Alison Grace Martin, artist (right), working on "Around a Torus" with students Quoc-Thinh Truong and Natalie Kratts.

Alison Grace Martin, artist (right), working on “Around a Torus” with students Quoc-Thinh Truong and Natalie Kratts.

At 11:30 AM on Friday I suddenly realized that time was running out to see the art show, which closed at noon. I could have spent quite a while there but had to be content with a targeted strike. First, I stopped to talk to Alison Grace Martin and Quoc-Thinh Truong, who were working hard on a very striking wood-strip sculpture of a torus. Martin traveled from Italy to create the collaborative sculpture at the JMM after her proposal was chosen from a field of submissions. Conference participants had been helping out with with the project all week. The first step was making ten-, twelve-, and fourteen-pointed stars out of wood strips. These stars were then assembled to create the torus. The positive curvature portions were assembled out of the ten- and twelve pointed stars (similar to a soccer ball’s hexagon/pentagon tiling), while the negative curvature on the inside of the torus was obtained by incorporating the fourteen-pointed stars (which could be correspondingly thought of as heptagons). Truong, a Junior double major in math and physics at Lenoir-Rhyne University, had put in a lot of time on the project.

A lot of the art on display was pretty impressive. Most of the artists were not present, so I was unfortunately not able to get permission to photograph their work, but I did find a link to some great shots.  However, I felt is was probably okay to take pictures of some people enjoying “Tessercraft”, created by Ben Signa, a student at San Francisco State University, since the photos don’t actually show the art. “Tessercraft” is a sort of virtual reality trip through the 3-d representations of 4-d shapes. A person wearing the headset can look in all directions and use a joystick to move around in a world of geometry. I couldn’t really explore the whole world because “walking” around in the world, especially climbing on and falling off of the Minecraft-like bricks, made me feel really disoriented and almost carsick. But it was awesome. Celes Woodruff, an Assistant Professor at James Madison University, was enjoying the headset, when her colleague Assistant Professor Cassie Williams took a picture and said, “This is going on the department webpage!“

“It’s worth it,” Woodruff said.

Celes Woodruff and Colin Weir enjoying "Tesseract", a creation of Ben Signa.

Celes Woodruff and Colin Weir enjoying “Tesseract”, a creation of Ben Signa.

More enjoying of "Tessercraft".

More “Tessercraft” fun.

After Math, Aftermath


Andrea Young as the department chair and Aaron Calderon as Mathematicus.

Andrea Young as the department chair and Aaron Calderon as brave Mathematicus.

Every year at the math meetings I notice the intriguing “Mathematically Bent Theatre” on the JMM schedule and plan to go. Until this year I had been thwarted by appointments and exhaustion. But no longer! Finally made it to watch Colin Adams and the Mobiusband players perform mathefied comedic skits. This was such a great choice for Friday night—the skits were really funny and the silliness was excellent brain balm after three days of hard math talks. Ah the joys of “Aftermath,” in which a dishonest mathematician (Tom Garrity) dies and is damned to help the devil (Adams) truly randomize the tortures of hell. “Mathematicus,” is the story of an uprising of brave students led by Mathematicus (Aaron Calderon) against the tyranny of mathematical logic, which forces the math professor-opressors (led by Andrea Young as department chair) to search for a new con. Finally, mean Professor Scourge’s heart grows three sizes in “A Pi Day Carol,” when he (Adams) is visited on Pi Day Eve by the spirits of Gauss, Noether, and his worst ever paper reviewer. These skits got me into the spirit and I will now have to go on a back issue reading binge of Adams’ Mathematically Bent column in Mathematical Intelligencer.

Drinking from a Fire Hose: the Current Events Bulletin

Yesterday, in a feat of mathematical endurance, I attended all four talks in the Current Events Bulletin. The Current Events Bulletin, started in 2003, is a neat idea for the joint meetings: prominent researchers give expository talks about important areas of research (generally not including their own research) for a general mathematical audience. This year, Carina Curto talked about topology and the neural code, Yuval Peres talked about the dynamics of abelian sandpiles, Timothy Gowers talked about probabilistic combinatorics and recent results in combinatorial design theory, and Amie Wilkinson talked about Lyapunov exponents. All four also contributed short overview papers to a booklet available to the audience. (The booklet is also available online as a pdf.) Interestingly, each talk was about 45 minutes long with a short break halfway through to allow people to escape if necessary without causing too much commotion.

These summaries are based only on my notes from the talks. I apologize for any errors or misinterpretations.

I probably got the most out of Carina Curto’s talk both because it was the first talk, so I was not yet fatigued, and because I went in knowing almost nothing about the topic. Topology comes into neuroscience in several guises, including topological data analysis, network theory, and understanding the neural code, the way the brain encodes information. Curto focused on the last topic, particularly as it relates to information about place. Neurons called place cells fire in response to an animal’s location in its environment. Even more interesting to me are grid cells. These cells fire based on the animal’s location in the environment, but the same neuron fires in response to several locations. The locations that cause one neuron to fire form a hexagonal lattice in the environment. Topology comes into studying both place and grid cells. For place cells, the places that stimulate each neuron form an open cover of the environment. (At the phrase “open cover,” topologists instinctively prick up their ears.) By looking at the firing of the place cells, the topology of the environment can be reconstructed based only on that information. For grid cells, there is a “fundamental domain” of the environment; the grid cells respond as if the environment is a torus. Weird!

Yuval Peres spoke about sandpiles. I’ve read a little about them before, including a great Nautilus article by Jordan Ellenberg. The idea of an abeliean sandpile is that you start with a number of grains of sand on one square in a grid. When the square has four grains of sand on it, it topples over, giving one grain of sand to each neighboring grid cell. (It’s abelian because, although it is not obvious, when two squares both have 4 or more grains of sand, the order of toppling doesn’t matter.) Peres talked about this problem and variations thereof. The starting grid can be seeded instead of empty. The “sand” can be made continuous instead of discrete, so a square can keep an integer amount of sand and equally distribute a fractional amount to each neighbor (in his article about it, he appealingly described this model as maple syrup in the grid of a waffle). Then there was “rotor routing,” which encodes a “nonrandom random walk” on the grid by labeling the squares with directions as the walker walks through them. Peres included a tantalizing open question at the end of the talk. He showed us a two-dimensional sandpile and a slice of a three-dimensional one. Outside of a neighborhood of the origin, the pictures were the same. Not similar, but pixel-by-pixel identical. There is currently no proof of why that would be.

Timothy Gowers spoke about Peter Keevash’s recent work on combinatorial design theory, especially ideas of probabilistic combinatorics. The standard example question is the Kirkman schoolgirl problem. There are 15 schoolgirls, and they walk to class in five rows of three. Is there a way they can walk so that over the course of the week, no two of them walk abreast twice? The general problem is to determine whether, in a set of n people, there is any way to form groups of people so that every set of s people is contained in exactly one set of r people? Gowers introduced some naive ways to approach the topic and explained why they didn’t quite work but how some of them could be tweaked to

Amie Wilkinson’s topic was the one I was most familiar with already, although because I am familiar due to more to my spouse’s rather than my own research (I am a dynamicist-in-law), my knowledge has some gaps. Her talk title is a question I have wondered about several times: What are Lyapunov exponents, and why are they interesting? The example she started with was the successive barycentric division of an equilateral triangle. As you find the barycenters of the triangles, is there some kind of limiting triangle shape? To make this question more precise, when you take a random walk through the triangle, choosing one of the six subtriangles at each step, how does the “aspect ratio” change? In this case, it goes to zero, so the triangles are getting very thin and needlelike. The rate at which they get needle-like is the Lyapunov exponent. After explaining what they were, Wilkinson gave a general overview of some of the places they pop up in Fields medalist Artur Avila’s work: ergodic theory, translation surfaces, and 1-dimensional Schrödinger operators.

On a practical note, the talks were in a dim, warm room, and by the end of the second one, I was sore from sitting so long. But I took a deep breath and powered through. In other words, this was a professional on a closed course. Do not attempt at home.

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!

Maybe Stick to the Escalators

Escalators: the safer option.

Escalators: the safer option.

The escalators here at the Washington State Convention Center are impressively long and sometimes pretty crowded.  So I can see why people might be heading for the stairs.  However, the news from the press room is that getting out of the stairwells might not be as easy as getting in.  Apparently two people have found themselves stuck in the stairwells in the last two days, with at least one person needing a rescue by security.  Turns out that some of the stairway doors automatically lock and do not allow reentry to the center.  That’s my public service announcement for the day

Flash Origami


My finished bowl and some nice origami swag from the flash fold.

My finished bowl and some nice origami swag from the flash fold.

Dr. Ryuhei Uehara at today's origami flash fold.

Dr. Ryuhei Uehara at today’s origami flash fold. Origami coach for the day Denise Wood of the AMS is in the background.

The JMM is a good place to love origami this year.  At noon today, the JMM app sent sent me a message about an “origami flash fold” at the AMS booth in the exhibit hall.  I had no idea what to expect but made my way over.  I was hoping a mob would spring up out of nowhere to perform an amazing choreographed dance involving origami.  What I actually found was a big group of people gathered around a table folding origami bowls.  I dug in and started working on a bowl myself.  The folding was a little tricky and I sought advice from my neighbor, who turned out to be Ryuhei Uehara, of the Japan Advanced Institute of Science and Technology, an editor of and contributor to Origami6.  An example of Dr. Uehara’s work in the area is depicted on the book’s cover–a single polygon that can be folded into two very different boxes.

Dr. Uehara's versatile polygon.

Dr. Uehara’s versatile polygon.

Math's best and brightest, folding paper at the flash fold.

Folding at the flash fold.





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