## Math blog roundup

Lots of good stuff happening in math blogging!

## Puzzle: low-height points in general position

I have no direct reason to need the answer to, but have wondered about, the following question.

We say a set of points $P_1, \ldots, P_N$ in $\mathbf{A}^2$ are in general position if the Hilbert function of any subset S of the points is equal to the Hilbert function of a generic set of $|S|$ points in $\mathbf{A}^n$.  In other words, there are no curves which contain more of the points than a curve of their degree “ought” to.  No three lie on a line, no six on a conic, etc.

Anyway, here’s a question.  Let H(N) be the minimum, over all N-tuples $P_1, \ldots, P_N \in \mathbf{A}^2(\mathbf{Q})$ of points in general position, of

$\max H(P_i)$

where H denotes Weil height.  What are the asymptotics of H(N)?  If you take the N lowest-height points, you will have lots of collinearity, coconicity, etc.  Does the Bombieri-Pila / Heath-Brown method say anything here?

## Yakov Sinai wins the Abel Prize

And I had the job of delivering, in a format suitable for non-mathematicians, a half-hour summary of Sinai’s work.  A tough task, especially since you can’t ask any experts for help without breaking the secrecy!  I like what Tim Gowers wrote in 2011 about doing the same job the year Milnor won.

I was very happy when I learned (after agreeing to make the presentation) that Sinai had won — mainly for the obvious reason that he’s such a deserving recipient, but selfishly because he didn’t realize either of my main two fears.  On the one hand, I feared that the laureate would be someone whose mathematics was so deeply different from anything I know that I would really struggle to say anything at all that I felt confident was correct.  On the other hand, if the winner were someone in number theory, I would feel an intense responsibility to convey the full picture of the winner’s work and how it fit into the entire sweep of the subject, and I would feel terribly guilty about any simplifications I made, and the thing would be a mess.  As it is, the talk was not exactly easy to prepare but I never worried I actually couldn’t do it.  And I learned a lot!

Anyway, the video of the whole ceremony, including my talk starting at about 9:00, is here.

(Note:  All the sound on this is coming from my mike.  So I know it seems like every joke I crack on here is followed by some seconds of uncomfortable silence, but no, seriously, some people laughed, you just couldn’t hear it!)

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## Math Bracket 2014

It’s that time of year again!  Presenting the 2014 math bracket.  School with the best math department wins every game.  As always, all rulings were made by a group, so don’t yell at me if your department loses to one you consider worse.  Also, this year the bracket team was entirely number theorists, so the rankings are no doubt biased to overweight the people we know.  (Previously:  Math Bracket 2013.)

## We are all Brian Conrad now

The quality of streaming conference talks has improved a ton, to the point where it’s now really worthwhile to watch them, albeit not the same as being there.  Our graduate students and I have been getting together and watching some of the talks from the soiree of the season, the MSRI perfectoid spaces conference.  This has been great and I highly recommend it.

One good thing about watching at home is that you can stop the stream whenever anybody has a question, or whenever you want to expand on a point made by the speaker!  We usually spend 90-100 minutes to watch an hour talk.  One amusing phenomenon:  when we have a question or don’t understand something, we stop and talk it out.  Then, when we start the stream again, we usually see that the speaker has also stopped, because someone in the audience has asked the same question.  This is very reassuring to the graduate students!  What’s confusing to us is invariably also confusing to someone else, even to Brian Conrad, because we decided to always presume that the unseen, unheard questioner was Brian, which is pretty safe, right?  (One time we could sort of hear the question and I’m pretty sure it was Akshay, though.)

## The trouble with billionaires

Cathy blogs today about the enthusiasm for billionaires displayed at the AMS public face of math panel, and her misgivings about it.  Cathy points out that, while gifts from big donors obviously accomplish real, useful, worthwhile goals for mathematics, they have a way of crowding out the public support we might otherwise have gotten, and sapping our will to fight for that support.

I think there’s an even deeper problem.  When we’re talking about putting up buildings or paying people’s salaries, we’re talking about things that require many millions of dollars, and asking:  who’s going to pay for them?  It’s not crazy that the answer “a rich person” is one of the things that comes to mind.

But when we talk about improving the public image of mathematics, we are not talking about something that automatically costs lots of money.  We’re talking about something that we can do on social media, something we can do in the newspaper, something we can — and frankly, should — do in the classroom.  Cathy describes the conversation as centering on “How can we get someone to hire a high-priced PR agent for mathematics?”  That means that the billionaire solution isn’t just crowding out other sources of money, it’s crowding out the very idea that there are ways to solve problems besides spending money.

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## The existence of designs

The big news in combinatorics is this new preprint by Peter Keevash, which proves the existence of Steiner systems, or more generally combinatorial designs, for essentially every system of parameters where the existence of such a design isn’t ruled out on divisibility grounds.  Remarkable!

I’m not going to say anything about this paper except to point out that it has even more in it than is contained in the top-billed theorem; the paper rests on the probabilistic method, which in this case means, more or less, that Keevash shows that you can choose a “partial combinatorial design” in an essentially random way, and with very high probability it will still be “close enough” that by very careful modifications (or, as Keevash says, “various applications of the nibble” — I love the names combinatorists give their techniques) you can get all the way to the desired combinatorial design.

This kind of argument is very robust!  For instance, Keevash gets the following result, which in a way I find just as handsome as the result on designs.  Take a random graph on n vertices — that is, each edge is present with probability 1/2, all edges independent.  Does that graph have a decomposition into disjoint triangles?  Well, probably not, right?  Because a union of triangles has to have even degree at each vertex, while the random graph is going to have n/2 of its vertices with odd degree. (This is the kind of divisibility obstruction I mentioned in the first paragraph.)  In fact, this divisibility argument shows that if the graph can be decomposed as a union of triangles with M extra edges, M has to be at least n/4 with high probability, since that’s how many edges you would need just to dispose of the odd-degree vertices.  And what Keevash’s theorem shows is there really is (with high probability) a union of disjoint triangles that leaves only (1+o(1))(n/4) edges of the random graph uncovered!

More details elsewhere from Vuhavan and Gil Kalai.

## What do Aner Shalev and Nike Vatsal have in common?

My mother-in-law was toting around a book of short stories translated from the Hebrew and I saw a familiar name on the front:  Aner Shalev.  Not the same Aner Shalev as the group theorist I know, surely — but no, I checked, and it’s him!  Good story, too, actually not a story but an excerpt from his 2004 novel Dark Matter (or I guess I should say Hachomer Haafel since it doesn’t seem to exist in English.)  It was good!

Sometime last year I was in a coffee shop in Berkeley doing math with Tom Church and on the bookshelf there was an old issue of Story, and in the table of contents I found Vinayak Vatsal.  Not the same Vinayak Vatsal as the number theorist I know, surely, but….  yep, it was him.  I only got to read the beginning of Nike’s story because I was supposed to be doing math, but that one was good too, what I read.

How many mathematicians are secretly placing stories in literary magazines, I’d like to know?

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## Proof school: it’s not just for math kids anymore

A while back I complained, I hope good-naturedly, about Proof School’s self-description as “a school just for math kids.”  A little Ravi told me that the website has since been revamped, and the new version, with tagline “For kids who love math,” is much more to my liking.  The phrase “math kids” is still around, but I think it presents them (us?) as less of a separate species, and more of an tribe bound by common culture:

By “math kids” we mean children who are truly talented and passionate about math. We say we’re looking for students who are internally pulled by math, not externally pushed into it. Of course, math kids have many interests beyond math or computer science–it’s more just a term of convenience and endearment, really–not an absolute. Almost a nickname. If you know any math kids, you know what we mean. Maybe you were one, once, too.

I’m OK with this!

I will say, though, that 6 occurrences of the words “passion” or “passionate” in the FAQ is too many.

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## An incidence conjecture of Bourgain over fields of positive characteristic (with Hablicsek)

Marci Hablicsek (a finishing Ph.D. student at UW) and I recently posted a new preprint, “An incidence conjecture of Bourgain over fields of finite characteristic.”

The theme of the paper is a beautiful theorem of Larry Guth and Nets Katz, one of the early successes of Dvir’s “polynomial method.”  They proved a conjecture of Bourgain:

Given a set S of points in R^3, and a set of N^2 lines such that

• No more than N lines are contained in any plane;
• Each line contains at least N points of S;

then S has at least cN^3 points.

In other words, the only way for a big family of lines to have lots of multiple intersections is for all those lines to be contained in a plane.  (In the worst case where all the lines are in a plane, the incidences between points and lines are governed by the Szemeredi-Trotter theorem.)

I saw Nets speak about this in Wisconsin, and I was puzzled by the fact that the theorem only applied to fields of characteristic 0, when the proof was entirely algebraic.  But you know the proof must fail somehow in characteristic p, because the statement isn’t true in characteristic p.  For example, over the field k with p^2 elements, one can check that the Heisenberg surface

$X: x - x^p + yz^p - zy^p = 0$

has a set of p^4 lines, no more than p lying on any plane, and such that each line contains at least p^2 elements of X(k).  If the Guth-Katz theorem were true over k, we could take N = p^2 and conclude that |X(k)| is at least p^6.  But in fact, it’s around p^5.

It turns out that there is one little nugget in the proof of Guth-Katz which is not purely algebraic.  Namely:  they show that a lot of the lines are contained in some surface S with the following property;  at every smooth point s of S, the tangent plane to S at s intersects S with multiplicity greater than 2.  They express this in the form of an assertion that a certain curvature form vanishes everywhere.  In characteristic 0, this implies that S is a plane.  But not so in characteristic p!  (As always, the fundamental issue is that a function in characteristic p can have zero derivative without being constant — viz., x^p.)  All of us who did the problems in Hartshorne know about the smooth plane curve over F_3 with every point an inflection point.  Well, there are surfaces like that too (the Heisenberg surface is one such) and the point of the new paper is to deal with them.  In fact, we show that the Guth-Katz theorem is true word for word as long as you prevent lines not only from piling up in planes but also from piling up in these “flexy” surfaces.

It turns out that any such surface must have degree at least p, and this enables us to show that the Guth-Katz theorem is actually true, word for word, over the prime field F_p.

If you like, you can think of this as a strengthening of Dvir’s theorem for the case of F_p^3.  Dvir proves that a set of p^2 lines with no two lines in the same direction fills up a positive-density subset of the whole space.  What we prove is that the p^2 lines don’t have to point in distinct directions; it is enough to impose the weaker condition that no more than p of them lie in any plane; this already implies that the union of the lines has positive density.  Again, this strengthening doesn’t hold for larger finite fields, thanks to the Heisenberg surface and its variants.

This is rather satisfying, in that there are other situations in this area (e.g. sum-product problems) where there are qualitatively different bounds depending on whether the field k in question has nontrivial subfields or not.  But it is hard to see how a purely algebraic argument can “see the difference” between F_p and F_{p^2}.  The argument in this paper shows there’s at least one way this can happen.

Satisfying, also, because it represents an unexpected application for some funky characteristic-p algebraic geometry!  I have certainly never needed to remember that particular Hartshorne problem in my life up to now.