Shadowplay

And now something for a change of pace. This is a somewhat older photo, but I still like it.

An expository paper on the Favard length problem

In case there’s anyone here who’s interested in the Favard length problem, I have just finished an expository paper written for the proceedings of the 2012 Abel Symposium. There has been a good deal recent progress on the subject in recent years, starting with this 2008 paper by Nazarov, Peres and Volberg, through my paper with Zhai, two papers by Bond and Volberg, and most recently my paper with Bond and Volberg. This exposition focuses especially on the number-theoretic aspects of the question for rational product sets, developed mostly in the BLV paper, although some of it goes back to the earlier papers. You can think of it as BLV-lite if you wish.

I have tried to keep the exposition as simple as possible, omitting many of the technicalities and focusing on examples where we can deal with just one number-theoretic issue at a time (as opposed to BLV, where we must combine the different methods together). I’ve also added a good deal of discussion and commentary. This makes the paper a bit more verbose than what I’m used to, but most of this was written in response to questions that I have actually been asked, so I hope that this will be a useful companion paper to BLV and the other references. Also, I did have a deadline for this, so a couple of things (notably the “Poisson lemma” in Section 3.1) got short shrift, and I probably would have found a few more typos and other such if I’d had more time to chase them. Oh well.

There are a couple of new things at the end of the paper. One is Conjecture 4.6. Matt Bond and I came up with this while trying to figure out whether the assumption on the cardinalities of product sets in BLV can be dropped. If the conjecture turns out to be true, than we can indeed drop that assumption. We have some supporting evidence for various special cases, but we don’t know how to prove it in general.

The second part that has not been published previously concerns “random 4-corner sets”. Peres and Solomyak (Pacific J. Math. 2002) proved that for a randomized version of the 4-corner set construction, the expected Favard length asymptotics is in fact C/n. This is a very nice geometric argument, but I found the original proof quite hard to read, so I reworked and simplified it some time ago. This is included here in Section 5.

The paper can be downloaded here. It will also be posted on the arXiv, if I can figure out how to post LATEX files with pictures.

A Knapp example for Salem sets on the line

The restriction phenomenon in harmonic analysis is best known for surface measures on manifolds. A classical example is the unit sphere, where on the one hand we have the Stein-Tomas restriction theorem for densities on the sphere, and on the other hand, Stein’s restriction conjecture for densities remains open. (Partial intermediate results are also available, but that is a longer story that will have to wait for another time.)

However, restriction estimates can also be proved for fractal sets.

Fall, again

In case you were wondering, I’m still alive. I’ve been working to finish several different projects – the kind that, unlike this blog, actually pay my bills. There will be updates on that soon, the first one probably in the next few days. In the meantime, here’s a photo I took yesterday.

Fall

Intellectual property at UBC

I write this blog at home, in my free time. I have never used my office computer for this purpose. My home computer, printer, internet connection, and the laptop I travel with are all paid from my own personal funds. In the past, I charged the cost of laptops to my research grants, on account of their being used primarily for research-related travel; I no longer do that. I have personal email addresses that I use for work-unrelated correspondence, and I have never put a personal snail-mail letter in the departmental mailbox.

There’s a reason why I’m telling you about it.

Via Faculty Association, we learn that UBC is proposing a new policy on intellectual property. Labelled as a “revision” of the existing Policy no. 88, “Patents and Licensing,” it represents a radical departure from long-established basic principles on intellectual property and academic work. Basically, it requires us to cede the ownership of all of our research and academic writing to UBC, except where industrial partners have a competing claim. In no case, if the policy passes, will the ownership rest with the faculty authors.

Here are the relevant links:

• The current Policy 88 (“Patents and Licensing”)
• The proposed new Policy 88 (“Intellectual Products”)
• The excellent response from the Faculty Association

The perils of changing the subject

(My previous post on the topic is here.)

The responses to last week’s PNAS study on gender bias in science have been satisfying, for the most part. I’ve gotten used to avalanches of knee-jerk reactions every time a study on science and gender comes out. This time, there is a good deal (relatively speaking) of subdued and contemplative silence, at least among the actual scientists; the denials seem diminished in quantity. The effect might not be obvious to a bystander, but is quite noticeable to someone who has been following the debates for a while. I hope that this is a good silence, that some of us are taking the time to sit down and actually think about it.

This of course doesn’t mean that the subject has suddenly become totally uncontroversial. As Sean Carroll says in comments:

At least the trolls have moved on from “there is no discrimination” to “discrimination is rationally justified.” Progress!

I’ll be more specific. The wonderful, wonderful thing about the Yale study is that it allows us to have this discussion without being called “paranoid,” “hypersensitive,” or “emotionally unbalanced.” It feels refreshing and different to read long, argumentative comment threads on the subject and never see those words.

The discrimination apologists argue that, given the same “official” credentials, the rational employer will give preference to a man over a woman, because babies, pregnancies, dolls, biological differences, innate abilities, bell curves, life priorities, and other similar perennials.

Then there are press responses. The New York Times ran an article on the Yale study, then followed up with a discussion page. Here’s what one of the participants contributed:

There is little to suggest that colleges and universities are systematically discriminating against women or discouraging them from pursuing STEM disciplines. [...]

Why should we focus on achieving balance in STEM fields, while ignoring the overall imbalance in higher education as men fall farther behind? Factors other than sexism are likely the cause as to why fewer women pursue STEM fields. When students choose majors, they take into account myriad factors, such as their interests, aptitudes and career aspirations. Some research suggests, for example, that women with high-levels of quantitative skills are also likely to have high aptitudes in other areas, while men with high STEM-aptitudes tend to be less talented in other areas.

That, right there, is why I usually stay away from this type of debates. Let’s recap what the study actually said: that given identical paperwork from two hypothetical job candidates, one male and one female, the woman was judged as less competent and offered a lower salary. This is not about whether girls, statistically speaking, are less interested in science. It’s about a specific candidate who had already met the prerequisites, got a degree, demonstrated interest and skill in research, stated his or her career priorities clearly and explicitly, and was received much better when his name was John instead of Jennifer.