New Installment of Math Book Club.  Also, Talking Math Life at BIRS with Richard Guy

I started this blog right after a hike around Lake Louise, in Banff National Park, AB Canada. It was so beautiful that I did not even think to break out my book.

GA Math Book Club Continues

My spring and summer reading has been dominated by Swedish noir and giant historical novels, most recently the incredible A Place of Greater Safety by Hilary Mantel. The French Revolution—Aaaaaaaaaaah! So brutal!  However, I needed enough breaks from the guillotine to make it through two very worthwhile mathy volumes in the last few months: Weapons of Math Destruction, by Cathy O’Neil, and Genius at Play: The Curious Mind of John Horton Conway, by Siobhan Roberts.  Neither of these books were what I expected, but I was pleased with both in their own ways.

Weapons of Math Destruction came out last year, and has been on my reading list since Evelyn Lamb reviewed it in the Scientific American blog. It hits some of my usually disconnected strong interests: interesting and unusual career paths in mathematics, the intersections of mathematics and social justice, and the ways in which acting with good or neutral intentions can lead to negative unforeseen and sometimes devastating consequences (I know, this is a weird interest, but where would all those depressing novels and plays be without this principle?). Cathy O’Neil has followed an unusual and interesting path, from academic life to Wall Street, data mining, and now her own company auditing algorithms. I enjoy reading O’Neil’s mathbabe blogWeapons of Math Destruction was an important book for me to read, because my usual vision of mathematics and social justice involves the ways that mathematical tools can be used to fight for fairness or justice (for example, to identify gerrymandered congressional districts).  However, O’Neil describes how the algorithms we design for efficiency and optimization, and even fairness, can treat people very unfairly and lead to systemic injustice.  She does this well, though without any real technical discussion. This book is truly accessible to a general audience. This spring, I gave it to some of my students who were taking summer jobs in finance or considering careers in industry.  This was not at all to discourage them from working in these areas, but to provide some context and some case studies. I want my students to love their careers, and I know that many of them have a strong sense of responsibility to the larger world.  Weapons of Math Destruction is, among other things, a manual for those who feel this sense of responsibility, of things we should not overlook or sweep aside as we try to build the better, faster, more efficient algorithm for everything.

Genius at Play also interests me as the story of an unusual mathematical life. John Horton Conway has, in some ways, followed a very narrow path in math: from early talent to Cambridge student, then Cambridge professor, and then on to Princeton.  However, his remarkable approach to mathematics and life, as well as his totally wacky charisma—captured in his own words whenever possible—make this a very different mathematical biography.  Siobhan Roberts does us the service of allowing us to meet Conway through their relationship as biographer (or “amanuensis”, as Conway sometimes refers to her) and subject.  I found the conversational and non-linear narrative style of Genius at Play uncomfortable at points, but I really liked it in the end. One reason I read this book was to get a bit of context on the history of cellular automata; I came away with much more than I bargained for.  I also learned that, if I ever get the chance to meet Conway, I should probably not ask him about the Game of Life.  Roberts walks a careful line to respect Conway’s privacy, but hints at some of his personal adventures.  This has the possibly unintentional consequence of making the reader, used to tell-all type biographies, even more curious about what is left out. On the other hand, a significant amount of math is kept in the book, and Roberts/Conway do a remarkably deft job of explaining big picture topics and even elucidating full proofs, while never seeming to oversimplify or condescend to the reader. This book gave me a new sense of the explosion of mathematical creativity brought on by the rise of computers, and the sense that I had experienced the extraordinary personality of John Conway first hand.


I hope that I can read a book about Richard Guy someday.

This week I am blogging about books from a conference in Diophantine Approximation and Algebraic Curves at the Banff international Research Station (BIRS).  This has been a great conference in several ways—seeing some of awesome math friends, meeting new people, very good talks, the chance to play exceedingly non-serious but very fun bridge.  I had a really valuable math discussion in which someone explained a difficult idea to me in an intuitive way (thanks, Benjamin Maschke!).  All this, plus BIRS is in a stunning natural setting in Banff National Park (in the Canadian Rockies). Finally, the conference was great because I got a chance to talk with Richard Guy.

Our group photo for the conference. Credit: Banff International Research Station.

Number theorist Richard Guy really deserves his own book.  He did just get his own 100th birthday conference, but that’s not going far enough.  Richard Guy collaborated extensively with Conway, and was mentioned many times in the GaP; however, there is so much more to his life and career than this collaboration. A few highlights: Guy was born in 1916 in England.  He excelled in mathematics, obtaining BA and MA degrees at Cambridge, then got a teaching certification from the University of Birmingham and became a mathematics teacher. He worked as a teacher in England for several years, though he was sent to Iceland and then Bermuda with the meteorological branch of the Royal Air Force for a period during World War II.  In the 1950s, Guy moved to Singapore and then to India, teaching at the University of Malaya and the Indian Institute of Technology. In 1960, Guy met Paul Erdös in Singapore.  Erdös encouraged Guy to pursue his interests in mathematics more seriously, and they began a mathematical collaboration that led to four joint papers.  Guy and his family moved to Calgary, Alberta, in 1965, where he joined the mathematics department at the University of Calgary and is still an active emeritus Professor.  He has written many books, including the classic Unsolved Problems in Number Theory.  He is also a very accomplished mountaineer, and… well, I could go on for a long time. I am a great admirer, of both his mathematics and adventurous spirit.  Sometimes it seemed that Richard Guy was essentially the star of the conference—the lecture room was full for his talk on the final afternoon, and all week people shared stories about their experiences with him and spoke with reverence about how he still proves theorems and walks up tall mountains.  Just before his talk, Jennifer Park said, “I hope I still love math as much as he does when I’m 100.” Agreed.

Thursday afternoon I found him in the lounge chatting with Andrew Bremner about Wimbledon, so I took this opportunity to introduce myself.  I asked Richard about mathematical life and I really loved his response.

Me: “What would you say to early career mathematicians about how to have a great life in math?”

Richard Guy: “Math is fun, and it is difficult, but you have to find your own level.  When you’re a small child, you learn to count and add and subtract and such.  There are lots of whiz kids at arithmetic, who then just kind of fold up when algebra, with all the x’s and such, comes along.  And it happens again with trigonometry, calculus and so on.  When you hit more and more abstraction as you go along, it’s easy to feel you’ve got beyond your depth.  But you can find your own level of abstraction and work there, because there are plenty of things to work on at each level.  The thing is not to get depressed when you don’t understand—and I mean sometimes I hardly understand—what people at conferences are talking about.”

Richard Guy, me, and Andrew Bremner. They were talking math and tennis when I dropped in on them.


Thoughts on the books?  Hopes for your relationship with math at 100?  Please share in the comments.

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Ximera Updates

Last year I wrote about a cool project I’d joined called Ximera. Bart Snapp and Jim Fowler at OSU have been developing a great set of tools to turn LaTeX documents into interactive websites for students. Within one of these sites, you can ask questions and get real-time answer checking, plot with desmos, and embed videos. OSU uses Ximera for their calculus courses, where you can get a good look at different types of questions that one can ask with this platform. With a small change in the header, these websites can be converted to handouts or even a book.

Example Ximera activity

When they announced another Ximera workshop, their fourth, I jumped at the chance with one goal in mind: writing documentation.

A few of you had reached out to me last year about how to get Ximera up and running locally, and I wasn’t much help. It takes some work to make all the moving parts work together correctly on any individual machine, and that is not my area of expertise. But we have a way to circumvent all of this now using CoCalc (formerly Sage Math Cloud).

You’ll need a CoCalc account with internet access (maybe only to get it started?), which costs $7/month. But if you can swing that, our getting started documents should let you edit your first assignments and set up your first course. And if something doesn’t work properly or if you spot a typo, let me know so I can make edits. Bart and Jim streamlined the workflow a bit since last year, but you will still need to use a terminal within CoCalc, and you’ll need to stay on the right side of the git gods, which can be frustrating. But once you get the hang of it, it’s easy to produce beautiful custom assignments that would take ages in an online homework site or a learning management system.

We also included some documentation on how to install Ximera locally but they might not work quite as well as I’d like. The Ubuntu directions are fairly solid (I think), but you’ll probably hit a snag on the Mac instructions when you get to the gpg key. Those problems are not unresolvable, but they’re technical and temperamental and beyond what was possible to write up nicely. PC instructions don’t exist at all, though if anyone would like to try to install and explain what you did, I’m sure we’d all be overjoyed to add them.

The dev team have worked out a couple more technical kinks since last time. They treat answer checking on things like factoring polynomials in a much more clever way. They’ve implemented non-numerical or -functional answers, so it’s easier to make things like fill-in-the-blank questions now. And they’re getting closer to a functioning free response question environment (one already exists, but the answers don’t really get stored anywhere that’s easily accessible right now).

There’s more documentation yet to write. Another workshop participant is working on describing all the environments and how they work, so once that’s ready I’ll pass it along. And we don’t have anything written on the instructor tools or how to sync to Canvas as far as I know.

The authors are looking toward an official release sometime next year, and I’ll update you when I know more. I’m excited to be a part of this project and I plan to use it a bit in my calc 2 class in the fall. When that’s ready I’m sure I’ll post about it so you can see how it comes together.

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Stuck in the South of France

    The Calanques, where I hiked and swam every day during the conference.

Valentijn Karemaker and Marius Vuille talking math outside the lecture hall at CIRM.

I love the variety of beautiful places (physical and intellectual) that math takes me.  This week math has taken me to the south of France.  What a wonderful place to do math, or really anything.  Not least because France has recently taken the extraordinarily cheering step of electing a Fields Medalist to Parliament.  I was at the Centre International de Recontres Mathematiques (CIRM), a math center near Marseille.  The conference: Arithmetic, Geometry, Cryptography, and Coding Theory (AGC^2T), a biannual conference which began in and has been held at CIRM since 1987. I hate to go overboard, but it might have been the nicest conference ever. It wasn’t just the daily hike to the nearby Calanques to swim in the Mediterranean, though that was incredible. The talks were great, which I say with full information because I went to all 35 of them. Even more, though, I just enjoyed talking with these very interesting people. The conference organizers, and really all the participants, deserve a great deal of credit for creating an exceptionally relaxed and positive atmosphere.

In graduate school I spent a lot of time reading Algebraic Curves over a Finite Field, by James Hirschfeld, Gabor Korchmaros, and Fernando Torres.  James Hirschfeld was a member of the scientific committee for the conference, so I got to meet him for the first time this week. We started chatting at a coffee break, and I didn’t realize at first that I had spent so much time in the company of his work. He was very personable, and I eventually told him that I had in fact purchased three copies of this book over the years (losing one in a move across the country and another in some unknown way). I asked how he came to start writing books, and he said that in fact he started his first book because his research wasn’t going well. He thought that writing a survey of the field would help him regroup. I’ll quote here something he had earlier written about this: “In 1972, when my research was going poorly, I decided to write a one-volume survey of the field of mathematics that I had been working in, that is, the combinatorics of finite projective spaces. … As the book was supposed to be a survey of the field, I decided to begin the book by compiling as complete a bibliography as possible.” He said that in compiling the bibliography, he read many papers and had a lot of ideas that helped him move forward with his research.

I found this narrative, well, not exactly surprising, because it makes sense—writing a deep survey like this would be incredibly enlightening. I would love to do it, and I do like writing survey/expository pieces. It seems impossible, at this juncture, that I will ever have the time and focused energy to do such a thing, but he had only been out of graduate school for 6 years at that point (the same as me, now). Ahem. Anyway, it was not surprising that writing the book worked, but it was somewhat unexpected and inspiring to me that he freely stated that his research had been going poorly.  Nobody ever says their research is going poorly in public! It is so rare! The market, the culture of the field, and our own dedication to positive thinking all encourage us to accentuate the positive, and greatly fear the negative. Since, when things aren’t going well, it becomes very easy to think that we actually aren’t smart enough and we probably shouldn’t be doing this after all. For those of us that are trying to get tenure or the next job, it seems like an act of self-destruction to send the message that we are stuck, or struggling in any significant way. So, I guess it really struck me to hear someone I respected casually mention that their research had been going poorly.

This is just another shot of the Calanques, because I can’t resist.

Now, poor is a relative term. We are each working from our own vision of what our research life should be, so one person’s poor could be another person’s ideal. But this anecdote, which Hirschfeld just mentioned in passing, really got me wondering what people do when things are going poorly. In particular, how do the mathematicians that I admire get unstuck or find inspiration in their research lives? I often imagine that the “mathematical experts” of the world don’t have this sort of trouble.  On the other hand, I spend some time every semester normalizing struggle and convincing my students that the experts struggle too. Students in algebra and number theory classes that I teach are generally assigned to watch The Proof, the Nova documentary about Andrew Wiles and the proof of Fermat’s Last Theorem (based on a book by Simon Singh). There are several reasons I share this film, but one of the things I always point out is just how long it took to prove this statement, and how many brilliant people got utterly stuck on the problem. The idea of being stuck on this scale, of making mistakes and struggling for years, seems to be a bit of a revelation and a comfort to my often-stuck students.

Ask the Experts: Getting Unstuck  

With all this in mind, I decided to ask some really bright, successful mathematicians about being stuck. What I wanted was to ask about times when they had really struggled, and felt that their research was going poorly. But, in the end, the specifics of this seemed too personal to ask people to share in someone else’s blog. I just couldn’t bring myself to put people on the spot about this. What I did manage, though, was to talk to some great people and get some more general advice about what some experts  have done when they were stuck or looking for inspiration. And of course, since being good at math means being good at this kind of struggle, they had really good suggestions.

I asked Bjorn Poonen, Clause Shannon Professor of Mathematics at MIT, if he had a good story about being stuck in research. After thinking a bit, he said that he had often been stuck but that there wasn’t really a good story.  He said, “At one point I was stuck for a year, and that was pretty much the story—I was stuck.” Bjorn says that when he has no good ideas to move forward on a problem, he finds it helpful to explain the problem to someone else, “not necessarily to get their advice, but just because the process of explaining it forces me to rethink it.”

Irene Bouw

Irene Bouw, Professor of Mathematics at Ulm University, had an immediate answer to my question of what she does when she is stalled on a problem: “Just leave it for a bit–that is the best thing.  After you have left it alone for some time, you have to go back over it from the start and check every single detail.” Irene is a realist about the situation, though, in a way that perhaps those looking back at the comparative wealth of time of graduate school can relate to. “There is no recipe that always works. Also, this method worked better when I was a PhD student and I had more time. I don’t have time to be stuck anymore. Now, maybe I come back to it, and I decide it is not so interesting, and leave it. Of course, this is more possible when you have tenure, and maybe you have more problems than you have time to work on.”  She gets to one of the hardest questions for a pre-tenure person—still there is no time, but there is great pressure to produce results. When should you push through on something that is less interesting to you now, for the sake of salvaging a publication, and when you should just cut your losses and move on?

Everett Howe

Everett Howe confirms that he, too, has been stuck, and in particular has sometimes felt stalled by a lack of inspiration.  He said that he has found that going to a conference and just talking to people about their problems has actually been a good way to find direction.  “When a friend or someone that I meet at a conference asks me something that they would like to know for their own research, and I have some ideas on how to answer, it gives me the incentive to work on their question. The personal connection provides the motivation.” I found it really interesting to hear Everett elucidate one of the ways that generally meeting people and building friendships within math, and particularly going to conferences, can be so worthwhile—these things can open up a whole new world of questions to care about, and maybe you actually know how to solve some of them!

Thank you very much to everyone who talked with me at the conference, and especially to James, Bjorn, Irene, and Everett.  Your thoughts on research setbacks? Conferences in the south of France?  Please share in the comments.

Addendum 

I should mention here that I was thinking about some of the ideas in this post weeks ago.  In an earlier comment thread, I asked Joseph Silverman some questions about his math life and how he dealt with being stuck in his research.  Just in case someone out there doesn’t read all the comments on every PhD + Epsilon post, I’ll share an excerpt here:

Me: “What do you do when you are stuck, and you may not know anyone who is interested in your problem? Is it strange to cold-call an expert, or to impose on your advisor? If you ask an expert for advice, is it your responsibility to add them to the project? ”

JS: “There are lots of strategies when stuck. Try to work out an example or a special case. Try to find a counterexample (which often leads to a proof). Try to read something that seems relevant to your problem. Talking to your advisor is fine. Put the problem aside and work on something else for a month or two. Cold-emailing an expert probably should be saved as a last resort, but talking to experts at conferences is a good idea. (Then, when you email them, they know who you are.) If there’s a specific fact you want to know, MathOverflow is good, but before posting, write out your question locally,read and re-read, wait a day or two, re-read again, then post. Also first search MO to see if your question has already been asked and answered. It’s amazing how much stuff is there.

“Generally, if anyone offers you advice, you should acknowledge that advice in the acknowledgements of the paper. But it has to be pretty substantial before you ask someone to be a co-author. There’s no hard and fast rule.”

Thanks again to Joe for taking the time to read the blog and answer my many, many questions!


Postscript: A few more thoughts from an expert

James Hirschfeld

I had such a good time talking with James Hirschfeld about his mathematical life that I wanted to share a little more of our conversation here. I asked him a few questions about how things had changed in mathematics, his early career, and how he found problems.  For your enjoyment, here is a sampling of things that that I found particularly interesting from his answers:

“Considering my career, I always remember that in my early career I was particularly fortunate: in the mid 1960s, if you had a PhD you could get a job anywhere.  There was a great post-war expansion of the university system.  So, I know that my experience is extremely different than it is today.

“But when I think back to the start of my research career, my MSc supervisor suggested a very particular problem—working on the double six theorem over the field with four elements. When I moved on to a PhD program, my supervisor didn’t suggest a new problem; he said just continue your previous work, with the double-six over larger finite fields. I did this, and of course it’s very nice to have a specific problem to work on.  The fault with this is that there is a larger context to consider. What I should have done is more wider reading.  Algebraic geometry has been through so many phases.  After the classical stuff, there was Andre Weil doing it more abstractly, then Grothendieck even more abstractly.  As a PhD student, it’s good to have a particular problem and get somewhere with it, but it’s important to acquire this broader knowledge, rather than just doing some calculation.

“And how do mathematical problems arise, anyway? Of course, there are a very few people who are doing something truly original. Otherwise there are two ways. You can generalize something or connect to similar problems. But equally well, we get problems from other subjects.  Physics or social sciences or biology. And there is an interconnectedness to look for within the fields: In my area of finite geometry, for example, it was the 1980s before geometers and coding theorists realized that they were working on the same problems.”

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