Category Archives: Day Two

And the Oscar Goes to…

The joint prize session is only 50 minutes long and doesn’t have a red carpet, but it is a chance for mathematicians to get together and congratulate each other for doing good math. The prize session and the AWM prize session the night before really show how varied* mathematics and the joints meetings themselves are. There are prizes for research accomplishments, research articles, expository writing, communicationservice to the profession, teaching, undergraduate research, and, if that were not enough, lifetime achievement. The recipients work on a huge range of topics in pure and applied mathematics.

AWM president Kristin Lauter congratulates my student Mackenzie Simper on the Alice T. Schafer award. Photo: Magnhild Lien.

AWM president Kristin Lauter congratulates my student Mackenzie Simper on the Alice T. Schafer award. Photo: Magnhild Lien.

The prize session recognizes a number of individuals, but reading the prize booklet and hearing their brief remarks started to make me feel warm and fuzzy. Even though we recognize them as individuals, they all seem to thank their families, their collaborators, and their students. Math is not a solo activity!

On a personal note, I can take no credit, but I am absolutely thrilled that one of my students, Mackenzie Simper, was awarded the Alice T. Schafer prize for excellence in mathematics by an undergraduate woman. I only met her last semester when she was in my complex analysis class, after she had done much of the work that would earn her the prize. Nevertheless, I can at least claim I knew her when…

You can read about all the prize awardees here (pdf).

*I originally wrote the word “diverse,” but a glance up at the stage showed a sea of white men in dark suits. There were a few people who didn’t fit that description, but I’m afraid we have a lot of work to do before I can use the word “diverse.”

A familiar story

movieposterThursday evening, I watched the MAA special screening of A Brilliant Young Mind, starring Sally Hawkins and Asa Butterfield. The movie follows a young man on the autism spectrum who finds solace in doing mathematics. His dream is to go to the International Mathematical Olympiad, and most of the movie follows this process and his training, first by an unconventional teacher in the UK, and then at a math camp in Taipei.

I enjoyed the movie, and it was well-made (shot beautifully and brilliantly scored), with incredible acting from the three main stars: Rafe Spall was very convincing as a bitter, but kind, man struggling with MS, Butterfield brought a good balance of detachment and neediness, and Hawkins is perfect in everything she does. The main message of the movie is easily summarized by its tagline: “True genius comes from opening your heart.” Most of the movie people are obsessed by external validation: the teacher wants his student to shine, the student wants an IMO medal, the mother wants her son to love her in a way he can’t. By the end, they find value in themselves. Especially Nathan Ellis, the young hero, finds connection and love with another math competitor, reconciles with his mother, and discovers that there are things far more important than being “clever”.

Overall, I do like the message. My problem is that the structure is a very well-tread upon series of events taking the young awkward boy, who is so socially awkward he can only do math, to a young man who gets a girlfriend and (maybe) stops caring about math. The implications are that math is only for those who cannot make emotional connections, and even more so awkward, white males. There are some girls in the math teams, one of whom becomes the love interest, but her role is weakened by the fact that she supposedly only makes the team because her uncle is the coach, and then decides to quit because she doesn’t believe she is good enough. The love story is very cute, but could have been connected to the math life a little better. For example, why can’t we show that you can be in young puppy-love and still like doing math? Why did the star have to be a boy, and not a girl? My main problem with the movie is that we have seen this story many times before (most notably in Good Will Hunting), and the setting, actors, and drama of the IMO could have been much better used to tell a story with the same message: that it’s important to make human connections, and that medals and recognition are not the reason to do math.

Overall it was a pleasant movie, but it could have been much better with a few tweaks. The movie comes out in DVD at the end of the month, in case you’re interested.

All the singularities*

Karen Smith.

Karen Smith (a little blurry).

Yesterday, I had the great pleasure of attending the AWM-AMS Noether Lecture, delivered by Karen Smith. Smith took us on a tour of modern algebraic geometry, and showed us how many contributions Emmy Noether made to this field of mathematics.

Smith introduced the audience to algebraic varieties (essentially sets of common zeros of polynomials), and the fact that they are everywhere in math. But her main goal was to show us the problems algebraic geometers are interested in, and in particular the question of deciding whether a variety is smooth, and if not how bad the singularities are. Her explanations of resolutions of singularities were great (and I appreciated the many pictures), and she has a level of energy and excitement that is really contagious.

The main technique for analyzing the “badness” of singularities is, instead of studying the variety itself, to study the ring of functions on the variety and reduce this to prime characteristic. This method of reducing this geometric problem to an algebraic one really goes back to Noether and the first isomorphism theorem (which Smith attributed to Noether even though the literature does not). Smith got a chuckle from the audience when she mentioned the “freshman’s dream”, in which reducing to characteristic p really allows you to say that (f+g)p=fp+gp. The upshot of this is that the p-th power map (whose fancy name is the Frobenius map) is actually a ring homomorphism (behaves nicely with addition and multiplication). By a Theorem of Kunz, a variety is smooth if the ring of functions decomposes in a nice way according to the Frobenius, so we really have reduced the problem of finding singularities to a simple algebraic problem! Finally, she mentioned some generalizations and other results by her and her collaborators.

As a fan of algebraic geometry, of course I liked this talk, but I think she did a great job for the general audience too. The link between algebra and geometry was clear, and Noether’s influence was adequately honored throughout. Really great talk indeed.

*To be sung to the tune of “Single Ladies”, by Beyonce.

All the singularities, all the singularities… If you liked it then you should have put a ring (of functions) on it!

Some Math Humor/Quotes

“My collaborators and I have been able to prove a variety… wait, that is probably not a great word to use in a math talk… a collection of results…” – Kate Thompson, AMS Contributed Paper Session in Number Theory. She later followed up with a similar joke when talking about something being “ideal”.

“I think we should do an experiment: change all the names of mathematicians, like Euler, Gauss, Lagrange, to female names, and see what happens.” – Karen Smith, AWM Noether Lecture.

Kubota standard L-series.

Kubota standard L-series.

“So, if you search for images related to standard L-series, this is what you get. This is the only image in my talk.” – Ellen Eischen, AMS Special Session in Number Theory and Cryptography, showing us some tractors.


“I’m really only going to talk about elliptic curves, but to me higher genus means any genus greater than 0.” – Dave Morrison, AMS Special Session on Higher Genus Curves and Fibrations of Higher Genus Curves in Physics and Arithmetic Geometry.

Evelyn Lamb (to Barry Cipra in the press room): Do you have any career advice?” Barry: “Marry well.”

I realize that these jokes/quotes are maybe only funny to me. Any other quotes you want to share? Post in the comments or tweet them with #jmm16.

Thinking about How and Why We Prove

William Thurston was the first example Thomas Hales gave in his talk on Thursday morning about formal proof. To be clear, Thurston was not an example of a formal prover but of the imprecision with which mathematicians often treat their subjects. Hales cited a passage from Thurston in which he used the phrase “subdivide and jiggle,” clearly not a rigorous way to describe mathematics.

Although I never met Thurston, I am one of his mathematical descendants. His approach to mathematics, particularly his emphasis on intuition, has permeated the culture in my extended mathematical family and has a great deal of influence on how I think about mathematics. That is why it was so refreshing for me to go to a session where intuition wasn’t really on the radar.

Hales was certainly not insinuating that Thurston was not a good mathematician. Thurston was the first mathematician he mentioned as an example of less-than-rigorously stated mathematics, but a few slides later he mentioned the Bourbaki book on set theory. Yes, even that paragon of formal mathematics sucked dry of every drop of intuition, falls short when it comes to formal proofs.

By formal proofs, Hales is not referring to Bourbaki-style mathematics but to proofs that can be input into a computer and verified all the way down to the foundations, whichever foundation one chooses. Hales is famous for his proof of the Kepler conjecture that says that the way grocers stack oranges is indeed the most efficient way to do it. The proof was a case-by-case exhaustion, and Hales was not satisfied with a referee report that said the referee was 99% sure the proof was correct. So he did what any* mathematician would do: he spent the next decade-plus writing and verifying a formal computer proof of the result. (Read more about this project, called Flyspeck, on the Aperiodical.)

Hales’ talk was for me a nice overview of the formal proof programs are out there, some mathematical results that have been proved formally (including some that were already known), and a nice introduction to where the field is going. I’m particularly interested in learning more about the QED manifesto and FABSTRACTS, a service that would formalize the abstracts of mathematical papers, a much more tractable goal than formalizing an entire paper.

The most amusing moment of the talk, at least to me, was a question from someone in the audience about the possibility of using a formal proof assistant to verify Shinichi Mochizuki’s proof of the abc conjecture. Hales replied that with the current technology, you do need to understand the proof as you enter it, so there aren’t many people who can do it. Perhaps Mochizuki can write it himself? Let’s just say I’m not holding my breath.

I attended two talks in the AMS special session on mathematical information in the digital age of science on Thursday morning. The first was Hales,’ and the second was Michael Shulman’s called “From the nLab to the HoTT book.” He talked about both the nLab, a category theory wiki, and the writing of the Homotopy Type Theory “research textbook,” a 600-page tome put together during an IAS semester about homotopy type theory. The theme of Shulman’s talk was “one size does not fit all,” either in the way people collaborate (contrasting the wiki and the textbook) or even in the foundations of mathematics (type theory versus set theory).

I don’t know if it was intended, but I thought Shulman’s talk was an interesting counterpoint to Hales,’ most relevantly to me in the way it answered one of the questions Hales posed: why don’t more mathematicians use proof assistants? Beyond the fact that proof assistants are currently too unwieldy for many of us, Shulman’s answer was that we do mathematics for understanding, not just truth. He said what I was thinking during Hales’ talk, which was that to many mathematicians, using a formal proof assistant does not “feel like” mathematics. I am not claiming moral high ground here. It is actually something of a surprise to me that the prospect of being able to find new truths more quickly is not more tantalizing.

You never know what you’re going to get when you wander into a talk that is well outside your mathematical comfort zone. In my case, I got some interesting challenges to my thinking about how and why we prove.

*almost no

Do not base a cryptosystem on the assumed hardness of discrete log in a Q-algebra

Alice Silverberg explaining the algorithm.

Alice Silverberg explaining the algorithm.

Alice Silverberg is a wonderful, formidable figure in Number Theory. Beyond her extensive research (look at her bibliography!), teaching, and professional service, she also worked as a mathematical consultant for the television show Numb3rs.  Her talk, based on joint work with Hendrik Lenstra, described their deterministic, polynomial-time algorithm to solve the discrete logarithm problem in Q-algebras.  This was the first talk in the AMS Special Session on Cryptography and Number Theory, happening today, tomorrow, and Saturday afternoon in room 606.

Kristin Lauter watching Alice Silverberg's talk.

Kristin Lauter watching Alice Silverberg’s talk.


Organized by Matilde Lalin, Michelle Manes, and Christelle Vincent in honor of Kristin Lauter’s AMS-MAA Invited Address, “How to Keep Your Genome Secret” (11:10 Friday in 6BC), the session features a really great slate of speakers.  I’m headed back there now for more math.

Good Morning, JMM. It’s still dark outside.

Hello!  Just starting my shift here in the press room, quite proud that I made it by 7:30.  Luckily I had the east coast advantage.  Still, it’s dark outside and I’m drinking coffee, trying to stay upright.  I will be here in room 613 all morning, with all of the other glamorous members of the math press, like Barry Cipra and Samuel Hansen.

Barry Cipra and Samuel Hansen in the press room. Math journalism in action!

Barry Cipra and Samuel Hansen in the press room. Math journalism in action!

Samuel Hansen was busily hosting the first episode of the second season of his podcast.  Not sure what Barry Cipra was working on, but I’m sure it will be excellent.

Why did they leave me in charge of the pressroom?  Because everybody important is working on Who Wants to Be a Mathematician.  There are 10 really impressive contestants who will be competing in semifinal rounds at 9:30.  The finalists will compete at 10:25.  Mike Breen hosts the game, and Ken Ono will be hosting the awards ceremony at 10:45.  This is all followed by a public lecture by Simon Singh, entitled “Fermat’s Last Theorem versus The Simpsons“.  All the festivities are happening in room 6A here at the convention center.