# Doing mathematics in a pandemic – Part I: AlCoVE

I’ll be writing up a series of posts on what I’ve learned so far about adapting my work to a pandemic-compatible lifestyle. This is the first, and focuses on math conferences. Stay safe out there!

It was March 15, 2020, and suddenly everything stopped.

This story likely sounds familiar, because the same thing probably happened to you. Classes went online. Conferences were cancelled. No more chatting with colleagues at department tea. Home life suddenly became radically different and also much more central.

The world had grinded to a halt, and yet… there was one thing that began. And that was an overwhelming sense of community and solidarity, because everyone else in the world had stopped too. And it seemed to me to be an excellent opportunity to try to create something new together.

## Mathematics and community

It is said that the most important aspect of conferences is not the talks, but the coffee breaks between them.

It sounds at first like a joke about how dependent mathematicians are on caffeine. But there is a real truth to it in a different sense. The coffee breaks are where connections are made, where new ideas are spawned, where the speaker meets the one person who just might have the right tools to crack that open problem that they posed on their last slide. They’re where pairs of mathematicians who find themselves in a deep conversation comparing each of their latest tableaux insertion algorithms awkwardly check their watches and schedules and both sheepishly admit that they weren’t really looking forward to the conference banquet anyway. They then grin and scurry off to an unoccupied whiteboard to make a new joint discovery.

When everything stopped, that stopped too. But did it have to, entirely?

This was a question I posed on the Facebook group for mathematicians who specialize in symmetric functions and related algebraic combinatorics (yes, there is a Facebook group for that!). I asked if anyone would want to help me organize an online conference that tried to re-create as many of those in-person networking aspects as possible. Something that could even potentially continue into the future, as flying to so many conferences all the time, while good for mathematical progress, is not really environmentally sustainable.

I got three enthusiastic responses within an hour. Laura Colmenarejo, Oliver Pechenik, and Liam Solus were on board, and we had an organizing committee!

## AlCoVE: an Algebraic Combinatorics Virtual Expedition

In order to capture the essential aspects of the conference, namely that it is about algebraic combinatorics and that it aims to capture as many of the in-person advantages of conferences as possible, we named it the Algebraic Combinatorics Virtual Expedition, or AlCoVE. It didn’t hurt that alcove walks are a highly useful and modern combinatorial construction that arise in the study of Coxeter groups, symmetric functions, and geometry (see these slides by Elizabeth Millićević for excellent illustrations of alcove walks). We had a name, and we had a pun. It was a good start.

Then came the design phase. Laura, Liam, Oliver, and I met on Zoom weekly to start planning, and started by trying to answer some of the basics:

1. What days should the conference be held?
We initially thought of holding a weekend conference, but then we considered that with home life being more central during the pandemic, perhaps we should have it during “work hours” so as not to overlap with participants’ family/life plans. So we decided on Monday and Tuesday, June 15-16, on a week in which participants at universities with either a semester or quarter schedule would be unlikely to be teaching.

I think it was the right choice in the end; in our post-conference feedback form, only 8 of the 71 respondents said they would have preferred a weekend conference. Another 24 were neutral, and the remaining 39 said they preferred the weekdays over weekend.

2. How do we account for differing time zones, given that participants are going to be in many different locations around the world?
Our solution to this was perhaps a bit biased towards the West, as our organizers were all in either America or Europe. But we planned the conference to be from 11 AM to 5 PM Eastern time, so that on the west coast of the USA it would be from 8 AM to 2 PM, and in Europe it would be an evening conference, for instance from 4 PM to 10 PM in London.

That being said, we had participants from India, South Africa, Australia, China, New Zealand, and more. The time zone barrier just didn’t matter as much as we thought it would. And according to the feedback form, most participants were happy with the time and scheduling of the conference.

3. How many speakers should we have and how long should each talk be?
Zoom fatigue is real, and it’s just harder to focus when staring at a screen than sitting in a lecture hall. In light of this fact, we decided to have talks be on the short side, a total of 30 minutes each including questions. This gave us space in the schedule for 12 talks (6 per day) with plenty of breaks and exciting social events in between.

We then came up with a list of potential speakers to invite. We were lucky to have a team four organizers with a diverse set of interests and geographical networks within algebraic combinatorics, and we tried to come up with a good balance of mathematical and geographical diversity among the speakers. While we didn’t initially consider gender diversity while creating our list, we were pleased to see that 6 out of 12 of the mathematicians we naturally thought of first were female. It was perhaps a reflection of the friendliness and diversity that already exists in the algebraic combinatorics community.

To our delight, everyone that we invited to speak accepted our invitation. There are perhaps some advantages of organizing a conference at a time when literally everything else is cancelled.

4. Should we have a poster session?
This took us a long time to decide on and subsequently plan; indeed, a virtual talk is one thing, but how do you run a virtual poster session? Then again, poster sessions are a great way to give younger participants, especially graduate students, the opportunity to share their work and ideas.

We did end up organizing a poster session, and limited the number of posters to 12 so that it would be more manageable in a virtual setting. We had a ton of excellent submissions that were very hard to choose between. The way we implemented it was by assigning one breakout room for each poster in Zoom, and then give every single participant “co-host” power in the meeting so that they can freely move between breakout rooms as if they are walking from one poster to another. (Non-co-hosts do not have this power in Zoom.)

It went well overall. See “Conference Day 2” below for details on how the poster session went, and ideas on how to make a poster session potentially run even more smoothly at future conferences.

5. What should “coffee breaks” or “lunch breaks” consist of, in order to optimize social and mathematical connection in a virtual environment?
I’m glad you asked! This was by far the most fun part of planning the conference, and there were many bouts of doubled-over, tears-streaming-down-face laughter among the organizing committee during our Zoom meetings as we brainstormed potential fun ideas for conference activities. Here was what we came up with, and the surprises involved in planning each:
• Polls. Fun, meaningless pseudo-mathematical polls, with multiple-choice questions like “What is the worst Coxeter group?” and “Do you consider yourself a combinatorist, a combinatorialist, or a combinatoricist?” were our first idea for a social event during the breaks.

We were inspired by Zoom’s “poll” feature, but we quickly realized that using Zoom’s built-in poll system was not ideal. We wanted to split participants into breakout rooms to take the poll, so that smaller discussions of the questions could take place. But Zoom’s polls do not show up when participants are in breakout rooms. So that eliminated Zoom’s feature as an option pretty quickly.

Instead, we used Google Forms to put together the polls. Here is one example:
AlCoVE Poll 1
We simply shared the link in the Zoom chat, then split participants up into breakout rooms randomly and gave them time to participate. We then called everyone back at the end to discuss the poll results, and it served as a predictably hilarious and relaxing break between talks.

• Escape Rooms. We created one short “virtual escape room”, again in Google Forms, which has a regular expression matching feature to check answers, so that you could prevent participants from going to the next “room” (page) until they have solved the riddle in the previous “room”.

(Tip: To enable this feature on a given question when creating a Google form, simply click on the three dots in the lower right of a question frame and click “Response Validation”. There are then options to make the answer have to match a regular expression of your choice, and return an error message if it is incorrect.)

Here was our conference escape room. Clearly none of us were professional puzzle writers, but when Team 2 escaped their breakout room and came back into the main Zoom room before any other team, they punched the air and cheered in victory, and we knew the social event had achieved its purpose.

• Scavenger hunt. We created a scavenger hunt, again in Google forms, that asked participants to find things in their home, such as math textbooks or conference T-shirts, to try to match or differ from their teammates in their breakout room to score the most possible points. We got some excellent pictures submitted to the scavenger hunt challenges, and it made for great “conference photos”.

• Virtual Excursions. These were intended as true breaks from participants’ home office desks, in which participants were split into breakout rooms and encouraged to walk around their house with their phone or laptop on Zoom to show their breakout room their local surroundings and just generally stretch their legs. The aim was to re-create the aspect of conferences in which participants walk from the conference building to the banquet hall and end up walking with a little group and chatting on their way.

It didn’t quite end up truly re-creating what we were hoping for, but it was an easy excursion to organize and was one step up from just putting participants in breakout rooms with no direction as to what would happen in that break, which can lead to awkward silences and a lot of turned-off videos.

• Make-Your-Own-Lunch breaks. There was a half hour “lunch break” in the middle of each conference day, in which participants were again split into breakout rooms and encouraged to make and eat lunch together over Zoom. It did lead to more interaction – who doesn’t like to talk about food? – and was the only official meal we scheduled for participants to have together.

• Happy Hour. At the end of the first day of the conference, we made everyone co-hosts (see Conference Day 1 or 2 below for some details on this process) and set up 11 breakout rooms. You can name breakout rooms manually in Zoom, and we called one room the “Lobby” and put everyone in the lobby to start out. The other rooms were called “Table 1” through “Table 10”. Since participants had co-host powers, they were able to go “sit down” at any table they chose.

This worked very well to mimic an actual happy hour in which there are a number of tables in a large conference room and participants mingle by moving from table to table to see old friends and meet new acquaintances. The only things we couldn’t provide virtually were drinks and appetizers!

## Conference Day 1: Success or disaster?

With all the talks, poster sessions, and social events planned out, it was finally time for the conference! Laura, Liam, Oliver, and I had several last-minute meetings to test everything and everything seemed to be in order.

Naturally, a major issue arose within the first half hour of the conference. As participants were signing in, we quickly realized it was capping the number of participants at 100, even though I had already bought the Zoom ability for my account to host 500 participants. Meanwhile, over 400 people had registered. The first talk was 5 minutes away, and I had no idea why Zoom was capping us at 100. What were we going to do?

We quickly sent emails directing everyone to switch to a different Zoom meeting number on Oliver’s university account, which had a 300 person capacity, and crossed our fingers and hoped that the number of participants did not exceed 300 at any given time that morning.

Luckily we capped out at about 290 during the first talk. Disaster averted!

In the meantime, I poked around Zoom and found the switch I needed to flip. Apparently even if your personal Zoom account is listed as being able to host large meetings, you are considered a “user” on your own Zoom account and you have to enable that “user” (yourself) to be able to use that power that the entire account bought. It makes no sense, but there it is.

I flipped the switch on zoom.us and we switched back to the original planned Zoom link after the lunch break. It was still glitchy; on both Oliver’s and my accounts, we had delays in the Zoom chat when people tried to post links and other information. It seemed that 200+ participants was simply getting a bit too large for Zoom to handle in one meeting, and their “large meeting” option was not entirely without issues yet.

The last thing that was awkward on the first day was the preparation for the happy hour. There is no way to assign users as co-hosts of the meeting before they log on, which means we had to manually make users co-hosts in advance of the happy hour. But there is also no button that makes everyone co-hosts at once on Zoom, so the only option is to manually make every participant a co-host one by one.

To make matters even more awkward, every time a participant is made a co-host, a little notification shows up on everyone’s screen. So the aim is to make them slowly enough that you don’t overwhelm the talk slides with notifications, but fast enough that everyone is a co-host by the time the happy hour starts so that they can all sit down at the “tables” (breakout rooms) of their choice. It was a tricky business but we got it done.

Aside from the technical issues, the first day went well. There were fantastic talks and funny polls and virtual excursions and a happy hour to cap it off at the end of the day.

## Conference Day 2: Success!

While we didn’t have beginner’s luck, we did learn from Day 1, because Day 2 went much more smoothly. The conference didn’t cap our participants at 100. There were fewer participants overall and therefore fewer glitches in the chat window. The talks were incredible again, and the social activities went smoothly.

The main new challenge was the poster session. This was far harder to prepare for than the happy hour, because not only did I have to make everyone co-hosts during the talk preceding the poster session, but I had to create breakout rooms according to the posters. I created one “Lobby” room and then one room per poster, and tried to put the speaker and name of the poster as the name of the breakout room.

What I didn’t realize was that Zoom has a character limit on the breakout room names. What that meant was that I couldn’t just copy and paste the names of the presenters and titles of the posters from our website into Zoom. I had to first abbreviate and edit the titles so that they were under Zoom’s character limit, in a way that the content of the poster would still be clear to a participant browsing the titles from within Zoom. And naturally if I was editing a title in Zoom but tabbed over to glance at the title again before hitting “save”, it would delete my work and I’d have to start over. It was an unbelievable pain and I’d definitely prepare the abbreviations in advance next time.

Luckily I just barely finished the naming and assigning and co-hosting by the time the poster session was about to begin. And it began! I mostly stayed in the main room and directed lost souls who lost internet connection for a bit, but my co-organizers said that the poster session went very smoothly overall.

## Video recordings and wrap-up

We recorded all the talks, and after the conference we used iMovie to do some basic processing (such as a title slide for each), and uploaded them to the new AlCoVE YouTube channel. We hope to add to this channel in future years!

Indeed, what was magical about AlCoVE is how it brought together so many mathematicians from all around the world so easily, and still re-created some of the social and networking advantages of in-person conferences. Moving more conferences online can not only drastically reduce the carbon footprint of academia, but even help with inclusivity and diversity in the community, as even those who ordinarily would not be able to travel were able to participate.

All in all, I believe AlCoVE was a very positive thing to come out of the worldwide shutdowns. I’m grateful to everyone who helped organize or speak or participate, and I hope (and will try to ensure) that it continues to run in future years, pandemic or no pandemic.

A few weeks after AlCoVE, I participated in FPSAC 2020 Online, the online pandemic version of an existing annual international conference called Formal Power Series in Algebraic Combinatorics (FPSAC). It was designed quite differently and also worked very well, and I learned about alternatives to Zoom breakout rooms like gather.town and Unhangout that could potentially be better for a happy hour or poster session than Zoom was. I’m excited to see where all of these recent virtual technologies lead the mathematical community in the long run.

A few weeks ago I attended the AWM (Association of Women in Mathematics) Research Symposium in Houston, TX. I gave a talk in my special session, speaking on queer supercrystals for the first time, to a room full of female mathematicians.

I was a bit disappointed when, at the end of my talk, no one raised their hand to ask any questions.  It’s usually the classic sign of an uninteresting or inappropriately aimed talk, so I figured that maybe I had to revisit my slides and make them more accessible for the next time I spoke on the subject.

Afterwards, however, several of the women in my session came up to me privately to ask specific questions about my research.  When I told my husband about this after the conference, he pointed out that perhaps they just were the kind of people to prefer asking questions one-on-one rather than raising their hands during or after the lecture.

“Did anyone in your session ask questions after the other talks?” he asked me, testing his theory.

I thought about it, and was surprised when I realized the answer.  “Woah, I think you’re right,” I said.  “I asked at least one question after nearly every talk.  But I think I was the only one.  Once in a while one other woman would ask something too.  But the rest kept their hands down and went up to the speaker during the break to ask their questions.”

Upon further reflection, I realized that this was even true during the plenary talks.  During an absolutely fantastic lecture by Chelsea Walton, I was intrigued by something she said.  She mentioned that the automorphism group of the noncommutative ring $$\mathbb{C}\langle x,y\rangle/(xy-qyx)$$ is $\mathbb{C}^{\times} \times \mathbb{C}^{\times}$ for all $q\neq \pm 1$, but the answer is different at $q=1$ and $q=-1$.  I knew that many of the standard $q$-analogs arise naturally in computations in this particular ring, such as the $q$-numbers $$[n]_q=1+q+q^2+\cdots +q^{n-1}.$$ So, I wondered if the exceptions at $q=1$ and $q=-1$ were happening because $q$ was a root of unity, making some of the $q$-numbers be zero.  So maybe she was considering $q$ as a real parameter?  I raised my hand to ask.

“Is $q$ real or complex in this setting?”

“It’s complex,” Chelsea answered.  “Any nonzero complex parameter $q$.”

“Really?” I asked. “And there are no exceptions at other roots of unity?”

“Nope!” she replied with a smile, getting excited now.  “Just at $1$ and $-1$.  The roots of unity get in your way when looking at the representation theory.  But for the automorphism group, there are only two exceptional values for $q$.”  Fascinating!

No one else asked any mathematical questions during or after that talk.

Now, I have the utmost faith in womankind.  And I would normally have chalked the lack of questions and outspokenness up to it being a less mathematically cohesive conference than most, because the participants were selected from only a small percentage of mathematicians (those that happened to be female).  But it reminded me of another time, several years ago, that I had been surprised to discover the same phenomenon among a group of women in mathematics.

One summer I was visiting the Duluth REU, a fantastic research program for undergraduates run by Joe Gallian in the beautiful and remote city of Duluth, Minnesota.  As a former student at the program myself, I visited for a couple of weeks to hang out and talk math with the students.  I attended all the weekly student talks, and as usual, participated heavily, raising my hand to ask questions and give suggestions.

The day before I left, Joe took me aside.  “I wanted to thank you for visiting,” he said.  “Before you came, the women never raised their hand during the other students’ talks.  But after they saw you doing it, suddenly all of them are participating and raising their hands!”

I was floored.  I didn’t know that being a woman had anything to do with asking questions.

I have always felt a little out of place at AWM meetings.  They are inevitably host to many conversations about the struggles faced by women in competitive male-dominant settings, which I have never really related to on a personal level.  I love the hyper-competitive setting of academia.  I live for competition; I thrive in it.  And it never occurs to me to hold back from raising my hand, especially when I’m genuinely curious about why $q$ can be a complex root of unity without breaking the computation.

But, clearly, many women are in the habit of holding back, staying in the shadows, asking their questions in a one-on-one setting and not drawing attention to themselves.  And I wonder how much this phenomenon plays a role in the gender imbalance and bias in mathematics.

At the reception before the dinner at the AWM conference, I spotted Chelsea.  She was, unsurprisingly, quite popular, constantly engaged in conversation with several people at once.  I eventually made my way into a conversation in a group setting with her in it, and I introduced myself.

“Hi, I just wanted to say I really enjoyed your talk!  I was the one asking you whether $q$ was real.”

Her expression suddenly shifted from ‘oh-no-not-another-random-person-I-have-to-meet’ to a warm, smiling face of recognition.  “Oh!  I liked your question!” she exclaimed.  The conversation immediately turned to math, and she was nice enough to walk me through enough computations to convince me that $q=\pm 1$ were special cases in computing the automorphism group of the noncommutative ring.  (See Page 2 of this post for the full computation!)

The entire experience got me thinking.  It was because I raised my hand that Chelsea recognized me, that she was happy to talk to me and mathematics was communicated.  It was because I raised my hand that I got the question out in the open so that other participants could think about it as well.  It was because I raised my hand that women were doing mathematics together.  And perhaps it is because I raise my hand that I have no problem interacting in a male-dominant environment.  After all, they raise their hands all the time.

It is tempting to want to ask the men in mathematics to take a step back and let the women have the limelight once in a while.  But I don’t think that’s the answer in this case.  Men should keep raising their hands.  It’s part of how mathematics gets done.  It helps to communicate ideas more efficiently, to the whole room at once rather than only in private one-on-one settings.  It draws visibility to the interesting aspects of a talk that other participants may not have thought of.

What we really need is for women to come out of the shadows.  So, to my fellow women in mathematics: I’m calling on all of us to ask all our questions, to engage with the seminar room, to not hold back in those immensely valuable times when we are confused.  And raise our hands!

# PhinisheD!

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.

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: and there are four closed walks of length two, from $v$ to any other vertex and back again:

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…

…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:

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

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:

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

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:

But if you get it wrong, red light!

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

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.

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!