly distributed”.

Theorem 1 (Dirichlet’s Theorem) If {(a, q) = 1}, then there are infinitely many primes {p} satisfying {p \equiv a \pmod q}.

Many ideas of the proof of Dirichlet’s Theorem were first seen in Euler’s proof of the infinitude of primes. Here, Euler actually showed that {\sum_p 1/p} diverges, and that is the approach that we will take for Dirichlet’s Theorem.

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REU

This post was written on an airplane two and a half weeks ago, but I forgot to post it until now. Therefore, it is somewhat delayed.

I was at the Penn State REU for seven weeks this summer, so I thought that I’d comment on it here. I feel that I should probably say something about my problem at the REU, but I can’t do much better than what others have already said. Therefore, for an explanation of the problem, go climb Mount Bourbaki.
I’ll instead make some general comments about research.

The goal of the REU is to provide a Research Experience for Undergraduates, and that’s precisely what was attempted: We were divided into groups and each group was given a problem to solve. At the beginning, nobody had the background to approach their problems, so most of the first few weeks were devoted to learning the background material. Only then could we start to attack the problems.

There are a number of ways in which the idea of undergraduate research in mathematics is not ideal. In the sciences, anyone can participate in a research project; every lab needs someone to do labor-intensive and mind-numbing tasks such as washing bottles. Mathematics, however, is entirely different.

There are no bottle washers in mathematics. Instead, mathematical research is highly individual and requires a large amount of preparation and prior knowledge. As a result, many attempts at undergraduate mathematical research never get past the preparation phase. In this sense, many REUs are like summer reading courses. But there’s one important difference: Because the stated purpose of an REU is to do research and not to learn, the reading courses are very rushed and incomplete. Not only do students never get to doing research, they also do not properly learn.

Of course, the idea of mathematical research for undergraduates is not completely meaningless. In fact, it’s probably healthy for students to understand that research is distinct from problem sets. However, they shouldn’t expect to magically understand research from an REU.

It might be more sensible to redefine REUs as reading courses and stop advertising the research component; it would better match what the students actually need to do. There are several reasons why such a suggestion is impractical, however. Students would find REUs less attractive as they would no longer look as good on their resumes. Maybe more importantly, The NSF would find REUs less attractive and might stop funding them.

The lesson here is that in the real world, appearances matter more than reality. Despite whatever I would like to believe, mathematics does have a nontrivial intersection with the real world.

A Live-TeXing Experiment

In the past two weeks, I’ve taken a mini-course taught by Yakov Pesin at the Penn State REU. The topic is “Fractal Geometry and Dynamics”. Instead of handwriting my notes, I tried to live-TeX notes during the lectures. This was inspired by Akhil Mathew; his thoughts on live-TeXing exist here.

The live-TeX experiment turned out much better that I thought it would. My first reaction to the idea of live-TeXing was that it would be infeasible and horribly inefficient. It seemed indisputable that I could write faster than I could type. Magically, however, I realized that my live-TeXed notes were more comprehensive and more legible than my typical handwritten notes. I thought that I would regularly fall behind while live-TeXing, but this did not happen; I rarely fell behind — certainly no more often than when I used paper and pencil. By this point, I think I’m going to live-TeX most if not all of my classes from now on.

It takes some effort to get used to Live-TeXing. The first few lectures were difficult; I struggled to keep up with all of the formulas. As the class continued, I became faster at typing and learned to think less about trivial matters such as formatting. I also used a number of LaTeX shortcuts for lazy people that sped up the typing process significantly. By the end of the class, I could keep up with even the most complicated formulae.

There are a number of benefits to live-TeXing. For example, having all of my notes on my computer means that it is less likely for me to lose my notes and (hopefully) less likely for me to spill water on them. Furthermore, they’re easier to use (because pdfs are searchable) and they save trees. With respect to content, my live-TeXed notes include all of the historical notes and heuristic remarks that I would have missed while writing manually. Furthermore, they can be read without a magnifying glass.

The main problem with live-TeXing is that it is difficult to copy down pictures during lectures; this was a particularly big problem in the fractal geometry class. I tried to describe each picture in words, though I’m sure that this was not an optimal solution. In the future, I might try to draw pictures by hand and scan them, or draw pictures in Paint. In addition, live-TeXing ended up taking up a bit more time; I spent a few minutes every evening cleaning up the day’s notes (fixing typos, dealing with formulae that run off the page, adding subsection headings). That’s probably a good thing, however; it’s good that I was forced to look back and think about the lecture.

Here are the results on my two weeks of live-TeXing. At two hours per day, this constitutes approximately 20 hours of lectures. If anyone took good handwritten notes with pictures from the class, I’d be happy to scan them and insert them into my file.

Here are the technical details of what I did to make live-TeXing happen.
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math.stackexchange.com

There exists a new website for mathematics:
http://math.stackexchange.com/

Users can post and answer questions about mathematics. Questions from all levels of mathematics are permitted and encouraged. The site is not a discussion forum; instead, through an efficient voting system, the site aims to obtain definitive answers.

A related site is MathOverflow. This site is an amazing tool used for mathematical research. Mathematicians ask and answer questions on the site, making communication in research much easier. The new Math.SE site is a more open relative of MathOverflow.

Math.SE was created through an open democratic process at Stack Exchange’s Area 51. The site entered private beta a week ago, and as of today, it is in public beta and everyone can use it. If you’re interested, go make an account. Spread the word and help this site succeed!