Paris June 2019

Back from nearly two weeks at the Institut Henri Poincare, where we were reinventing rational points, though they actually seem pretty much as they have always been. But lots of new ideas floating around and in particular lots of problems I see as potentially rich ones for students.

Last week featured the hottest temperatures ever recorded in France, reminding one that when you move the mean of a distribution even a little, the frequency of formerly rare events might jump quite a lot. Paris was spared the worst of the heat; after initial predictions of temperatures going over 100F, the hottest day of the conference was 97 and the rest of the week was in the mid-90s, regular old East Coast US summer weather. But of course France doesn’t have regular old East Coast US summer air-conditioning. Faiblement climatisé is the order of the day. The word for heatwave in French is “canicule,” which comes from the Italian word for Sirius, thought to be a bringer of hot weather.

It’s also the Women’s World Cup. Tickets for the US-France quarterfinal, held the night before I left, were going at 350 euros for the very cheapest, but I don’t think I’d have wanted to go, anyway. The Orioles are the only team I love enough to really enjoy rooting for them as the visiting team. Instead I went to Scotland-Argentina, which looked like a laugher 70 minutes in with Scotland up 3-0, but ended in a controversial tie after Scotland’s apparent save of a last-minute penalty kick was called back when VAR showed the goalie jumping off the line a moment before the ball was kicked. The ref called end of time directly after the second kick went in to tie the game, to the confusion and dismay of the players on the field; both teams needed a win to have a real chance of advancing past the group stage, and the tie left them both out. Scottish forward Erin Cuthbert pulled something out of her sock and kissed it after her goal; later I found out it was a picture of herself as a baby. I like her style!

I ate well. I ate whelks. They’re OK. I ate thiebou djienne at this place near IHP which was much better than OK. I ate a watermelon-chevre salad that was so good I went to a spice store and bought the pepper they used, piment d’espelette, and now I have a spice Penzey’s doesn’t sell. Favorite new cheese I ate on this trip was Soumaintrain.

I went to the museum of Jewish history where I saw this campaign poster:

And I saw the computer teen Blaise Pascal built for his dad in 1642, which is at the Musée des arts et métiers, along with a revolutionary 10-hour clock:

And right there at the museum, later that night, just by my good luck, there was a free Divine Comedy concert as part of the Fête de la Musique. It was sold out but, my good luck part deux, someone’s friend didn’t show up and in I went. Great set. Sort of a beautifully multinational moment to watch an Irish guy play a They Might Be Giants song in Paris in front of a cast of the Statue of Liberty:

I also learned on this trip that when French kids play Capture the Flag they use an actual French flag:

and that “Good Grief!” in French is “Bon sang!”

Cold Topics Workshop

I was in Berkeley the other day, chatting with David Eisenbud about an upcoming Hot Topics workshop at MSRI, and it made me wonder:  why don’t we have Cold Topics workshops?  In the sense of “cold cases.”  There are problems that the community has kind of drifted away from, because we don’t really know how to do them, but which are as authentically interesting as they ever were.  Maybe it would be good to programatically focus our attention on those cold topics from time to time, just to see whether the passage of time has given us any new ideas, or cast these cold old problems in a new and useful light.

If this idea catches on, we could even consider having an NSF center devoted to these problems.  The Institute for Unpopular Mathematics!

What cold topics workshops would you propose to me, the founding director of the IUM?

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.)

Applied Algebra Days this weekend in Madison

One of the exciting aspects of math at Wisconsin is the new emphasis on what I call “applied pure math” — that is, applied math that doesn’t involve PDEs or numerical analysis.  If you’re in town and want to see what this looks like, you can come to the first Applied Algebra Days conference, featuring Ronny Hadani, Pablo Parrilo, Olga Holtz, and lots of other interesting people.  And as an added bonus it’s in the shiny new Wisconsin Institutes for Discovery!  Schedule here.  Abstracts here.

Gendered conference campaign

A group of philosophers runs a gendered conference campaign, whose goal is to get conference organizers not to plan all-male rosters of speakers.

Does math need a campaign like this?

Discuss.

JMM, Golsefidy, Silverman, Scanlon

Like Emmanuel, I spent part of last week at the Joint Meetings in New Orleans, thanks to a generous invitation from Alireza Salefi Golsefidy and Alex Lubotzky to speak in their special session on expander graphs.  I was happy that Alireza was willing to violate a slight taboo and speak in his own session, since I got to hear about his work with Varju, which caps off a year of spectacular progress on expansion in quotients of Zariski-dense subgroups of arithmetic groups.  Emmanuel’s Bourbaki talk is your go-to expose.

I think I’m unlike most mathematicians in that I really like these twenty-minute talks.  They’re like little bonbons — you enjoy one and then before you’ve even finished chewing you have the next in hand!  One nice bonbon was provided by Joe Silverman, who talked about his recent work on Lehmer’s conjecture for polynomials satisfying special congruences.  For instance, he shows that a polynomial which is congruent mod m to a multiple of a large cyclotomic polynomial can’t have a root of small height, unless that root is itself a root of unity.  He has a similar result where the implicit G_m is replaced by an elliptic curve, and one gets a lower bound for algebraic points on E which are congruent mod m to a lot of torsion points.  This result, to my eye, has the flavor of the work of Bombieri, Pila, and Heath-Brown on rational points.  Namely, it obeys the slogan:  Low-height rational points repel each other. More precisely — the global condition (low height) is in tension with a bunch of local conditions (p-adic closeness.)  This is the engine that drives the upper bounds in Bombieri-Pila and Heath-Brown:  if you have too many low-height points, there’s just not enough room for them to repel each other modulo every prime!

Anyway, in Silverman’s situation, the points are forced to nestle very close to torsion points — the lowest-height points of all!  So it seems quite natural that their own heights should be bounded away from 0 to some extent.  I wonder whether one can combine Silverman’s argument with an argument of the Bombieri-Pila-Heath-Brown type to get good bounds on the number of counterexamples to Lehmer’s conjecture….?

One piece of candy I didn’t get to try was Tom Scanlon’s Current Events Bulletin talk about the work of Pila and Willkie on problems of Manin-Mumford type.  Happily, he’s made the notes available and I read it on the plane home.  Tom gives a beautifully clear exposition of ideas that are rather alien to most number theorists, but which speak to issues of fundamental importance to us.  In particular, I now understand at last what “o-minimality” is, and how Pila’s work in this area grows naturally out of the Bombieri-Pila method mentioned above.  Highly recommended!

In which I attend a conference on fundamental groups in arithmetic geometry from the comfort of my own home

I don’t watch videotaped lectures — in general I’ve found the difficulty of seeing the board and hearing the lecturer makes it impossible for me to maintain enough focus to engage with the mathematics and take good notes.  In fact, I think the only online video lecture I’ve ever viewed all the way through was one of my own, because I somehow lost the notes I’d used and needed to generate a new set so I could give the talk again.

But I was really sorry not to be able to make last week’s introductory workshop for the Newton Institute’s special semester on non-abelian fundamental groups in arithmetic geometry — so sorry that I decided to try watching the recorded lectures on my laptop.  And they’re great!  Crisp sound and visuals, appropriately timed close-ups on the board, and even a camera pointed at the audience so you can see the people asking questions.  And you can download the talks to your iPod!   Three cheers for the A/V team at the Newton Institute.

As of tonight just the Monday and Tuesday talks are up, which is already plenty to keep me busy.  I just watched Deligne talk about counting l-adic local systems on curves over finite fields; highly recommended.

When I was first giving public lectures, someone gave me the hoary advice that I should quell nervousness by imagining the members of the audience in their underwear.  Strange to think that, in this new broadband world, most of them actually are.

One-second precis of Deligne’s talk:  starting with Drinfel’d in the early 80s, you can count the number of l-adic local systems on a curve over F_q by applying whatever version of the Langlands correspondence you have available and then using an appropriate trace formula to count automorphic forms.  It turns out that the number of rank-d l-adic local systems “defined over F_{q^n}” seems to behave as if it were governed by a Lefschetz fixed point formula, i.e. as if it were the number of F_{q^n}-rational points on some variety.  But what variety?  Not the moduli space of rank-d vector bundles with connection on the curve; that has dimension twice as large as the dimension of the purported variety suggested by the result of the counting problem.  But one still may hope — bolstered to some extent by recent work of Arinkin and Flicker — that the point count is reasonably legible and has something to do with the hyperkahler geometry of that moduli space.  I don’t think that summary made tons of sense — so watch the video!

Do you have a photographic memory for math talks?

I had a rather startling conversation with a colleague of mine today. We were discussing a conference we’d both attended in 2006, and my colleague’s account revealed — as if it were nothing out of the ordinary — that he could provide a complete catalogue of all the lectures delivered there. (With rather pungent evaluative commentary, too, but that sort of thing isn’t for blogging.) This colleague went on to tell me that he more or less remembers every serious math conversation he’s ever had.

I’m completely different; I often can’t remember the contents of a talk I saw the previous day. Or, more precisely, I can’t recall the contents — the material is familiar to me if I hear it again.

Anyway, readers, who’s normal, me or my colleague? Can you reel off the list of speakers from conferences of yesteryear?