Daily Archives: January 8, 2016

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

For the WIN

The AWM panel (with cool slideshow in the background).

The AWM panel (with cool slideshow in the background).

Among the few non-committee things I did yesterday was attending the AWM Panel Discussion on “Research Collaboration Conferences for Women: Who, What, Where, When, Why and How?”. Moderated by Michelle Manes, and featuring panelists Maria Basterra, Susanne Brenner, Ellen Eischen, Kristin Lauter, Kathryn Leonard, and Ami Radunskaya, the panel mostly focused on spreading information about existing conferences for women. These conferences are not AWM-specific, but they have partly been sponsored by the organization.

I had the privilege of attending two of the WIN (Women in Numbers) conferences, and I was glad to see a packed audience and to hear many questions about these opportunities. These conferences started with WIN, held at Banff in 2008 and organized by Kristin Lauter, Rachel Pries, and Renate Scheidler. According to Lauter, the three were sitting at a number theory conference and realized how few women were in attendance. They wondered if it was possible that there are just not very many women in Number Theory. During lunch that same day, they decided to write down names of female number theorists off the top of their heads, and by the end they had a list of about 75 people. They decided that there was clearly something causing women not to attend, be it availability, inclusivity, and appeal of existing conferences. The goal of the conference was to focus on talking about and doing mathematics, with senior mathematicians at the helm of various projects, and mentoring early-career mathematicians and advanced graduate students. They figured that was the critical transition period in which women were dropping off research mathematics. Another goal was to have a proceedings volume attached to the meetings, in part to have some end result for the participants, but also to encourage continued collaboration and research from each of the groups.

Now, the Women In (Blank) conferences have spread to other areas, like WIT (Women in Topology), WISh (Women in Shape – about shape modeling), WhAM (I forget this acronym), and others. In fact, I have been having fun thinking of other acronyms and how to fit them to this conference (someone needs to come up with a conference for WICKED or WIRED).  All of these conferences have followed similar formats: focus on research, pick problems and groups ahead of time, create and maintain an email list and network, and publish a proceedings volume.

Two recent developments make these conferences much easier to plan. The first is that there is now a Springer series devoted to AWM proceedings. The second, more exciting, and more recent one is the award of an ADVANCE grant to the AWM by the NSF. This grant is intended for the sole purpose of creating, supporting, and encouraging more of these types of conferences. What was once an isolated endeavor of motivated individuals is now supported by an organization whose goal is to promote participation in mathematics by women. How cool, right?

Of course, there are many objections to things like this. Some of the common ones are: is it OK to exclude men from this? Are women going to be able to collaborate with men if they only go to these conferences? Is it detrimental to graduate students to have publications in these proceedings rather than by themselves on a “real” journal? These questions have been asked many times.

To the first one, someone in the audience (I forget who, my apologies) gave the best answer: it would be unfair to have conferences for only women if the system was actually fair. But, the mathematics world is not fair in its treatment of men and women (even though it has gotten better), so giving the underprivileged group a small advantage can only tip things in the direction of fairness. Of course, many people will disagree with this statement, but I found it very pleasing.

The second question was a little silly if you think about it: what is the problem if a woman somehow decided to publish only with other women? How is this different from what men have been doing for centuries? I really didn’t understand the point of this question.

To the third, Lauter gave a great answer. These proceedings are actually peer reviewed very seriously, and many of the papers published are high caliber research. And how can a publication hurt you, really?

Anyway, I left excited about coming up with a new acronym and organizing a new conference myself. Would you?

The panel shows off some of the proceedings volumes.

The panel shows off some of the proceedings volumes.