Another First Day

The view from the Mathematics and Computer Science Department at Colorado College.

Monday was the first day of classes for me, and also the first day of a new job.  This fall I started as an assistant professor at Colorado College, a liberal arts college of about 2000 students in Colorado Springs, Colorado.  Colorado Springs is about 60 miles south of Denver, at the base of the truly awesome 14,000-foot Pike’s Peak.  Colorado College is great: beautiful setting and campus, my colleagues are interested in their work, but also laid back and simpatico, and the students are creative, curious, and motivated.  It is like many other schools in those ways, though of course CC has its own distinctive personality that I particularly like.  However, one truly unusual thing about Colorado College: the school year here is broken not into semesters or quarters, but into 8 blocks of 3.5 weeks each.  Students take one class per block, with all of their attention on that one course (at least in theory).  Professors teach only one class at a time, too.  But the teaching is intense—most classes meet for around 3 hours every weekday morning, with office hours, problem sessions, and/or labs in the afternoon.  It’s easy to spend 5-6 hours a day in intensive student contact, plus there’s always preparation for the next day and grading.

The intensity is intense.  Did I mention it was intense?  That word comes up constantly when describing the “block plan”.  No question, this schedule can be tiring.  Luckily, each block is followed by a block break, two and a half days plus a normal weekend, when students are completely free to go camping, read novels, play video games, binge watch TV, or whatever.  Of course, if you’re a professor, you might spend a couple of those days grading and a couple getting ready for your next course, which doesn’t leave much of a break when you’re teaching successive blocks.  Again luckily, professors don’t teach during every block—different departments on campus have different teaching loads, but professors generally teach from 4 to 6 out of 8 blocks.  In non-teaching blocks, professors keep up with service, but have a remarkable amount of freedom to travel for research, go to conferences, or do field work.  So professors also have some real breaks built in.

When I tell people about this system, they often tell me that they think it is crazy.  I assure you, I have thought it was crazy at moments as well. However, I knew what I was getting into when I came, since I spent two years as a visiting assistant professor at Colorado College.  That was a really positive experience, and I am very happy to be back here this fall.  Yes, there are aspects of the block plan that are difficult, for both students and teachers.  I think exhaustion is the biggest issue, followed by time management. Many people ask me whether knowledge retention is a problem—how do students remember anything when they completely switch focus every 4 weeks?  My experience has been that this is no more of a problem than on the semester system; Calc 2 students everywhere will claim that they have never heard of the chain rule.

The fact is that there are also many aspects of the block plan that are deeply rewarding and advantageous to teaching and learning. For example, the fact that students have only one class means that very few students totally blow off the class they are in. It happens, but very rarely.  Taking students on field trips and getting out of the classroom is much easier.  Something about the bigger chunks of time spent together can make it easier to build strong relationships with students. If a student is really passionate about math, they can take several courses in succession, moving through the entire calculus sequence, linear algebra, and number theory in one year if they like.  Though students have other appointments, it’s generally ok if class goes over by a few minutes sometimes.  Seeing the students every day means I get far fewer emails.  Non-teaching blocks make extended research travel possible during the semester, without having to cover or cancel classes.  For me, all these positives more than make up for the difficulties.


Getting off campus with classes is a great perk of this job. Colorado College has a satellite campus near Crestone, Colorado. I took this picture there a couple years ago when I took my students to “Number Theory Camp”.

So, to return to the moment: I have been teaching for 5 days and I am more than a quarter of the way through a Linear Algebra course.  I have 23 students, I know all of their names, and class has been exceptionally fun so far. We have covered solving linear systems, row reduction, geometric interpretations of solution sets, spans, linear independence, and linear transformations, and we have already done one Mathematica lab.  The first test is Thursday of next week, and will also cover matrix operations, invertibility, and determinants.  I have done almost nothing for the last week and half but prepare, teach, meet with students, and grade. I am soooo excited that it is Saturday.  But, of course, I woke up at 7 AM thinking about how to explain linear transformations differently.  I always obsess a bit when I am teaching, and it is one of my great life projects to learn to let go of whatever just happened in the classroom and live in the present.  Maybe that’s one reason the block plan works for me, though—it leaves me less time to obsess, since I’m spending so much of my time during the block actually teaching.


Really missing my friend, collaborator, and Villanova office neighbor Katie Haymaker right now.

One week in, things are mostly excellent here.  I love being back in Colorado. I have to say that I also miss my friends, colleagues, and students from Villanova.  I miss being part of the Graterford program, my students at Community Learning Center, the energy and food of Philadelphia, and the dense mathematical network of the east coast.  I LOVED living in Philly and working at Villanova. Starting over is exciting but moving is horrible.  It takes a long time to make connections in a community, to have real friends and feel at home, and, having finally made those connections, it is really sad and hard to leave those people and start all over again in a new place.  So why did I leave, even for a great job like this?  A big reason was to be close to my partner.  I had been in a long-distance relationship for the last three years, which was simply not sustainable for me.  I am also much closer to my family here.  The importance of these factors overrode the sadness of leaving for me.

This constant starting over is one of the worst parts about early career life in an academic job.  It is poignant to move again, when many of my friends from grad school and all my past jobs are buying houses or having babies or forming bands.  And here I am again, giving away my plants, shoving half my belongings into storage, moving into a new office, and going to new faculty orientation to meet the rare other people my age that are looking for new friends.  However, there are so many great people in exactly this position, and at least this time I have already started over once in this place—I mean, I have a dentist at least!  Friends, community, and a sense of belonging can’t be too far behind.

Starting a new job? Interested in the block plan or teaching in some other unusual way?  Please share in the comments.

Posted in moving, social aspects of math life, teaching | Tagged , , | 1 Comment

Category Theory and Context: An Interview with Emily Riehl

Emily Riehl

Emily Riehl is an incredibly accomplished early-career mathematician, working at the interface of category theory and homotopy theory. She is also a stunning number of other things, including a creative interdisciplinary scholar, working musician, and high-level athlete. A brief career outline: she did her undergraduate work at Harvard University, graduate work at Cambridge and the University of Chicago, was an NSF and Benjamin Peirce Postdoctoral Fellow at Harvard from 2011-2015, and is currently an Assistant Professor at Johns Hopkins University. Emily has been awarded an NSF standard grant and a CAREER award to support her work. She is the author of 21 published research articles, two books (Categorical Homotopy Theory and Category Theory in Context), and many other expository works. All this, and she also performs as a rock/alternative bass player and plays on the US women’s national Australian Rules football team. I recently learned about Emily’s work and profile while looking for women mathematicians to interview for the Association for Women in Mathematics newsletter.  I thought maybe PhD+epsilon readers would also be interested to hear about an early-career mathematician doing some really cool things.  The following interview is a compilation of email and Skype conversations from August 2017, while Emily was in Australia to compete in the AFL International Cup.

A longer version of this interview will appear in an upcoming issue of the Association for Women in Mathematics newsletter.

Question: How and why did you get into category theory? Is there a basic result that you can share that gives the flavor of what you love about it?

Emily Riehl: For graduate school, I deferred from the University of Chicago for a year to go to Cambridge and do what they call a Part III. One of the courses they offered at Cambridge was in category theory, and I liked it instantly; I fell in love. I feel like it chose me as much as I chose it. And it was for the reason that I think that everyone chooses their field, ultimately: the proofs felt like the right way of thinking about mathematics. I felt right away that this is the sort of argument that I wanted to delve into.

Category theory can sound intimidating because it’s highly abstract, but it’s actually not that hard. Several of the most important definitions are quite elementary, and you can start stating and proving the theorems pretty quickly. Indeed, there’s a common belief in category theory that once you understand the statement of the theorem, you can probably supply the proof yourself. Identifying the correct definitions is really the harder thing. The only reason that you typically don’t learn category theory until graduate school is that it requires a rather high degree of mathematical sophistication to appreciate what it’s for.

One of my favorite theorems in category theory is that right adjoints preserve limits—or, since you always get a dual theorem in category theory by simply “turning all the arrows around”—that left adjoints preserve colimits. This result specializes to explain why tensor products distribute over direct sums, why inverse images preserve intersections and unions while direct images only preserve unions, why quotients of topological spaces are formed by first identifying the appropriate points and then topologizing this quotient set. It’s not so much that I appreciate having one proof instead of having to repeat the argument in each context but I feel that the category theoretic proof—which uses the fact that limits are characterized by a “mapping in” universal property, while colimits are characterized by a “mapping out” universal property—is the right one.

Q: You are early in your career, but you have written many, many papers, two books, and a lot of shorter expository work (like posts on the n-category cafe).  How do you do so much stuff?  Do you have any insights into how/why you are so productive?

ER: I read Hardy’s A Mathematician’s Apology in high school and my main takeaway was from the forward written by C P Snow, who described Hardy’s typical day: he devoted four hours in the morning, from 8-12, doing math, and then spent the afternoon watching cricket. It struck me as a particularly aspirational life style and so I’ve always focused more on working well than on working long hours. My main time management strategy is to start work on the thing that is due the soonest last, when I’ll be the most focused. So, for example, if I have a referee report due in three months, I wait until almost three months have passed, and then start to read the paper. I also do the preparation for my teaching in the hour or hour and a half before class, in what often feels like a race to figure out how to prove all the theorems before I rush across campus. Occasionally this gets me in to trouble, for instance when I was trying set up a transfinite induction over the reals and couldn’t understand why the intermediate stages were all “countable” (aside: I’m now firmly in the camp that believes that the axiom of choice is clearly true, while the well-ordering principle is clearly false). But this approach is very effective at reserving time for research and other long-term projects.

Q: What do you think are the best/worst parts of a life in math overall?

ER: The worst thing is how intellectually isolated we all are, how few people there are with whom we can share the insights that we find the most exciting, even among other mathematicians. For me personally I feel very frustrated that there is this huge part of my emotional life that most of the people whom I care about have no access to.

My favorite part of my job has always been giving talks. Research talks are my favorite, for the reasons alluded to above, but I also get some of that same thrill from giving colloquia or even from teaching. Even in high school, I enjoyed the performative aspects of lecturing. When I ran for student body president, my only real interest in the job was to give the campaign speech in front of the entire school.

Q: You begin your book Categorical Homotopy Theory with a quote from ‘On proof and progress in Mathematics,’ by William Thurston: “…what we are doing is finding ways for people to understand and think about mathematics.” How has Thurston’s perspective on mathematics as a community endeavor, with human understanding at its core, influenced your mathematical life?

ER: I’ve wondered at various points whether I should be concerned about the amount of time I end up devoting to expository projects, such as the books, because it does certainly eat into research time. This is one of many instances where I’ve found Thurston’s essay, which I’ve re-read a few times now, to be helpful for keeping these kinds of projects in perspective. The passage you quote above is his definition of mathematical progress, which he sees as much broader than simply proving theorems. I happen to particularly enjoy mathematical exposition, so I think it makes sense—or as the economists would say, is a comparative advantage—for me to play that role in the broader community.

I read from a different section of this essay—on the difficulties of mathematical communication—at the introductory meeting for an AMS sponsored Mathematics Research Community workshop in Homotopy Type Theory that I co-organized this past June as a way of framing our goals for the week, which were largely to provide an opportunity for people who are not currently a part of that community (e.g., because they’re doing their PhD at a place that doesn’t have a faculty member working in that area) to find their way in.

Q: What is next for you, in math and life?

ER: One of my favorite things about academia is that the job changes all the time, or at least it can, if you want it to. Right now I’m focused on growing the category theory group at Johns Hopkins and a few long-term research projects that I’d love to get through before an MSRI semester on Higher Categories and Categorification that will take place in 2020. In a decade’s time, I hope I’m working on projects that I can’t even imagine now and have found a way to be a part of larger mathematical and public conversations.

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“Bridges” from, of course. Mouseover text: “And is says a lot about you that when your friends jump off a bridge en masse, your first thought is apparently ‘my friends are all foolish and I won’t be like them,’ and not ‘are my friends okay?’.” If all your friends start playing bridge, what then?

One of my friends recently suggested that I write a blog about how to deal with the stress of going to conferences when you are an introvert. This is a great topic, and I am not the most outgoing person in the world, but I find the social side of conferences stressful for a slightly different reason—not so much because there are too many people, but because there is only one obvious topic of conversation (math!), and it is a topic fraught with peril. It is great to talk about math and ask and answer questions, but talking only about math can be very discouraging—there is so much struggle inherent in internalizing very abstract objects and difficult mathematical concepts, and people are specialists in such small areas, that math talk can be isolating and high-pressure. It’s enough to make me hide out in my hotel room all day, like a true introvert. Full disclosure: that is what I’m doing right this minute.

So here is my conference social stress question: how do you comfortably hang out with mathematicians that you like but don’t know well, without talking about math? I would like to get to know math people beyond their math, because that is the usual human way to know people, and because it makes talking about math more fun. So it would be great to find a low-stress way to get to know people, that doesn’t involve math, when in fact math may be the only thing we know that we have in common.

One answer: drinking beer. Unfortunately there are a few flaws to this approach: I can’t keep up with this as I get further from graduate school; there are many people who don’t drink, and this leaves them out; plus sometimes it stresses me out even more to have multiple drinks with professional acquaintances I don’t know very well. What if I say something weird?

Another solution: games. Yes, this is fairly obvious—where would math graduate students be without board game nights? But somehow I forgot about the power of games to ease math hangouts until I spent some time last summer at a math institute that featured a daily hour of tea, bad cookies, and games. Unless some dire circumstance prevented it, pretty much everybody came to the common room every day at 2 pm and spent an hour or half playing/watching other people play chess, bridge, go, backgammon, Civilization, Innovation, Settlers of Catan, only-down-clues group crossword, the to-me-totally-mystifying cryptic crossword, or several other games that I never learned the names of. Game tea-time was fun, but it was also a bit magical, math-wise—people talked to each other over games. Suddenly they were more comfortable with each other, and were more comfortable talking about math. I wished that I could carry that atmosphere with me to research conferences, so I started looking around for people to play games with. Which brings me to…

My current favorite math hangout game: bridge. Again, bridge is nothing new, except to me. Apparently almost everyone used to know how to play bridge. However, since it’s not as common with early-career people, one motivation for this blog is to advertise bridge for those who do not already play. Because bridge is kind of complicated and intimidating, but ultimately so fun and worth the effort of learning (ooh, remind you of anything? math, anyone?). For those who have never played, bridge is a card game for four people, in two sets of partners. Each hand involves a round of bidding and then a round of play, which takes about 5-10 minutes total. The bidding round involves a system of encoded communication with your partner across the table. The playing round is about taking tricks. Simple right? Well, kind of. It can also be pretty hard, and as rich and complicated as you can stand.

Okay, so why is bridge great, for math conferences and otherwise? Bridge is a group game, but for a small group, so you get to engage with everybody. It is substantial enough that you can play your whole life and still keep improving and enjoying it. It involves chance and skill, so it is not the perfect-knowledge, bare-intellect grudge-match of chess or go, or the total blank probability fest of bingo or dice. If you don’t care to keep score (and you don’t really have to), you can play it in 5-minute chunks, so there’s no big commitment. A large number of people all over the world already know how to play. It can involve a great deal of fairly sophisticated reasoning, and lots of math people enjoy that, so it’s even easier to find math people to play. And, conferences aside, it’s especially likely that you can get a table going among senior people in a math department, so bridge is a nice way to spend some lunch hours bonding with your senior colleagues.

This blog was directly inspired by the phenomenal time I had playing “non-serious” bridge at a recent conference. At this conference, I found another beginning player and a couple of more experienced players who agreed to play very non-seriously. The only deck of cards we could find was kind of a disaster—anyone who wanted to identify the jack of spades (which was actually a modified joker) from the back could easily have picked it out at three paces. But we had a great time, and two hours passed like absolutely nothing. The next night we played again, though we had to take turns because there were several more people who wanted to play (and alas, still only one deck of cards). Playing was fun, and I felt like I got to talk to some people in a different way than I generally would have at a conference. It was awesome. Hooked, my fellow beginner and I decided to get a math group together for online Skype-bridge. We Skyped and used the website Bridge Base Online and some pretty basic bidding conventions. We made a lot of mistakes, but again, it was totally fun. I think this is the clearest way in which playing bridge is like talking math—you have to be willing to make mistakes in public, but the payoff can be great when you get going. I can’t wait for the next one.

I think my only caveat here is that, as in a mathematical collaboration, matching expectations is key—I am still a beginner, and a good, serious bridge player who unwittingly got stuck with me as a partner would hate it. However, while there are many competitive, skilled bridge players in math, it seems that even really good bridge players can have fun playing with beginners, as long as they know what they are getting into.  Playing online, though, people can be very unkind.  I’m talking to you, guy who all caps yell-chatted at me for playing too slow.

So, the moral of the story is that I think everyone reading this should learn to play some very basic bridge. When we meet at a conference, we should have a great time playing and getting to know each other. We can then ask each other math questions, and the math world will be that much better connected, and happier. Problem solved.

Okay, the real moral of the story is that I learned again that doing non-mathematical activities at conferences can be surprisingly and wonderfully worthwhile. I’m selfishly advocating that everyone learn bridge, but I also had a great time trying bouldering and going hiking with people at the same conference. When we are not doing math, these others are actually just people. Somehow, building awareness of this fact makes it infinitely easier to talk to them about math. With that in mind, maybe it’s time for me to head to the conference reception.

Best conference hangout strategies? Games you like? Bridge tips? Please share in the comments.

How can I leave my hotel room when I am the guest of the day? My prize was this certificate, displayed at the front desk all day, this tomato soup, and a $7 credit at the hotel shop. I must say that I was really quite thrilled by all this.

Posted in conferences, social aspects of math life | Tagged , , , | 10 Comments