Maryam Mirzakhani makes history

The IMU has just announced this year’s Fields medal winners. For the first time ever, a Fields medal has been awarded to a woman, Maryam Mirzakhani. I will have the honour of attending the ceremony this morning.

The official press release on Mirzakhani’s research is available, as are the citations for the other Fields medalists. I’d like to speak to what the selection of a female Fields medalist means to me as a woman and a mathematician. In that, I would like to paraphrase something that Melissa Harris-Perry has said about the election of President Obama. Mirzakhani’s selection does exactly nothing to convince me that women are capable of doing mathematical research at the same level as men. I have never had any doubt about that in the first place, and I have said so here many times. What I take from it instead is that we as a society, men and women alike, are becoming better at encouraging and nurturing mathematical talent in women, and more capable of recognizing excellence in women’s work. I’ve said here before that the highest level of achievement within the age limit set for the Fields medals requires a confluence of both exceptional talent and favourable circumstances. Talent must be recognized, nourished, directed in productive ways, accomplishment must be acknowledged and promoted. Among the setbacks I experience every day and hear about from other women, Mirzakhani’s award offers a reason for guarded optimism, a point of evidence that sufficient dents have been made in the many layers of glass ceilings that a woman could break through all of them to the highest level.

Szemeredi wins the Abel Prize

Congratulations to Endre Szemerédi on winning this year’s Abel Prize in mathematics! Ever since my work took a turn towards combinatorics years ago, I have been constantly awed by the breadth, depth and vision in Szemerédi’s work. His impact on mathematics has been beyond profound, not just in combinatorics, but also in many adjacent fields, from harmonic analysis and number theory to computer science. I am especially happy to see his name alongside those of the previous prize winners, such as Serre, Atiyah and Singer, Carleson, Lax, Gromov, or Milnor. He has long deserved this level of recognition.

Tim Gowers, who spoke on Szemerédi’s work following today’s announcement, has posted a written version of his presentation here.

Various and sundry

In no particular order:

♦ The Harmonic Analysis group at UBC has a new website. We’re still adding content and working out the kinks, but that’s basically what it’s going to look like. The website was created by Tom Hollai and Krista Pravetz of eMarketing Vancouver. We’ve enjoyed working with them and would recommend them to anyone needing to upgrade their web presence.

♦ The Erwin Schrödinger Institute, recently threatened with extinction, has been granted a reprieve:

The ESI will continue operation in its present form until May 31, 2011. Thereafter it will form a ‘Research Platform’ of the University of Vienna with the (unchanged) name ‘Erwin Schrödinger Institute for Mathematical Physics’.

Funding of the ESI research programmes and workshops in 2011 and 2012 seems assured. Unfortunately the Junior Research Programme of the ESI has to be suspended until further notice due to lack of financial support. This suspension does not affect current Junior Research Fellowships.

The Directors of the ESI would like to take this opportunity to express their gratitude to the scientific community for their overwhelming support of the ESI during these difficult negotiations.

♦ NSERC long range plan: I ended up not attending the information session at the CMS meeting. I’d been planning to go, but then other activities got in the way. Turns out, if you invite 20+ people from all over the continent to come and speak in your session, it’s hard to tell them once they’re here that you won’t hang out with them because you have a policy meeting to catch – and an information session at that, as opposed to a committee meeting where decisions are actually made.

It’s not clear how much I really missed, though. The steering committee has posted the slides from the presentation along with some FAQ answers. Compared to the expanded terms of reference, there’s some additional information about the procedure but (as far as I can tell) not about the substance of what they’re doing. All the substantive information is phrased in frustratingly general terms, for instance “How is research in mathematics and statistics impacting science.” Really? Where do we even start?

The FAQ answers look like so:

“Why isn’t there representation from ______ on the steering committee?”

• It was important to keep the committee small enough to function effectively, and there are many different aspects to try to balance. Everyone on the committee wears several hats!

• Because the committee is limited in size, it is very important that we get input from ________, ideally in the form of discussion papers, but comments to the committee via the website are also welcome.

While I’m sitting here and waiting for someone to ask me for a discussion paper, I’ve spent some time browsing the committee website, and (in case any committee members are reading this) I’d like to suggest a few improvements to its interactive functionality. Right now, the page setup does not encourage discussion of any of the specific issues on the agenda. The only places where comments can be posted are the three lonely blog entries (look under “recent posts”), so that if I wanted for example to submit feedback on the mathematics institutes – which I do – I’d have to post it under some completely unrelated article where no one would know to look for it. Now, I happen to have a reasonably popular (by math standards) blog where I can post whatever I like and there is a good chance that people will see it. But in terms of engaging the community, having designated comment threads for specific topics on the committee website would work much better.

The men who stare at formulas

The n-category Café has a post about “Dangerous Knowledge”, the BBC documentary I reviewed here some time ago; there’s also a discussion in the comments on whether mathematicians (or academics, or creative types) are really different from “normal” people. If you came here from the link over there, welcome, and here’s hoping that you’ll enjoy this recent interview with John Nash. (Hat tip to 3QD.)

Around the 6-minute mark in the second video, Nash is asked explicitly whether his mental illness might have in some way contributed to his creativity and enabled his mathematical work. He points out in response that his work in game theory was all done before the onset of his mental problems and that he “did not develop any ideas, particularly on game theory, while being mentally irrational”. He also recalls a mistake in a published paper that he completed shortly before the breakdown and suggests that it may have been due to a malfunction of his mind.

Continue reading “The men who stare at formulas”

The Piatetski-Shapiro theorem

I have just learned that Ilya Piatetski-Shapiro died on February 21, 2009, just a month short of his 80th birthday. Most of his research has been in algebraic number theory and representation theory. I’m not a number theorist, and I know even less about representation theory, so I can’t tell you much about his work in those areas. However, I would like to tell you about an early result of his on the summability of Fourier series, known as the Piatetski-Shapiro theorem in harmonic analysis.

Suppose that c_k,\ k\in{\mathbb Z}, is a sequence with the property that \sum_{k=-\infty}^\infty c_ke^{2\pi i kx}=0 almost everywhere on [0,1]. Does it follow that c_k=0 for all k? It turns out (due to Menshov) that the answer is negative. Hence the following definition.

A set E\subset [0,1] is called a set of uniqueness if the only sequence c_k such that \sum_{k=-\infty}^\infty c_ke^{2\pi i kx}=0 for all x\in [0,1]\setminus E is c_k=0 for all k. Otherwise, E is called a set of multiplicity.

If E is closed, it is known that E is a set of multiplicity if and only if it supports a distribution whose Fourier coefficients tend to 0 at infinity.

It was thought for a while that the word “distribution” in the last sentence can be replaced by “measure”. This is what Piatetski-Shapiro disproved.

Theorem (Piatetski-Shapiro). There is a closed set E\subset[0,1] such that E is a set of multiplicity, but does not support any measure \mu with \widehat{\mu}(k)\to 0 as |k|\to\infty.

In other words, E supports a distribution whose Fourier coefficients vanish at infinity, but does not support a measure with the same property!

Piatetski-Shapiro proved that one can take E to be the set of all numbers in [0,1] whose dyadic expansion \sum_{j=1}^\infty r_j2^{-j} obeys n^{-1}\sum_{j=1}^n r_j\leq r, where r is a fixed number in (0,1/2).

Alternative proofs of the Piatetski-Shapiro theorem were given by Kaufman, Körner and others. The following brief sketch of the Kaufman-Körner argument is based on an exposition by Nir Lev. See the introduction to his thesis for the full length version.

Continue reading “The Piatetski-Shapiro theorem”

Analysis at UBC: an update

If you’re applying for admission to graduate schools, or if you’re graduating this year and applying for a postdoctoral positions, and if your research interests are in harmonic analysis, you may want to consider applying here. Our new harmonic analysis group is looking for new students and postdocs starting next year; therefore, here’s a brief description of what we have to offer.

As of this year, our group consists of Mahta Khosravi, Akos Magyar, Malabika Pramanik and myself. Mahta’s research area is quasiclassical asymptotics on manifolds: she uses harmonic analysis to address a family of questions with roots in both quantum mechanics and analytic number theory. Akos, Malabika and I have a wide range of interests, from classical harmonic analysis to geometric measure theory, analytic number theory and additive combinatorics. Speaking of the latter, we’re very lucky to have Jozsef Solymosi as a colleague.

We now have a respectable level of activity in both harmonic analysis and additive combinatorics. There’s a small group of graduate students and postdocs that we hope will grow a little bit more in the next few years. Short-term visitors come reasonably often. We’re working to arrange for research-level graduate courses in harmonic analysis and related areas to be offered on a regular basis. This year, there are two: my course in Fourier-analytic methods in additive combinatorics, and Akos’s course in harmonic analysis next semester. Three, if you also count Jozsef’s course in additive combinatorics next semester – it’s not harmonic analysis, but it’s not exactly unrelated, either. We’re hoping to have at least one topics course in analysis next year, two if possible. We’ve also started a weekly working seminar several weeks ago.

(After the lean and unhappy years I’ve had here from 2000 to 2006, this is quite a change…)

If you’re interested in applying here and would like more information, please feel free to drop us a line.

György Elekes

I have just heard the sad news that György Elekes passed away.  He was 60.

If you work in additive combinatorics or anywhere close to it, there are some names that you just have to know.  Elekes is one of them, for proving many beautiful results in combinatorics and combinatorial geometry, and especially for his amazing 1997 proof of a sum-product estimate based on the Szemerédi-Trotter theorem.  In just a few lines, Elekes improved the previously known bounds and accomplished a great deal besides: by making the connection between incidence theorems in combinatorial geometry and the number-theoretic questions about sums and products, he changed fundamentally the way we think of those questions. Elekes’s bound remained the best known until 2005, when it was improved by Solymosi; that result, as well as Solymosi’s more recent improvement, are both based on geometric methods.

I have not had a chance to meet Elekes, but Jozsef Solymosi calls him “an exceptional talent and a great teacher.” (I hope he doesn’t mind if I quote him on that.)


André-Aisenstadt Prize

Congratulations to Jozsef Solymosi on winning the 2008 André-Aisenstadt Prize! The citation on the CRM web page praises him for “the extraordinary efficiency and elegance of his results at the cutting edge of a new field, additive combinatorics (sometimes called arithmetic combinatorics)” and “the simplicity and deep insight in each of his works.” What this means is that where other authors – absolutely no slouches by any standards – write 30-page papers using sophisticated technology, Jozsef will prove a better result by much simpler methods, throw in additional insights that alone make the paper worth reading, and he’ll do it all in, say, 4-5 pages. Well done!