PhinisheD!

Sometimes it’s the missteps in life that lead to the greatest adventures down the road.

For me, my pursuit of a Ph.D. in mathematics, specifically in algebraic combinatorics, might be traced back to my freshman year as an undergraduate at MIT. Coming off of a series of successes in high school math competitions and other science-related endeavors (thanks to my loving and very mathematical family!), I was a confident and excited 18-year old whose dream was to become a physicist and use my mathematical skills to, I don’t know, come up with a unified field theory or something.

Me at the age of 18-ish.

Me at the age of 18-ish.

But I loved pure math too, and a number of my friends were signed up for the undergraduate Algebraic Combinatorics class in the spring, so my young ambitious self added it to my already packed course load. I had no idea what “Algebraic Combinatorics” even meant, but I did hear that it was being taught by Richard Stanley, a world expert in the area. How could I pass up that chance? What if he didn’t teach it again before I left MIT?

On the first day of the class, Stanley started with a simple combinatorial question. It was something like the following: In a complete graph with $n$ vertices, how many walks of length $k$ starting at vertex $v$ end up back at vertex $v$ on the last step? For instance, if $n=5$ and $k=2$, the graph looks like: graph5-noblue and there are four closed walks of length two, from $v$ to any other vertex and back again:

graph5

There is an elementary (though messy) way to solve this, but Stanley went forth with an algebraic proof. He considered the adjacency matrix $A$ of the complete graph, and showed that the total number of loops of length $k$ starting from any vertex is the trace of $A^k$. One can then compute this trace using eigenvalues and divide by $n$ to get the number of loops starting at $v$. Beautiful!

I remember sitting in my seat, wide-eyed, watching Richard Stanley quietly but authoritatively discuss the technique. It was incredible to me that advanced tools from linear algebra could be used to so elegantly solve such a simple, concrete problem. To use a term from another area of algebraic combinatorics, I was hooked.

But I was also a freshman, and didn’t yet have a strong grasp of some of the other algebraic concepts being used in the course. I studied hard but wound up with a B+ in the class. Me, get a B+ in a math class? I was horrified, my 18-year-old Little-Miss-Perfect confidence shattered. Now, not only was I fascinated with the subject, I gained respect for it. It was a worthy challenge, and I couldn’t help but come back for more.

In the years that followed, I took more courses on similar subjects and wrote several undergraduate research papers. I dabbled in other areas as well, but was always drawn back to the interplay between combinatorics and algebra. I now find myself, as of Friday, May 20, 2016, having completed my Ph.D. at UC Berkeley on a topic in algebraic combinatorics…

Maria Grad - Web - 19

…and I often wonder how much that silly little B+ motivated me throughout the years.

(See page 2 for a summary of my thesis. My full thesis can be found here.)

Expii: Learning, connected

It’s been several months since I posted a gemstone, and the main reason is that much of my free-time mathematics energy recently became channeled into a new project: Expii.

Expii (currently beta) is a new online crowdsourced learning site that aims to fill the gaps in users’ understanding of topics, with the goal of making math, science, and other topics easy for everyone in the universe. Its motto? Learning, connected.

With an addictive, game-like format (hence the XP pun) in which users are awarded “fame points” for writing good explanations and “experience points” for successfully making it through tutorials, Expii is more interactive and community oriented than other online learning resources like Wikipedia. It is also more structured than question-and-answer sites like Quora or Stack Exchange, in that the primary “graph structure” for the topics is organized by our team, and users fill in the content in the nodes.

The first thing you see when you go to www.expii.com is the highest-level Universe graph:

Screenshot - 06012014 - 05:48:02 PM

This currently has two disjoint subgraphs: Expii Guide and Calculus. One can scroll or click to zoom in on Calculus:

ZoomInCalculus

And magically, other smaller subgraphs appear! Keep zooming in and eventually you get to the lowest level of detail, which has Topic nodes that you can click on:

TopicNodes

Let’s click on Lines and Slopes. This brings us to a user-written explanation of lines and slopes!

LinesAndSlopes

It is easy for an explainer to write interactive questions, for the student to answer before moving on to the next part of the explanation. For instance, if you answer the first question correctly here, it gives you a green light and reveals the next part of the explanation:

OneRight

But if you get it wrong, red light!

WrongAnswer

When you’re done with a topic or simply feel like browsing, you can scroll down for a seamless transition to a related topic:

NextTopic

And this is just the beginning. Expii was founded by Po-Shen Loh and Ray Li only a handful of months ago. They quickly drew in a fantastic team of mathematicians of scientists (including me) that care about education, outreach, and spreading the love of learning in a way that is fun and engaging. It will be exciting to see what Expii becomes over the next few years.

If you’d like to experiment with Expii yourself, write some explanations, or contribute to the project, contact me and I can get you a referral code so that you can log in. It’s the newest and shiniest gemstone in mathematics education!

FindStat and combinatorial statistics

Last semester, I attended Sage Days 54 at UC Davis. In addition to learning about Sage development (perhaps a topic for a later blog post), I was introduced to FindStat, a new online database of combinatorial statistics.

You may be familiar with the Online Encyclopedia of Integer Sequences; the idea of FindStat is similar, and somewhat more general. The Online Encyclopedia of Integer Sequences is a database of mathematically significant sequences, and to search the database you can simply enter a list of numbers. It will return all the sequences containing your list as a consecutive subsequence, along with the mathematical significance of each such sequence and any other relevant information.

FindStat does the same thing, but with combinatorial statistics instead of sequences. A combinatorial statistic is any integer-valued function defined on a set of combinatorial objects (such as graphs, permutations, posets, and so on). Some common examples of combinatorial statistics are:

  • The number of edges of a finite simple graph,
  • The length of a permutation, that is, the smallest length of a decomposition of the permutation into transpositions,
  • The number of parts of a partition,
  • The diameter of a tree.

The FindStat database has a number of combinatorial objects programmed in, with various statistics assigned to them, which can all be viewed in the Statistics Database tab. The search functionality is under Statistic Finder, in which you can choose a combinatorial object, say graphs, and enter some values for some of the graphs. It will then tell you what statistics, if any, on graphs match the values you have entered.

So this is strictly more general than OEIS: we can think of integer sequences as combinatorial statistics on some collection of combinatorial objects represented by the nonnegative integers, such as finite collections of indistinguishable balls. Not that FindStat should be used for integer sequences – OEIS already does a splendid job of that – but FindStat provides something that OEIS cannot: an organized database of mathematical data that doesn’t necessarily have a natural linear ordering.

The last, and most interesting, feature of FindStat is its “maps” functionality. There are many known natural maps of combinatorial objects, such as the map $\phi:P\to B$ sending a permutation to its corresponding binary search tree, where $P$ denotes the set of all permutations and $B$ the set of all binary search trees. (See here for all the maps currently implemented on the Permutations class in FindStat.) Now, given a statistic $s:B\to \mathbb{Z}$ on $B$, we automatically get a statistic $$s\circ \phi:P\to \mathbb{Z}.$$ FindStat uses this fact to give the user more information: it will give you not only the matching statistics on the combinatorial object that you chose, but the matching statistics on all other possible combinatorial objects linked by any relevant map in the database! This can help the working combinatorialist discover new ways of thinking about their statistics.

Olympiad vs. research mathematics

Recently, some interesting discussions on math education and mathematical philosophy have taken place on, of all places, my Facebook wall. Since Facebook is a rather restrictive medium, I feel it’s time to widen the scope of my blog to include such topics.

I am currently sitting in the main common area in our residence halls at the Math Olympiad Summer program (abbreviated MOP) in the middle of Lincoln, Nebraska. It is rather quiet, as the students are all taking their 5th “MOP test” in the last two weeks, and my fellow instructors and graders are either proctoring the tests or preparing more material to teach these brilliant kids.

It got me thinking about a topic that is often discussed among mathematicians and among those in math contest circles: How correlated is mathematical contest ability with mathematical research ability? Does one help or hinder the other? Is the social environment that math competitions create hostile to noncompetitive students, especially women?

Math, Marathons, and Spelling Bees

The first two questions are somewhat easier to address, and many successful mathematicians have expressed their thoughts. Terrence Tao, who earned a gold medal at the International Math Olympiad at the age of 13, has written a blog post on the topic. He links to this post on LessWrong, which lists a number of quotes by great mathematicians regarding math competitions.

Given how many prominent mathematicians were successful at math contests themselves, I was surprised that the general consensus among these mathematicians seems to be that success in math contests hardly correlates at all with the ability to do the slower, deeper, more tedious work required for mathematical research. Some go on to claim that being immersed in “math contest culture” may actually harm one’s ability to produce novel mathematical ideas, since it encourages a kind of impatience in the mathematician. William Thurston even goes so far as to compare math contests with spelling bees:

These contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.

In an ideal world, I would agree. That is, if schools gave as good of a sense of what doing mathematics was as they give a sense of what writing is, then math Olympiads would not be as necessary in terms of encouraging students to pursue mathematics. As it is, though, math contests such as the AMC, which at least require some level of mathematical thinking, are the closest thing most American high school students can get to experiencing a glimpse of what mathematicians really do.

Additionally, although I am in no position to disagree with William Thurston, and my opinion may change as I dive deeper into the research world, I find further problems with the spelling bee analogy. Math contests, especially at the Olympiad level, require much more meaningful work than memorizing a dictionary and spelling some words correctly. At MOP, we teach the students modular arithmetic, generating functions, projective geometry, the probabilistic method… all things that a professional mathematician might spend a lifetime studying. And the students who really understand these subjects deeply are generally the ones who perform better on the tests we give them. So what is going on?

Let’s consider an alternative analogy: Math Olympiads are to mathematical research as a 5K road race is to a marathon. Both are challenging endeavors that take a lot of training and practice. The 5K requires more speed, strategy, raw strength, and head-to-head competing, while the marathon is more about patience and diligence.

But are they uncorrelated? Are 5K’s just the spelling bees of running, and don’t necessarily predict future marathon or ultramarathon success? This is where I disagree. The best marathoners in the world would leave most runners in the dust in a 5K, and the best 5K runners in the world can place highly in any marathon after only a few months of endurance training. Similarly, I have no doubt that most of the best mathematicians in the world would do just fine on the USAMO, and I have seen firsthand that the best Olympiad students can usually produce some good research after just a few months at a summer REU.

Of course people who have never run a 5K in high school can start running at the age of 25 and run a marathon; I am not disagreeing on that point. But I do disagree that math Olympiad training gives no significant mathematical advantage; it’s always going to be easier for that high school track star to go on to do marathons later in life. Whether they choose to do so is a different matter and depends on a lot of personal factors that are hard to quantify.

Social Environment and Women

The third question – whether the social environment created by math contest culture is hostile to noncompetitive students and girls in particular – is a trickier one. I bring it up because I just read a blog post from last year by “MathBabe” Cathy O’Neil. A quote:

The reason I claim math contests are bad for math is that women are particularly susceptible to feelings that they aren’t good enough or talented enough to do things, and of course they are susceptible to negative girls-in-math stereotypes to begin with. It’s not really a mystery to me, considering this, that fewer girls than boys win these contests – they don’t practice them as much, partly because they aren’t expected by others, nor do they expect themselves, to be good at them. It’s even possible that boys brains develop differently which makes them faster at certain things earlier- I don’t know and I don’t care, because I don’t think that the speed issue is correlated to later deep thought or mathematical creativity.

As a woman who has excelled at math competitions and is now pursuing a Ph.D. in math, I find this comment both interesting and very hard to relate to. Math contests, which started for me in middle school, were always a joy to me, because I loved the mathematics so much. Yes, I practiced to get faster, but I mainly practiced because I was amazed at how you can use modular arithmetic to find the units digit of $7^{2002}$ without calculating the whole number, and how the area of a triangle was equal to the product of these mysterious quantities, the inradius and the semiperimeter. My father would have me prove the Pythagorean theorem, or derive the quadratic formula from scratch, and with each new understanding I appreciated mathematics even more.

I made friends through math teams and programs, and the community support spurred me on to continue to practice and study. Rather than it being a hostile environment, I found the social circle of math geeks to be much more welcoming than the bullying crowd of “popular” kids that dominated my high school. So wherever the negative social reinforcement is coming from for girls, I don’t think it can possibly be coming from math contests themselves.

One interesting thing about Cathy’s comment, though, is that perhaps boys’ brains do develop in a different way. Perhaps we should have different divisions for women in more of our math contests, just as there is always a womens’ division in any 5K race.

But even as is, I find that math Olympiad training is useful and encouraging for students in mathematics, women included. I would never have gone so far with mathematics if it hadn’t been for the math contests that helped me realize that math was more than just memorizing your multiplication tables. Or the dictionary.

Rota’s Indiscrete Thoughts

I am a huge fan of Gian-Carlo Rota, who has been said to be the founding father of modern algebraic combinatorics. (He is also my mathematical grandfather-to-be.)

Rota was a philosopher as well as a mathematician, and wrote an entire book primarily concerning the philosophy of mathematics. His book is called Indiscrete Thoughts.

I’ve been reading this recently, and I highly recommend it. It reads like a novel; he motivates everything with enticing examples regarding mathematicians that he has known or familiar mathematical theorems and proofs. He brings up a lot of interesting points and questions, including:

  • Is mathematics “created” or “discovered”? This is a common point of debate among mathematicians, and Rota addresses it beautifully. He gives clear and precise examples of mathematical work that is obviously one or the other, and then goes on to show how the two notions can, and do, naturally coexist.
  • How can we make rigorous some of the notions that mathematicians use all the time, but can never formally write about? There are plenty of processes that go on in our mind, leaps of faith and intuition, that we cannot easily talk about and use in a formal mathematical setting, because they are not part of established formal logic.
  • What is mathematical beauty, and why does it seem to depend on context and historical era?

Even if you don’t agree with Rota’s conclusions, his examples are so vivid and revealing that it’s impossible not to get something out of this book. I personally am coming away with a clearer perspective on mathematics and what it actually is.

TeXStudio

First post in several weeks; term has hit. But in the midst of the hustle and bustle of the start of the semester, I’ve discovered a gemstone within the mathematical software world that was too good not to share: TeXStudio.

I discovered it while preparing for the “LaTeX Tricks” seminar this week, which I am organizing as part of the UC Berkeley Toolbox Seminar. (The seminar is looking to be quite exciting if you’re in the area – we will have a variety of speakers give 10-minute talks on their favorite LaTeX tool or package. It will be this Wednesday from 2:30-4PM in room 891 Evans Hall.)

Since I will be giving the introductory talk, I was looking around at LaTeX front-ends today to see which ones to recommend. My first conclusion was that Wikipedia is awesome. They have a nice comparison chart of TeX editors here:

http://en.wikipedia.org/wiki/Comparison_of_TeX_editors

I then went through and tried out a few of the more “green” editors on that chart. A long-time Gedit user myself, I first tried the LaTeX Gedit plugin. It looked nice, but after about 10 minutes it mysteriously stopped working. It turns out it isn’t compatible with the new version of Gedit, and you can’t go back to the old version of Gedit sicne it doesn’t run on the new version of Ubuntu (12.04). Sigh.

So I tried a few of the others. One was LyX, which has a nice GUI that automatically renders your math mode code inline. But I find it to be slower to type in, since there are odd cursor placings after the automatic rendering. (Perhaps one just needs to get used to it.) Other highly-rated editors on the comparison chart were AUCTEX, a plugin for Emacs, and TeXlipse, a plugin for Eclipse. However, I haven’t yet gotten past the Emacs learning curve, and I don’t need the full power of Eclipse.

Finally, I tried out TeXStudio. It’s great! It has:

  • Nice auto-complete features based on your most-used commands. So, you’ll begin typing \begin{it… and it will ask you if you want a \begin{itemize}...\end{itemize} block with an \item command in the middle. Yes, I do, thanks.
  • It aligns your code nicely. The next line is automatically tabbed like the last one. Particularly useful in a series of \item lines.
  • It has a collapsible toolbar on the left that can display a number of things, the most important being an outline of your entire document by sections and subsections. You can click on the outline to jump to a section of your document.
  • A nice pdf previewer that you can search, and you can jump to that point in your code by clicking on the pdf! A wonderful feature.
  • Helpful menus for things like Asymptote, PStricks, TikZ, and all the really weird symbols that you always forget the commands for, like $\wp$ (\wp).
  • Last but not least, it is highly customizable. You can make your own macros for strings that you commonly type, you can change the font and size of the code, and the side and bottom toolbars can be easily collapsed (and then brought back) if you don’t feel like looking at them.

Here is a screenshot that demonstrates many of the above features:

Well, I’m sold. I will be switching to TeXStudio.

Introduction

Welcome!  The purpose of this blog is to record some of the particularly beautiful mathematical ideas I have seen or invented, and share them with you.

The process of doing mathematics is much like a quest to uncover mathematical truths.  Sometimes, such a truth may be valid but uninteresting, just another pebble or grain of sand along the beach.  But other times, you will uncover a gemstone – a particularly aesthetic, beautiful, or useful truth hiding in the vast sandpiles of information.

This blog is devoted to the gemstones of my mathematical investigations.  Enjoy!