Setting up a virtual conferencing room

What if we didn’t have to emit quite as much carbon to do mathematics effectively?

As the latest heat wave sweeps the globe, it’s nice and cool down here in the basement of the mathematics building at Colorado State University. Thanks to a combination of my NSF grant and contributions from the CSU College of Natural Sciences, we have set up an easy-to-use, high quality virtual conferencing room here!

While there is certainly a benefit to in-person networking at conferences, some other aspects of academic work can arguably be improved by building high quality virtual spaces. For one, there is a significant carbon footprint caused by academic travel, and the COVID-19 pandemic has given us the opportunity to think about ways to slash our carbon footprint. For another, there are disabilities, economic disparities between regions, and caregiving situations that make it difficult for some academics to travel as much as others. Improving virtual options could improve accessibility and enable such academics to more easily share their ideas.

And finally, playing with virtual technology is fun!

The Setup

The first thing a virtual conferencing room needs is a designated computer and webcam. But how do you store such a computer in a secure way?

Our solution was to buy a laptop and store it in this wall-mounted lock box on a wall in the (small) room:

The box opens with a custom keypad combination to reveal a laptop:

Included in the box is a webcam and (not pictured) a USB lavalier microphone with a 20 foot cord. Open up the laptop and the following software is easily accessible to all users:
  • Zoom
  • Microsoft Teams
  • LogiTune, which allows one to easily adjust the zoom, focus, and settings on the Logitech webcam
  • Open Broadcasting Software (see my previous post on using OBS for live virtual talks or recordings)
  • VB Audio Cable for virtual audio to use with OBS

Directly across the room from the laptop box is a wall with several options for giving a talk. First, there is a whiteboard on the wall for low-tech virtual math presentations:

What’s that hanging from the ceiling? It’s not a projector screen… why, it’s a pull-down green screen!

Stand in front of the green screen, turn on OBS and upload your slides to the computer, and you can give a slide presentation that makes it look like your slides on the laptop are actually displayed behind you on the green screen! Here is an example of a recent talk I gave using this method:

The green screen area can be recognized by the computer to be filled in with a slide presentation; this makes the mathematics look much more crisp and readable than if one were to point a webcam at an actual slide presentation being given via a projector.

Finally, there are also extra hooks installed on either side of the green screen that allow another screen to be hung – a black sheet. This is the backdrop for use of a Lightboard, which my colleague Alexander Hulpke obtained and added to the room with the help of the College of Natural Sciences:

As you can see, the presenter can stand between the Lightboard and the black sheet and write mathematics on the back side of it. Then you can record using the laptop and flip the camera orientation (flipping the camera can also be done for livestreaming on Zoom) and create an excellent presentation with mathematics seeming to float in the air before your eyes:

To use the Lightboard, the room has to be completely dark; it is best to use a room with no windows if a Lightboard is going to be set up. The Lightboard has its own lights that illuminate the speaker’s face, but otherwise if there is light in the room then there will be too much reflection in the glass board.

The darkness does make the video a little choppy, but the mathematics is crisp and clear. You can also see the lavalier mic attached to my collar in the video above, making for high-quality sound recording. This is essential for the Lightboard presentations in particular, because otherwise the sound will be muffled behind the glass board and not be recorded.

There are certainly improvements that can still be made to the room in terms of both video quality and ease of use (a better camera may eliminate the choppiness on the Lightboard video, for instance), and suggestions are welcome! In the meantime, I hope this post inspires other academic departments and instituitions to set up similar technology so that we can continue to share ideas more sustainably in the years to come.

Doing mathematics in a pandemic – Part IV: Talks with OBS

This is the final post in a four-part series on adapting to the pandemic as a mathematician. See Part I – AlCoVEPart II – Collaboration, and Part III – Teaching.

As conferences moved online, a number of different methods of giving a remote talk became commonplace. One was to simply point a webcam at a chalkboard and lecture as usual. Another is to make slides and use the “Share Screen” option on Zoom to show the slides to the audience. Another popular method, which I have used a number of times, is to make partial handwritten “slides” in Notability or GoodNotes on an iPad, with space left for doing examples and computations, and then share the iPad screen over Zoom and walk the audience through.

Today I’ll be explaining how to use Open Broadcasting Software (OBS) to give a talk from home in which your slides show up behind you as if you were standing next to a projector screen, but are nearly as crisp as if you were reading the PDF on your own computer screen. François Bergeron at UQAM first introduced me to this method, and his COVID-19 page features excellent explanations of how he creates his own virtual talks.

First, here is what the output of my first and only attempt at using OBS in a virtual talk looked like:

The above talk was given at the Enumerative Combinatorics session of the virtual Canadian Math Society winter meeting in 2020. Other videos from this session are available here, including another example using OBS by Marni Mishna.

There are two steps to getting this working: (1) setting up your video sources in OBS, and (2) feeding the video output to Zoom.

Step 1: Video sources in OBS

The first step is to install OBS from Once you install and open it, you’ll see a window with a preview of what your video project looks like. At the bottom of the windows are various menus: Scenes, Sources, Audio Mixer, etc.

You’ll only need one Scene, and you’ll add various Sources to put together the Scene, using the + button at the bottom of the Sources box. Here were the Sources I used:

  • Color Source. This allows you to essentially set a “Background color” for your scene. It defaults to a dark grey. Once you add the color source, you can right click it and click “Properties” to change its color.
  • Window Capture. This is the source that you can use to make your slides appear. Create a Window Capture source, open your slides in your pdf viewer in another window, and then go back to OBS and right click on Window Capture to select Properties. There you can tell it to use the pdf viewer as the window that you’re capturing in this source. Finally, go back to the pdf viewer and go into presentation mode, so that you can flip through the presentation with a clicker or keyboard as you would in a real classroom.

    Tip: If you’re on a Mac or another operating system that has multiple desktops, you may need to open your slides on the same desktop window as OBS is opened in for OBS to find the pdf viewer as a source. In particular, you need to feed it to Window Capture before going to full screen or presentation mode; otherwise OBS will not recognize it as a source.
  • Video Capture. This uses your laptop webcam or other webcam to capture your face. You can again right click to get to Properties to change the webcam you’re using if you wish. To filter out everything except your head, you want to use:
    • Chroma Key filter. Right click on the Video Capture source and go to “Filters”. Then click the + under the “Effect filters” box and click “Chroma Key”. You can then click on the Chroma Key filter to choose which color you want to filter out.

      Since I had a green screen (see Part III), I filtered out the color of my green screen behind my head, and voila, the video capture source only showed the outline of my head and nothing else. If you don’t have a green screen, make sure you’re positioned in front of a blank wall and then just choose the color of that wall to filter out.

Finally, make sure you order the above sources in the Sources box so that Video Capture is highest up, then Window Capture is second, then Color Source is third. This way your head appears in front of the slides which appears in front of the background color.

I resized my Window Capture layer by dragging its outline in the OBS preview window so that it sat in the upper left of the screen, leaving room for my head on the right as if I were standing next to the projector screen.

Here is a screen shot of OBS after setting up everything as above (plus an audio input source – see Optional Step 3 below):

Step 2: Connecting OBS output to Zoom

For this step, you’ll need to install the OBS Virtual Camera plugin. It’s easy to install, and once you do, if you restart OBS there will now be a “Start Virtual Camera” option under the Tools menu.

If you click “Start Virtual Camera”, this creates a virtual camera device recognized by Zoom. Now, log into Zoom, start a meeting, and click on the up arrow next to the “Stop Video” (or “Start Video”) button. There should now be an option to select “OBS Virtual Camera” as your video camera in Zoom. Select it, and you should see your OBS creation being streamed over Zoom!

Now comes a tricky technical issue. Suppose you want to use a clicker to flip through your slides in presentation mode. Then you need your laptop monitor to be focused on the displayed slides so that the click registers as a slide advancement. But then that means that you can’t see OBS or Zoom on your laptop screen, so you can’t see yourself as you’re gesturing to things on the virtual screen, and can’t aim appropriately or make sure your head isn’t blocking the words.

To solve this, what I did was to log into Zoom on both my laptop and iPad, and position the iPad in front of my laptop webcam so that the webcam still captured my head, but so that I could see myself on the iPad while my clicker clicked through the slides on my laptop. There are other solutions as well; it should be possible to port the iPad screen itself into Window Capture in OBS as well, though I haven’t personally figured out how to do this. But as long as you have one screen to click through your slides and another to view yourself, you’re good to go.

At this point, you’re nearly ready. Zoom will capture your audio as normal, and displays your new video setup from OBS, so you can give your talk!

Optional Step 3: Audio via OBS

There was one minor issue when I practiced this setup: OBS does some processing which makes the video feed into Zoom lag behind the audio that is also being captured by Zoom. It was only about a half-second lag, and people who I practiced with said it was noticeable but not a major issue.

I did find a way to fix the lag issue, however, and that was to pipe the audio through OBS as well, so that OBS took in both audio and video inputs from me and output both in sync to Zoom. Here are the steps I took to do so:

  1. Create an Audio Source in OBS. Go to Sources again and add an Audio Input Capture source. You can go to its Properties to set it to capture whichever microphone you prefer (in my case, the lapel mic that I described in Part III).
  2. Install a Virtual Audio Cable. This will have to be third party software, as at the moment OBS does not have a virtual audio plugin that resembles its virtual camera feature. I installed VB-Audio Cable, which is a program that can take output audio from one software source (in our case, OBS) and “plug it in” as the input audio to another software (Zoom).
  3. Turn on audio monitoring in OBS. To do so, in the Audio Mixer box at the bottom of your OBS screen, click the Settings wheel next to the volume button on Audio Input Capture, and go to Advanced Audio Properties. Then in the Audio Monitoring column, set both settings (for both Audio Input Capture and Mic/Aux) to “Monitor and Output”.
  4. Launch VB-Audio Cable. It needs to be running for the next steps to work. Besides this step, you don’t need to interact with the VB-Audio Cable program at all.
  5. Set OBS’s audio monitor to VB-Cable. Go to the main OBS menu at the top and click on Preferences. Click on the Audio tab, then scroll down to Advanced. There, set the Monitoring Device to VB-Cable.
  6. In your Zoom meeting, set your microphone to VB-Cable, which should appear as an option for your Zoom microphone now.

Now you’re all set! Time to go give that awesome virtual talk.

Please share your own tips for giving virtual talks in the comments below!

Doing mathematics in a pandemic – Part III: Teaching

This is the third post in a four-part series on adapting to the pandemic as a mathematician. See Part I – AlCoVE, Part II – Collaboration, and Part IV – Talks with OBS.

Of all the things I had to figure out how to adapt to the pandemic reality, I found teaching to be the most challenging by far. So much of the value of teaching comes from the in-person connection between students and teachers, and between peers in the classroom. How can you replicate an entire classroom experience on a 14 inch computer screen? How can you pull off hybrid teaching without diminishing the experience for those students who take the course remotely?

I taught two courses in the fall of 2020 – a small graduate-level class on advanced combinatorics topics, and a larger undergraduate class of 30 students on introductory combinatorics.

Graduate class: outdoor teaching

When, over the summer, the studies came out showing that outdoor transmission of the coronavirus was minimal, I decided to see if I could get outdoor teaching set up for my graduate class for at least the first half of term, with the plan of moving the class online once it got too cold.

There were a lot of considerations to take into account when setting up a good outdoor learning environment. What do you write on? What if it rained? How do you record the lectures outdoors, with good sound quality, to make sure everyone can still participate even if they have to quarantine due to a COVID-19 exposure? How do you make sure the students can hear you over the noise of nearby traffic and birds and other outdoor distractions? How do you ensure student comfort when taking notes, without having traditional desks?

Here were the tools I used to solve – or at least attempt to solve – each of these issues.

  • Rolling whiteboards. The CSU math department ordered lightweight rolling whiteboards with weather-resistant aluminum frames specifically for this purpose. There were at least three instructors who started the semester teaching outdoors, and we made a schedule of who would roll it out and who would roll it back in each day. They worked well outdoors, and as long as you could guarantee you’d be in a shady spot, there was no glare.
  • A good location. Behind the math building on my campus was a shady spot on the grass next to a large parking lot with very little daytime traffic going in and out. A generator nearby provided some ambient white noise that drowned out the traffic from a nearby road. Two trees provided a feel of being somewhat removed from the bustling campus sidewalk on the other side. It wasn’t too much effort to roll a whiteboard there. We really lucked out on that front – it was pretty much ideal.

    Not all campuses may have such a spot, and some universities solved this using outdoor tents.

    Here is a picture of the location and the whiteboard (from a meeting with a grad student, not from class):
  • USB lapel mic for recording. A lapel mic, also known as a lavalier mic – one that clips to your collar – is the best way to pick up only your voice and filter out other noises when recording outdoors. I got this one mainly for its 20-foot cable that allowed me to walk around freely at the board, and there are plenty of options out there like it.

    I plugged the mic into my laptop and did the recording using the macbook webcam and Quicktime. Nothing fancy, but it did the trick for helping students catch up or participate remotely.
  • Personal amplifier for sound projection. It’s hard to project your own voice sufficiently in an outdoor setting, especially with masks. So I got a personal amplifier that would help project my speech to the class. I found that tucking the microphone under my mask and turning the volume low was a good way to get the sound to amplify; it didn’t pick up the sound so well when it was on the other side of the mask.

    If teaching outdoors post-pandemic, I highly recommend it; without a mask it would be even better at getting accurate sound and projecting it to the class.
  • Weatherproof box and laptop stand. What if it started raining, and all my recording equipment and laptop got rained on? And how do you set your laptop or webcam up at the right height to record yourself writing on the whiteboard?

    I solved both of these issues with this large Husky storage box. It allowed me to carry out all my gadgets and whiteboard markers from my office at the start of each class all at once, and then it doubled as a stand to put my laptop on to record my lectures. It’s about the right height – you don’t want something too tall so it doesn’t block the students’ views. And then if it rains, you quickly throw everything back into the storage box.

    Luckily, my class was at 1 pm and located in the Colorado front range. The late summer/early fall weather patterns are very predictable, with thunderstorms and rain usually rolling in from the mountains in late afternoon, around 3 pm or later. So rain wasn’t generally an issue. There was one day that it started drizzling in class, but not enough that it wasn’t still pleasant to be outside or possible to take notes. Luckily everyone was there that day, so I put the recording devices back in the box and finished up the lecture without issue.
  • Lap desks for student comfort. My graduate class was very small, so I got a couple of cheap lap desks for anyone who wanted to use them. They turned out to be perfect; a simple solution worked in this case. Some students brought folding chairs, others opted to sit on the grass. Either way, student comfort was never a complaint.

There were two aspects of outdoor teaching that I hadn’t accounted for in my planning. One was the record-setting wildfires that hit our region of Colorado starting two days before class started. Some days, the air quality was simply too hazardous to spend a long period of time outside. On those days I sent an email in the morning and moved class online.

The other aspect didn’t really have to do with outdoor teaching per se, but was about in-person vs remote. At CSU, some classes were online and others were in person. It meant that some of my students really liked being in person outdoors, since then they could just stay there for their next outdoor math class in the same location. But others had to sprint to campus from their apartment to make it to my class, since they had an online class just before it that they needed to be at home for.

In the end, the students’ scheduling issues lined up in such a way that it made more sense to go fully remote after the first few weeks. But the outdoor teaching, for the short amount of time it happened, went fairly smoothly.

Graduate class: remote teaching

Ah, teaching from home. That peaceful, relaxed setting in which, halfway through the class, you hear your two-year-old twins running down the hall towards your office screaming “DIAPER FACE!!!” at the top of their lungs and cracking up, then throwing a double tantrum when Daddy frantically tries to drag them away from Mommy’s office.

Home distractions aside, in many ways the teaching setup became simpler once we moved the course online. My setup consisted of:

  • A dedicated Zoom classroom. Zoom now requires either a waiting room or a password, and I prefer the password so that students can go into the classroom early and chat before I get there, like in an in-person classroom environment. I put the Zoom ID everywhere on the website, the syllabus, on Canvas, etc. so that students can easily find it each time they try to log in.
  • An iPad and Apple Pencil for writing math real-time during class. See Part II for more details on these.
  • Notability. This is a note-taking app for the iPad that is worth every penny of it’s $10.00 price tag. It has a good note organization system as well as a nice selection of colors, pen sizes, stroke erasing, and continuous scrolling. I logged into Zoom on both my iPad and my laptop, and shared the screen on my iPad so that I could write the lecture on Notability as my “white board”. Then after the class, you can even share the Notability file as a “recording” of the class notes for the students to use – and for you to use in the future.
  • A Canvas course. My university uses Canvas as their online course organization system, and it’s become a good way to organize online classes too. I use the Announcement feature quite a bit, as well as putting all the homeworks and tests as Assignments with appropriately weighted grades so that Canvas would average their grades automatically. It also allowed a designated upload space so that students could upload their homework assignments as pdf files directly to Canvas, where I could use a tool called SpeedGrader to grade on the screen.
  • University storage space for video and note uploads. Videos take up a lot of space, and our course Canvas pages themselves were not large enough to hold a semester’s worth. Luckily, we had plenty of space on our campus OneDrive folders, and so I created a folder there for video uploads, and then linked to them on the Canvas page.
  • A tech support staff person, if possible. As the semester wore on, most instructors who were teaching remotely agreed on one thing: it was extremely time consuming and exhausting, far more than teaching in person.

    Eventually I realized why: it was the high volume of what I call “button-clicking” tasks. You have to find the Zoom lecture video recordings, rename them, and upload them to OneDrive. You have to wait for that upload to finish. You have to link to those videos from Canvas. Then you do the same for the Notability files. There are homeworks to upload and link to. There are homework solutions to grade in an online format that requires logging into Canvas with your university ID, which takes extra two-step verification because you’re working off campus. There are emails to respond to. Just a lot of very draining screen time spent clicking buttons in isolation.

    If the pandemic was going to be a long-term thing, I think it would make sense for universities to invest in a tech support crew, or perhaps hired undergraduates, to take this burden off of the professors. In my case, my tech support person was my husband, Bryan Gillespie. He had made the admirable decision to be the primary parent to our twin two-year-olds during the pandemic, which allowed me to keep up my momentum on the tenure track while we avoided the covid risks that came with childcare. But Bryan also has a Ph.D. in mathematics, so he was more than capable of uploading video files to a website and naming them appropriately.

    And he did so, so that I could play with my kids a little more and he could feel a little more connected to the world. Interesting times indeed.
Above: Bryan Gillespie, my personal tech support staff.

Undergraduate class: Online

My undergraduate class, Introduction to Combinatorics, was one of two sections, the other being taught by Rachel Pries. Both sections were run fully online, and Rachel and I teamed up to make a plan for how we would run the course. It consisted of both synchronous classes and asynchronous components.

The asynchronous components were assigned readings as well as 10-minute videos that we created to go along with each lecture. The idea was that the students would watch the 10 minute video before coming to class, and then in-class time could be devoted to discussions, student presentations, and problems. It worked very well for a first combinatorics class, which doesn’t require a lot of theory but does benefit from a lot of practice.

The videos were a lot of work to make, but I wanted to make a resource that I or other professors could use in the future, so I put some effort into making them high-quality. Here is one of them, which I uploaded to YouTube in order to share here on this post:

As you can see, I created the videos in a way that would most closely resemble an actual lecture format, with me standing in front of the mathematics and speaking to the audience, but without the glare and blurriness of me actually standing at a physical whiteboard and pointing a webcam at it. Here was how I created them:

  1. A green screen. I purchased a green screen for my home office. A green screen allows video software like Zoom to more crisply cut you out and put a chosen background behind you, since it allows the video processor to just filter out everything that is the shade of the green screen. So as long as you aren’t wearing green, it works very well and you don’t get the blurriness of the default Zoom background feature.

    In order to use Zoom with a green screen, open a meeting, and click on the little up arrow next to “Stop Video”. Click on “Choose virtual background”, and then check the little box at the bottom that says “I have a green screen.” Finally, upload your preferred background photo or video with the little + button in the upper right of the Choose Virtual Background window, and click on it to set it as your background.
  2. Math in background. The math you see in my background is a pre-recorded video that I upload into Zoom and set as my Zoom background as above. I recorded it using Notability on the iPad (see above), using the following steps:
    • First I write out partial notes in Notability that don’t have all the computations done, alternating whether the notes are in a column on the left half or the right half of each page, to keep space for my head on the Zoom screen.
    • I then make a screen recording on the iPad by swiping down from the upper right corner to get the iPad recording menu, and pressing the Record circle. It gives three seconds to tap back to Notability before it starts recording.
    • I speak out loud as if I’m giving the lecture to time it, and fill in the examples and computations on Notability with my Apple Pencil as I slowly scroll through the notes. The iPad only records the visual screen, not my sound, but speaking helps me make sure it’s timed well for my recording with my face and audio.
    • I finally swipe down from the upper right again to press stop on the recording, and it generates a video in the Photos app on the iPad. The video needs to be compressed to a smaller size to use as a Zoom background, so I got a free compression app called Compressor and I feed the screen recording to the Compressor app. Finally, I save the output to Dropbox so that I can access it from my computer.
  3. Zoom recording. With the background video ready, it’s time to open Zoom to record the final video. I open Zoom to a new meeting with just myself, and upload the video to the Choose Virtual Background menu as described in Step 1. Then I select my default CSU background to start, press “record” on Zoom, and after saying the intro piece, I click on the video to set it as my virtual background during the recording, and start talking about and pointing to the math as it shows up behind my head.
  4. Uploading. When you close the Zoom meeting, the folder with the recording pops up. At that point I rename the video and upload it to OneDrive.

So yes, it’s a lot of steps, but for only 10-15 minute videos, it wasn’t too bad to do the recording twice to make it look really nice and professional. Here are a few other tricks I discovered that improved the setup over time:

  • Proper height stand for laptop or webcam. I noticed I looked more natural and teacher-like if I was standing while recording rather than sitting. So I put one of the Husky storage boxes that I mentioned above (see outdoor teaching) on top of my desk and sat my laptop on top of a little box on top of that. That put the webcam at the right height, with my green screen behind me, so that my head was perfectly framed next to the notes as I stood and gestured to the mathematics.
  • Lapel mic. As I mentioned above, my house was not exactly quiet at all times with twin 2-year-olds running around. So I plugged in the lapel mic that I had gotten for outdoor recording, and went to audio settings in Zoom to choose that as my microphone. This way it only picked up my voice and not the noise around me.
  • Lighting. There is one window in my office and it’s good for illuminating the left hand side of my face during the day. I needed a light on my right to illuminate the rest, and cancel the shadow on my green screen (which can make the background less crisp). So I set up an LED lamp like this one on the right hand side of my laptop. Not the most professional lighting setup, but it worked.
  • Makeup. Bright lights and a white background can really make for a washed-out appearance. I rarely wear makeup, but I noticed that some red lipstick and blush and eyeshadow really made a difference in making me look human even with the lighting.
  • Practice. The one difficult thing about watching yourself on the screen to try to gesture to the math is that the video is flipped, opposite of what you would see in a mirror. So if you move your hand to the right, it goes left on the screen. There’s no way around this if you want to see the mathematics in the correct orientation as you’re recording, so you just have to practice the weatherman gesture technique. It was tough for the first 2 or 3 videos, but I got used to it.

    The other thing that required practice was speaking in time with the background recording as it did math and scrolled on its own behind me. Like the gesturing, you get used to it. You learn little phrases you can say to delay if it’s not scrolling as fast as you thought it would, and you learn how to wrap up a sentence quickly and segue into the next topic if it scrolls before you were expecting.

It was a lot of work, but I had fun doing it. Here’s a picture of the home green screen setup I described above:

If you have tips of your own on remote, outdoor, hybrid, or asynchronous teaching, please share them in the comments below!

Doing mathematics in a pandemic – Part II: Collaboration

This is the second post in a four-part series on adapting to the pandemic as a mathematician. See Part I – AlCoVE, Part III – Teaching, and Part IV – Talks with OBS

The first aspect of academia to be affected by the pandemic was conferences; the second was in-person collaborative projects. That research collaborator you invited to speak in your seminar can no longer visit, and the potential for a two-day intense collaboration to kick off a new project diminishes drastically. You can no longer meet in person with your graduate students, at least not as easily. Little things like deciding when you’re going to hold the fall Putnam club meetings suddenly turn from a quick conversation in the math department hallway into a five-email exchange.

So I, like all other mathematicians, found ways to adapt. I’ll share a few things that really worked, a few things that really didn’t, and a few extra tools that made things nicer. If you have tips of your own, please share them in the comments below!

Things that really worked

  • iPad with an Apple Pencil. Tablets have turned out to be an essential tool for remote research collaboration. The Apple Pencil stylus mimics writing on paper very well, and it’s great for writing shared scratchwork real-time, like you would when working alongside someone on a whiteboard or at a desk. I immediately purchased an iPad at the start of the pandemic (thanks CSU!), and I opted for the large-screen 12.9 inch size so that I had plenty of space to write mathematics and share it virtually.
  • Zoom. This almost goes without saying at this point, but it’s the best videoconferencing software I’ve tried so far. Its video and audio quality and the lack of lag really are impressive, and important for the natural flow of conversation.
  • The Zoom whiteboard. If you click “Share Screen” in a Zoom meeting, the first option is to share a whiteboard that other collaborators can write on as well. It’s a little finicky, but Zoom has been improving it, and here are some tips:
    • Saving. Always remember to click “Save” on the whiteboard annotation menu before ending the Zoom meeting. I wish Zoom did this automatically, but it unfortunately does not. It saves them as .png files, one for each page, in your Zoom folder, which should pop up after you leave the meeting.
    • Names popping up. If you see the annoying feature of someone’s Zoom name popping up by where they’re writing, click on the three dots dropdown on the annotation menu and click “Hide names of annotators”.
    • New page. You can click in the lower right corner of the whiteboard screen to make a new page.
    • Unsharing and resharing. Zoom used to have the extremely annoying feature of forgetting what was on the whiteboards when you stop sharing, so that if you share something else and then go back to sharing the whiteboard, it’s blank again. They have recently fixed this, and now it remembers the whiteboard for the duration of a Zoom meeting even after it is unshared and shared again!
  • Overleaf. Overleaf is a great tool for writing collaboratively in LaTeX, complete with an online editor with a preview window that is well-synced with the LaTeX code.
    Its file sharing system uses git for version control, so mathematicians who prefer working locally to working on the cloud can clone the git repository to a local folder. I recommend using the Overleaf project to share files, as opposed to say Dropbox (see below), so that everything is in one place.

Things that didn’t really work

  • A physical whiteboard to point your webcam at. This was the first thing I tried in order to collaborate over Zoom, before obtaining an iPad. It’s hard to set up in a way that it’s easily visible with no glare, and you end up getting back pain from hunching over so your face is in the screen sometimes as well as the whiteboard. I believe it can be done correctly if you have the right webcam and office setup though.
  • Writing something down on a piece of paper and holding it shakily up to your webcam. I admit, I’ve done it. We’ve all done it. But no. Just no.
  • Email. An email is great for setting up a Zoom meeting. Not so great for doing collaborative mathematics. It’s slow and cumbersome and a Zoom meeting is almost always better. The exception was when an email served as a way to share a quick idea before you forgot it, so that you can bring it up again at the next meeting.
  • Dropbox. In my experience, any time a Dropbox folder is set up and shared, it’s later forgotten about and then everyone has to ask each other what the Dropbox folder was called. Someone would make an extra Dropbox folder containing a single file consisting of a picture of a diagram they drew, and then after viewing it everyone forgets where the picture went. It’s also not very good for simultaneous editing of papers, in terms of version control. (See Overleaf above.)
  • Google Drive. See Dropbox.
  • Any video chat client that is not Zoom. I have heard some people saying they like Microsoft Teams, but I think it’s safe to say that avoiding Hangouts or Facetime or Facebook video chat is a good idea. The lag and connection issues alone make these alternatives very inconvenient, and they don’t have sharing or whiteboard capabilities.

Little things that are worth it

  • Paper feel screen protector for iPad. I didn’t even know this existed until the holiday season, when my husband surprised me with this. It is an iPad screen protector that makes writing with the Apple Pencil feel actually like writing on paper. After installing it, I’ve found that writing on the tablet actually is preferable to me to writing on paper, and this is coming from someone who loves paper and doubted tablets would ever truly replace them. It’s a little thing, but it made a huge difference to me.
  • Zoom chat or Google Hangouts (outside of meetings). A chat client to send quick ideas and messages, start impromptu Zoom meetings, and just say hi once in a while, is in my experience very useful, and can help avoid some of the email overload of the pandemic era.
  • Zulip, Slack, and Discord. For larger groups, an organized chat client like Zulip, Slack, or Discord can be very helpful. Threads can be sorted by topic and it is easier to follow what is going on. Zulip is my personal favorite, but I’ve had good experiences with all three.
  • Google docs/sheets. Sometimes you just need something a little simpler than Overleaf to manage tasks or jot down ideas. Google docs has pulled through for me in such situations.
  • Being kind, being silly, and having fun. There’s a real lack of human connection these days, and it’s always good to check in with collaborators to see how they’re doing, put up a funny Zoom background, or watch someone’s cat walk across their keyboard. Little things like this, for me, help to keep my job fun and worthwhile.

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!
For the other posts in this series, see Part II – Collaboration, Part III – Teaching, and Part IV – Talks with OBS.

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 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.

Addendum: FPSAC 2020

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 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.

On Raising Your Hand

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!


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:


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 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:


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.

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.