This post is for Hillary Clinton.

Now even in the matter of homesteads women are not allowed free land unless they are widows with the care of minor children [...] The alleged reason for this discrimination is that women cannot perform the required duties and so, to save them from the temptation of trying, the government in its fatherly wisdom denies them the chance.

But women are doing homestead duties whenever homestead duties are being done. Women suffer the hardships - cold, hunger, loneliness - against which there is no law; and, when the homestead is “proved,” all the scrub cleared, and the land broken, the husband may sell the whole thing without his wife’s knowledge, and he can take the money and depart, without a word. Against this there is no law wither!

No person objects to the homesteader’s wife having to get out wood, or break up scrub land, so long as she is not doing these things for herself and has no legal claim on the result of her labour.

- Nellie McClung, May 1916

The laws have changed over the last 90 years, but that business with claiming the fruits of our labour for ourselves is still unfinished. We may be legally entitled to equal pay, but just recently the U.S. senate blocked a bill that would have actually allowed women to enforce it. Hillary Clinton and Barack Obama both voted for the bill. John McCain did not vote, but was happy about the outcome.

Even when the results of our labour are legally ours, it’s just, well, impolite of us to insist on claiming them. Hillary has experienced it. I have experienced it. Many of us have.

I’ve never believed that women should support Hillary because of their experience of discrimination or backlash. There are better reasons to vote for a candidate, Senator Clinton or otherwise. What are her policies? How exactly is she going to help us? But it is because of that experience that we understand a little bit better what she is doing and why, regardless of whether we agree with her or not.

She didn’t quit when she was told she should? If you had given up a fight for the sake of collegiality, only to have it rubbed in your face, if you had been asked to concede time and again for no apparent reason except that it would be nice of you to do so, and if you didn’t see any of that happening to your male colleagues, you too would develop a gag reflex when it comes to voluntary concessions. You might, in fact, aim to go too far in the other direction, because that’s what you do first if you want to find a good balance.

She was negative and divisive in her campaign? Yes, she was, and that has cost her. She has tested everyone’s patience, mine included, with the Bosnian sniper fire stories and the endless speculations on how she would be winning if the rules were different. The rules are what they are, and she, of all people, should know that well enough. She’s spent too many years learning and following the rules.

But back to being divisive? Just recently, I came upon a study that claims that women don’t get elected to public offices because they’re not really interested. There will be a separate post about this, because I’m a little bit familiar with the “ambition gap,” but for now I’ll just say that this perception is a real problem for us. If Hillary had been less negative, the default assumption would have been that she didn’t really want the job all that much. She did want it, and she was negative, and that got her in another kind of trouble.

Barack Obama had a whole different set of default assumptions to fight. He handled it better, and he won. I hope that he wins in November. But this post is about Hillary.

Next time that a woman runs for the U.S. presidency, I hope that she will run a better organized campaign and that she will not have a history of politically expedient, but ultimately misguided, votes. But also, that she will not have to spend much time establishing that, yes, a woman might actually run more than a token campaign. That she will be judged on her policies, not her clothing. Perhaps she will even get to the point of being able to run without first having to make all those compromises that become a liability later on. Then she might be able to win.

And that just might be possible because of Hillary’s campaign. In the end, she did change the rules. A female candidate is no longer just a token candidate, it’s someone who might actually get the nomination. It’s someone who might have a very close fight with one of the best politicians, of either gender, that we’ve seen in a long time. Someone who does want the job and will likely be good at it if elected.

The glass ceiling is made of presumptions and beliefs. Hillary did break that.

Here’s how Nellie McClung ends her essay:

Women will make mistakes, of course, and pay for them. That will be nothing new - they have always paid for men’s mistakes. It will be a change to pay for their own; and in paying for them they will learn wisdom.

Last Spring, the NSERC Discovery Grants applicants received the following letter from NSERC (I quote from mine):

The 2007-08 Discovery Grants program budget was under great pressure despite the injection of close to 6 million of new funds. This situation was due to the substantial increase in the number of applications which could not be matched by a corresponding growth in the program’s budget. Therefore, the budget was insufficient to meet the needs of the large number of new and returning applicants. For the Pure and Applied Mathematics “A” Grant Selection Committee 336, this resulted in the budget awarded to renewal applicants being approximately 16% lower than the amount previously held by that group of researchers. This made the competition more difficult than in previous years and the resulting recommendations are reflective of this extra pressure.

You can’t just have two posts about research grants back to back. So, here’s a little bit of storytelling songwriting at its best, from 25 years ago.

As some of the readers here may know, the NSERC Discovery Grants program is under review.  An international committee was struck last year to evaluate the effectiveness of the program.  The report of that committee has now been posted on the NSERC web site.

In a nutshell, the committee says: the program works just fine. There is room for a little bit of improvement (and the report makes specific suggestions in that regard), but no major overhaul is either needed or recommended. Also, the program could use more money.

The main issue specifically addressed in the report is the following. The “success rate” in the Discovery Grants competition is roughly 70% (i.e., in any given year, 70% of applicants are funded). This is much higher than, for example, the 30% success rate in the NSF individual research grant competition. Does it mean that NSERC is not applying high enough standards? Should the Discovery Grants program support instead a smaller group of the top-ranked applicants, funding them at a higher level, and cut off the “long tail”?

Here’s what it looks like from a mathematician’s perspective.

It’s spring at last, even in Toronto…

… Yorkville is not too crowded on a weekday afternoon, but weekends are a different story…

… and if you’re at the right place on Bloor Street at the right time, you might see Aimee Mann playing a live show for free.

It’s been a good semester. I have enjoyed the conferences and other organized activities (that’s where you learn things and get ideas), but also the quiet periods in between (that’s when you actually get the work done). It’s pretty quiet right now, and I have nothing else left to organize, so that I can actually focus on research.

There will likely be a separate post on thematic programs and on the joys of organizing one. Right now I’ll just say that this has been great for my own research program. When you’re isolated - as I was for several years - it’s hard to keep up with what everyone else is doing, and perhaps even harder to stay motivated. Now I’m enjoying mathematics more than I have in years.

A few more Toronto photos under the cut.

Academic politics in England circa 1908, according to F.M. Cornford:

You think (do you not?) that you have only to state a reasonable case, and people must listen to reason and act upon at once. It is just this conviction that makes you so unpleasant. There is little hope of dissuading you; but has it occurred to you that nothing is ever done until every one is convinced that it ought to be done, and has been convinced for so long that it is now time to do something else? And are you not aware that conviction has never yet been produced by an appeal to reason, which only makes people uncomfortable? If you want to move them, you must address your arguments to prejudice and the political motive, which I will presently describe.

And what might these be? Let’s examine a few important political principles:

The Principle of the Wedge is that you should not act justly now for fear of raising expectations that you may act still more justly in the future — expectations which you are afraid you will not have the courage to satisfy. A little reflection will make it evident that the Wedge argument implies the admission that the persons who use it cannot prove that the action is not just. If they could, that would be the sole and sufficient reason for not doing it, and this argument would be superfluous.

The Principle of the Dangerous Precedent is that you should not now do an admittedly right action for fear you, or your equally timid successors, should not have the courage to do right in some future case, which, ex hypothesi, is essentially different, but superficially resembles the present one. Every public action which is not customary, either is wrong, or, if it is right, is a dangerous precedent. It follows that nothing should ever be done for the first time. [...] a Fair Trial ought only to be given to systems which already exist, not to proposed alternatives.

Another argument is that ‘the Time is not Ripe’. The Principle of Unripe Time is that people should not do at the present moment what they think right at that moment, because the moment at which they think it right has not yet arrived. [...] Time, by the way, is like the medlar; it has a trick of going rotten before it is ripe.

But surely, there must be a way to get something done?

This most important branch of political activity is, of course, closely connected with Jobs. These fall into two classes, My Jobs and Your Jobs. My Jobs are public-spirited proposals, which happen (much to my regret) to involve the advancement of a personal friend, or (still more to my regret) of myself. Your Jobs are insidious intrigues for the advancement of yourself and your friends, speciously disguised as public-spirited proposals. The term Job is more commonly applied to the second class. When you and I have, each of us, a job on hand, we shall proceed to go on the Square.

Squaring can be carried on at lunch; but it is better that we should meet casually. The proper course to pursue is to walk, between 2 and 4 p.m., up and down the King’s Parade, and more particularly that part of it which lies between the Colleges of Pembroke and Caius. When we have succeeded in meeting accidentally, it is etiquette to talk about indifferent matters for ten minutes and then part. After walking five paces in the opposite direction you should call me back, and begin with the words, ‘Oh, by the way, if you should happen …’ The nature of Your Job must then be vaguely indicated, without mentioning names; and it should be treated by both parties as a matter of very small importance. You should hint that I am a very influential person, and that the whole thing is a secret between us. Then we shall part as before, and I shall call you back and introduce the subject of My Job, in the same formula. By observing this procedure we shall emphasise the fact that there is no connection whatever between my supporting your Job and your supporting mine. This absence of connection is the essential feature of Squaring.

Remember this: the men who get things done are the men who walk up and down King’s Parade, from two to four, every day of their lives. You can either join them, and become a powerful person; or you can join the great throng of those who spend all their time in preventing them from getting things done, and in the larger task of preventing one another from doing anything whatever. This is the Choice of Hercules, when Hercules takes to politics.

Link found in the comments on Crooked Timber.

The New York Times reports on a recent study just published in Science:

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”

Instead of promoting better understanding, the concrete examples might only distract and confuse the students:

In the experiment [conducted by Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler], the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. [...]

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

 

Better late than never… here’s the belated second part of the update on the Clay-Fields conference. We were very lucky to have Tim Gowers give the Distinguished Lecture Series during the conference, and in this post I will try to give a brief account of his lectures.

There were of course many other interesting talks - in fact this may have been the first time in my life that I attended every single lecture at a conference. I wish I had the time to write about them all. That won’t be happening now, but many of the conference topics are close to my research interests and it’s likely that they will come up in future posts.

This was also the first time that I was the main organizer of a conference this size. It’s been a lot of work, but it has also been very rewarding and totally worth it. I’ve had a lot of help from Andrew Granville, Bryna Kra and Trevor Wooley, and the Fields staff did a fantastic job running everything so smoothly. We’d had new audiovisual equipment installed in the conference room just days before the conference… and everything worked perfectly from day one. Isn’t that something? We even had a screening of two short experimental movies, courtesy of Andrew Granville.

Thanks again to all participants for coming!

The rest of this post will be about Tim’s lectures. It’s a little bit long, so I will put it under the cut. The subject is ”quadratic Fourier analysis”.

… bad things happen to umbrellas.

Photo taken at Bay and Wellesley, quite possibly the windiest intersection in this part of town.

Conference blogging will resume once I’ve caught up on sleep.

Here’s a small sample of what we’ve had so far.

Before the conference even began, Vitaly Bergelson gave two introductory talks on the ergodic approach to additive number theory problems: the first one about the Poincaré recurrence theorem, the second about equidistribution, the “van der Corput trick” and Weyl differencing. While the lectures were accessible to students, they also included a good number of things that the senior mathematicians in the audience might not have heard before, from the history of the subject to a somewhat unexpected “quadratic” point of view on the theorems of Roth on arithmetic progressions and Sárközy on square differences in dense sets. 

Vitaly’s conference talk started with an introduction to Szemerédi’s theorem.  (There were reportedly people in the audience who didn’t know Szemerédi’s theorem - which is great!  We’re always happy to see new people around and to introduce them to the subject.)  He then went on to discuss his work with Alexander Leibman and Randall McCutcheon on ergodic theorems and an extension of the polynomial Szemerédi theorem to “generalized polynomials” - functions that can be built by iterating polynomials and the floor function.

To continue with the ergodic theory theme - Tamar Ziegler talked about her work with Terry Tao on the polynomial Szemerédi theorem in the primes, and Bernard Host gave a lecture on his joint work with Bryna Kra on nilsequences and their applications in ergodic theory and additive combinatorics.  We have two more “ergodic” talks scheduled tomorrow, by Nikos Frantzikinakis and Maté Wierdl.

Ben Green and Terry Tao both gave talks about the progress on their program to prove the Dickson (or Hardy-Littlewood) conjecture on the asymptotic number of solutions to linear equations in the primes. In an earlier paper, they reduced the ”non-degenerate” case of the conjecture to proving two statements, which they dubbed the inverse Gowers norm conjecture and the Möbius-nilsequences conjecture. (”Non-degenerate” means that the system does not either include or encode implicitly any equation in two variables.  For instance, the twin primes conjecture involves the equation , which is degenerate.) 

According to Terry Tao, they have now resolved the Möbius-nilsequences conjecture - this was the subject of his two lectures.  The proof consists of a “number-theoretic” part where specific information about the Möbius function is exploited via the circle method, and a “dynamical” part involving a new Ratner-type theorem for nilmanifolds.  I’m sure that we will be hearing more about this soon!

Ben Green gave an update on the inverse Gowers conjecture.  For the norm, this conjecture was proved by Green and Tao in 2005 in and in finite fields of characteristics , and by Samorodnitsky in characteristics 2.  Last year, Lovett-Meshulam-Samorodnitsky and (independently and about the same time) Green-Tao found finite fields counterexamples for norms with .  However, these examples only work in finite fields of low characteristics.  In a recent paper on distribution of polynomials over finite fields, Green and Tao proved that this particular type of counterexamples can’t occur if the characteristic of the fields is large enough; in particular there is still a good chance that the conjecture will be true in .  (See Terry’s blog post on the subject.)

Akshay Venkatesh gave a “speculative” (his words) lecture about his joint work with Jordan Ellenberg on modelling number-theoretic phenomena by the statistics of seemingly unrelated random objects. For instance, the distribution of zeroes of L-functions can be modelled by the distribution of eigenvalues of random matrices chosen from a fixed subgroup of SL(N) for some large N; this goes back to Montgomery in the case of the Riemann zeta function, and to Katz-Sarnak and Cohen-Lenstra for more general L-functions. Another example: there are parallels between the statistics of arithmetic functions (e.g. partition or divisor functions) and certain phenomena in algebraic geometry. Most of this is heuristic rather than rigorous, but the numerical evidence is reported to be quite compelling and the heuristic considerations do suggest intriguing questions. In a follow-up Math Department colloquium talk, Akshay described some specific results of this type (joint with Jordan Ellenberg and Craig Westerland) at the interface of number theory, algebraic geometry and topology.

To be continued…