On every cloud a magic charm she sees

In the words of General MacArthur said, “We are not retreating. We are advancing in another direction”.

Well. Dyzma's closing speech was shorter and more to the point.

Update: For the correct attribution and context of that quotation, see here.

File under: unexpected sights

I’m back in Vancouver, trying to finish up the revised version of the paper with Malabika Pramanik on differentiation theorems. There have been a couple of developments since the preprint was posted on the arXiv and we have included these in the new version. More on that soon, hopefully before the end of this week.

Meanwhile, here are a few more photos from my trip to Spain, mostly on the odd side. The first one is from the Campo del Moro gardens in Madrid. With all due respect to Nassim Taleb…

This is a small cafe in Barcelona, near Plaza de Catalunya.

The next three were taken at the Madrid-Atocha train station.

Conferences, peer review, and political interference

Our science minister Gary Goodyear is getting involved with organization and funding of scientific conferences. Last week he asked the Social Sciences and Humanities Research Council to reconsider funding for an upcoming conference at York University. Specifically, he recommended conducting a “second peer review”. Here is an excerpt from his official statement:

It has come to my attention that following a recommendation of a peer review board earlier this year, the Social Sciences and Humanities Research Council provided $19,750 under its Aid to Research Workshops and Conferences Program to a conference at York University entitled “Israel/Palestine: mapping models of statehood and prospects for peace”.

Approval of this funding was based on an initial proposal that did not include detailed information on the speakers at the conference. Since funding was provided, the organizers of the conference have added a number of speakers to their agenda.

Several individuals and organizations have expressed their grave concerns that some of the speakers have, in the past, made comments that have been seen to be anti-Israeli and anti-Semitic. Some have also expressed concerns that the event is no longer an academic research-focussed [sic] event.

The SSHRC did request an update from the conference organizers, then issued a statement to the effect that everything is in fine order, thank you very much, and the conference will be funded as planned.

The Canadian Association of University Teachers has called on Goodyear to step down:

“It’s unprecedented for a minister – let alone a minister from the department that funds the granting councils – to intervene personally with a granting council president to suggest that he review funding for an academic conference,” said CAUT executive director James Turk. “This kind of direct political interference in a funding decision made through an independent, peer-reviewed process is unacceptable and sets a very dangerous precedent.”

This blog is not an appropriate venue to discuss the Israeli-Palestinian situation, for more reasons than I could list here (and please keep that in mind if you would like to comment). Neither do I want to discuss the “hate speech” accusations such as those quoted in this article. The conference abstracts are over here. If any of them qualify as hate speech under Canadian law (see here), then there are appropriate procedures in place. Intimidation by political interference in the peer review process isn’t one of them.

I do want to repeat what I said in an earlier post: that academic freedom applies to all views expressed in the context of academic dialogue, including those we disagree with. Especially those we disagree with. That’s pretty much the point of it. And ultimately, academic freedom leads to better science. The correctness and significance of scientific ideas isn’t always clear right away and we’re better off if all such ideas are allowed to compete on their merits.

Of course, if an academic conference became a political event instead, then that would be a problem. However, a political science conference does not become a political event just because opinions about politics are being expressed. After all, that’s what political scientists do for a living. Political action – now that would be another matter. I don’t think, though, that we’ve seen any evidence of that.

But the main purpose of this post is to clarify several aspects of the organization and funding of academic conferences for those readers who have never been involved with that.

La Sagrada Familia and the hyperbolic paraboloid

I’m travelling in Spain this month – mostly for mathematical reasons, but, well, it’s Spain. Last week I was fortunate to see La Sagrada Familia.

La Sagrada Familia is the opus magnum of the great Catalan architect and artist Antoni Gaudí. Gaudí was named to be in charge of the project in 1883, at the age of 31, and continued in that role for the rest of his life. From 1914 until his death in 1926 he worked exclusively on the iconic temple, abandoning all other projects and living in a workshop on site.

The construction is still in progress and expected to continue for at least another 20-30 years. The cranes and scaffolding enveloping the temple have almost become an integral part of it. That’s not exactly surprising, given the scale and complexity of the project together with the level of attention to detail that’s evident at every step. Almost every stone is carved separately according to different specifications. Here, for example, is the gorgeous Nativity portal. (Click on the photos for somewhat larger images.)

To call Gaudí’s work unconventional would be a major understatement. To call it novelty – don’t even think about it. His buildings are organic and coherent. Everything about them is thought out, reinvented and then put back together, from the overall plan to the layout of the interior, the design of each room, the furnishings, down to such details as the shape of the railings or the window shutters with little moving flaps to allow ventilation.

Gaudí’s inspiration came from many sources, including nature, philosophy, art and literature, and mathematics.

Maximal estimates and differentiation theorems for sparse sets

Malabika Pramanik and I have just uploaded to the arXiv our paper Maximal operators and differentiation theorems for sparse sets. You can also download the PDF file from my web page.

The main result is as follows.

Theorem 1. There is a decreasing sequence of sets with the following properties:

• each is a disjoint union of finitely many intervals,
• as ,
• the densities converge to a weak limit ,
• the maximal operators

  

  and

  

  are bounded on for .

It turns out that the set does not even have to have Hausdorff dimension 1 – our current methods allow us to construct so that can have any dimension greater than 2/3. We also have $L^p\to L^q$ estimates as well as improvements in the range of exponents for the “restricted” maximal operators with . See the preprint for details.

Theorem 1 allows us to prove a differentiation theorem for sparse sets, conjectured by Aversa and Preiss in the 1990s (see this post for a longer discussion).

Theorem 2. There is a sequence of compact sets of positive measure with such that differentiates . More explicitly, for every we have

for a.e.

You go, girl!

See also here and especially here.

Bourgain’s circular maximal theorem: an exposition

The following spherical maximal theorem was proved by E.M. Stein in the 1970s in dimensions 3 and higher, and by Bourgain in the 1980s in dimension 2.

Theorem 1. Define the spherical maximal operator in by

where is the normalized Lebesgue measure on the unit sphere . Then

for all

The purpose of this post is to explain some of the main ideas behind Bourgain’s proof. It’s a beautiful geometric argument that deserves to be well known; I will also have to refer to it when I get around to describing my recent joint work with Malabika Pramanik on density theorems. Among other things, I will try to explain why the case of Theorem 1 is in fact the hardest.

Note that Theorem 1 is trivial for ; the challenge is to prove it for some finite . It is known that the range of in the theorem is the best possible, but we will not worry about optimizing it in this exposition. (Not much, anyway.)

Let’s first try to get a general idea of what kind of geometric considerations might be relevant. Fix and . For the sake of the argument, let's pretend that we are looking for a counterexample to Theorem 1, i.e. a function with small but large. Let's also restrict our attention for the moment to characteristic functions of sets, so that for some set . Then . On the other hand, let be the set of all for which there exists a circle centered at such that a fixed proportion (say, 1/10-th) of is contained in . Then

for all ,

and in particular . If we could construct examples of such sets with fixed, but arbitrarily small, this would contradict Theorem 1. In particular, if we could construct a set of measure 0 such that for every (or some other set of positive measure) there is a circle centered at and contained in , Theorem 1 would fail spectacularly. Thus one of the consequences of Bourgain’s circular maximal theorem is that such sets can't exist. (This was also proved independently by Marstrand.)

Let’s now see if we can use this type of arguments to prove the theorem.

Just close your eyes and think of me

In research supported by the National Science Foundation and scheduled to be published in the July issue of the Journal of Experimental Social Psychology, psychologist Joshua Aronson of New York University and colleagues come to the opposite conclusion. Studying college students from across the country, they find that when black students are prompted to think about Obama before they take a challenging standardized verbal test, their scores did not improve relative to white students’ compared to when they did not receive the prompt. And they did no better than black students not prompted to think about Obama. “Their test scores weren’t affected by prompts to think about Obama,” Aronson tells me. “We didn’t find any relationship between test performance and being prompted to think positive thoughts about Obama, although we absolutely expected to. [...]

That may read like The Onion, but actually it’s Newsweek.

That’s all I would have had to say about it, really, except that then the article makes reference to a matter of particular interest to this blog:

[...] years of research on stereotype threat had shown that being reminded that you belong to a group that is stereotyped as being inferior at some task tends to make you do worse on that task [...] So you’d think that focusing on Obama might have the opposite effect: “I belong to a group that includes the brainy president of the U.S.!” Indeed, female students do significantly better on math tests when the tests are given by a female rather than male mathematician, apparently because seeing a female mathematician undermines the “girls can’t do math” stereotype.

I’m assuming that the authors of the study are actually trying to be helpful. That they would like to find a way to improve the academic performance of black students, especially if it might be something as easy to do as, say, displaying pictures of Obama in classrooms. That they’re genuinly disappointed that something they thought promising doesn’t actually work. They had really hoped that getting the students to think about Obama would raise their scores – not enough to close the race gap, mind you, but at least a little bit.

But from our point of view – I’m saying this as part of a stereotyped group – any such work should begin with a very fundamental premise. We’re not all the same. Different groups respond differently to different situations and there is no reason to expect otherwise. Having female students write a test in what they likely see as a less threatening environment is not the same as having black students fill out a questionnaire about Obama. Really, it’s not. What women in math and blacks in higher education have in common is that there aren’t a lot of us. Beyond that, there are more differences than similarities. What works for one group doesn’t have to work for another, and that’s without even looking at the variations within each group. If you don’t notice or acknowledge these differences, you’re engaging in a big, fat piece of stereotyping, even as you’re trying to improve our lot.

I’ve seen – can’t remember where – inorganic chemistry compared to “the study of all animals that are not elephants”. All of us who are not white men are saying hello.

Hat tip to Coates.

Playtime!

In which the ancestry of the house cat is proved beyond all reasonable doubt.

TIGERS LEOPARDS Vs Pumpkins! – More videos at Metacafe

Density and differentiation theorems for sparse sets

Over the next couple of weeks, I will be posting short expositions of various parts of an upcoming paper by Malabika Pramanik and myself on maximal estimates associated with sparse sets in . I’ll start by explaining some of the questions that motivated us to do this work. We first learned about them from Nir Lev. We are grateful to him for the many conversations we had at the Fields Institute and for pointing us to references that would otherwise be very hard to find.

The following question was raised and investigated by Vincenzo Aversa and David Preiss in the 1980s and 90s: to what extent can the Lebesgue density theorem be viewed as “canonical” in , in the sense that any other density theorem that takes into account the affine structure of the reals must follow from the Lebesgue density theorem?

Let’s make this more precise. For the purpose of this post, we will say that family of measurable subsets of has the density property if for every measurable set we have

for a.e. .

This is slightly different from standard terminology, but there should be no danger of confusion, as we will not use any other density properties here. We write .

The Lebesgue density theorem states that the collection of intervals has this property. It also implies that collections such as or have it, just because the intervals in question occupy a positive and bounded from below proportion of .

But that does not exhaust all examples. For instance, consider the family , where . We have and , hence the Lebesgue argument no longer works. Nonetheless, this collection does have the density property, by the hearts density theorem of Preiss and Aversa-Preiss.

Note, however, that the collection in the last example is not closed under scaling , $r>0$. Aversa and Preiss have in fact proved that if a family of intervals is invariant under such scaling and has the density property, then its density property must follow from the Lebesgue theorem in the manner described above.

On the other hand, if we consider more general sets than intervals, then it turns out that there are indeed scaling-invariant density theorems that are independent of the Lebesgue theorem. This was announced by Aversa and Preiss in 1987; the proof (via a probabilistic construction) was published in a 1995 preprint.

Theorem 1 (Aversa-Preiss): There is a sequence of compact sets of positive measure such that and:

• is a Lebesgue density point for , and in particular we have

• the family has the density property.

The analogous question for differentiation theorems turned out to be much more difficult.

We will say that differentiates for some if for every we have

for a.e. .

For example, the Lebesgue differentiation theorem states that the collection differentiates .

The differentiation property is formally stronger than the density property, by letting range over characteristic functions of measurable sets. However, there is no automatic implication in the other direction.

The following theorem was conjectured by Aversa and Preiss in 1995, and proved very recently by Malabika Pramanik and myself (paper in preparation).

Theorem 2. There is a sequence of compact sets of positive measure with such that differentiates . More explicitly, for every we have

for a.e.

Our construction of , like that of Aversa and Preiss, is probabilistic. We prove that the sequence can be chosen so that the maximal operator associated with it is bounded on appropriate spaces. This in particular implies the differentiation theorem.

The exact statement of the maximal estimate, and some of the ideas from the proof, will follow in the next installment.