Author Archives: evelynjlamb

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.”

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

The Swag Guide to the Exhibit Hall

You know why you’re really here: exhibit hall swag. No trip to the Joint Meetings is complete without a trip around the exhibit hall. You’ll probably run into an old friend, and you can pick up some free stuff. You can get pens and candy pretty much everywhere, but some of the booths have some more exciting items. If you only have a few minutes to make a pass through, here are my picks for the top swag.

My JMM swag haul. Not pictured: large quantities of candy. (The Pi Mu Epsilon booth has Heath bars!)

My JMM swag haul. Not pictured: large quantities of candy. (The Pi Mu Epsilon booth has Heath bars!)

Right at the entrance, Maplesoft is handing out blinky light clip things. The nice man at the booth said you can use it to be a little more visible if you’re out walking in the dark or just to spice up a boring outfit. Maplesoft is also holding drawing for three gömböcs, odd little solids with two equilibrium points, one stable and one unstable. You can enter the drawing at the entrance to the Networking Center on the 4th floor (4D).

DeShaw is handing out umbrellas, and Pearson has some reusable water bottles. Sage is handing out some stickers, Tessellations has free bookmarks, and the Heidelberg Laureate Forum will hook you up with a keychain. The NSA booth has Cubebots (existence of a surveillance device inside is neither confirmed nor denied, mostly because I didn’t ask). 

IMG_1449

Cubebot may be watching you.

The Legacy of R.L. Moore booth has lots of free resources about inquiry-based learning and the man himself. I (and many others) feel uncomfortable about how much reverence is shown to Moore, a famous racist who deliberately put up barriers to black students, but there’s no doubt that the Moore method and IBL have had a huge impact on my mathematical life.

As they have the past few years, Elsevier is providing free massages. (If you’re participating in the Elsevier boycott, you might have to decide whether it extends to free massages.)

The MAA doesn’t have any outstanding swag, but while supplies last, they are selling some slightly used books for $5 or $8. I picked up a copy of Studies in Global Geometry and Analysis, edited by S.S. Chern.

The American Statistical Association has a pretty good spread. I got a copy of Significance magazine and a math-themed magnetic poetry set (to complement the one I already have at home!). I passed on the cloud-shaped stress ball, but it was cute.

As usual, the AMS booth packs a swaggy punch. I saw yo-yos, magnets, sticky notes, coasters, foam hands, and calendars, including my favorite, the calendar of mathematical imagery. But my favorite item might be the maze pen, available in a wide range of colors.

The perfect pen for after you get lost in a talk.

The perfect pen for after you get lost in a talk.

Until 5 pm tonight (January 6), you can also enter a drawing to win a $100 AMS bookstore gift certificate.

I somehow missed the Math Reviews booth, but I have it on good authority that they are handing out fold-up frisbees, post-it notes, and even t-shirts if you agree to have your picture taken and posted on MathSciNet.

While you’re in the exhibit hall, make sure to enjoy the art gallery as well. If you want to follow along at home, the art exhibit has a page on the Bridges website.

Did I miss your favorite swag? Let me know where I can pick it up!