Jordan Ellenberg's Blog, page 13

That’s it. That’s the post.

[Dean Kremer allows six runs in 1/3 of an inning including a grand slam. Yesterday we lost 13-0. The day before that we lost 10-2 in a game where we were getting no-hit into the 8th. Two days before that we took a 7-4 lead into the ninth and lost 10-7.]

A remark of Simone Weil, which I learned from Karen Olsson’s book The Weil Conjectures.

In the original, “Pour moi, je pense bien que Dieu, selon la parole pythagoricienne, est un géomètre perpétuel – mais non pas un algébriste.” Here’s the letter to her brother from which this is taken.

I wanted to write more in Shape about the complicated moral weights people assigned to the difference/tension between algebra and geometry. I wrote a little about what Poincaré thought about it, who saw the two subjects as representing two temperaments, both indispensable, though any given mathematician might possess them in different proportions.

But then there is this, from S.I. Segal’s “Topologists in Hitler’s Germany”

“the … Nazi movement saw “truly German” mathematics as intuitive and tied to nature, often geometrical, and certainly not axiomatic. Axiomatics, “logic chopping”, too great abstraction, was Franco-Jewish.”

It is grimly funny to imagine the Nazi ideologues locating in abstract algebra and the insolubility of certain equations in radicals the ultimate origins of rootless cosmopolitanism.

When I took creative writing in high school my idea of writing a poem was writing down some thoughts that felt expressive to me and organizing those thoughts into lines of various lengths. Our teacher gave us assignments to write poems in form: sonnets, pantoums, villanelles. This seemed artificial and out-of-date and absurdly restrictive. Why should line 2 have to rhyme with line 5?

What our teacher said was that the absurd restrictions are there to be restrictions. If you sit down with the goal of expressing yourself you only say what you intend to say and this is rarely interesting. The restrictions of form force you into a channel you’re not used to and then you might find yourself saying something you didn’t know you wanted to say.

So maybe pandemic social life was like that? It sort of was, for me. I wasn’t in the office so I didn’t see math people and chat with them there as usual. I wasn’t running into people at the coffeeshop. So I did some things I didn’t usually do. I was on Zoom calls with groups of people from my class in high school. I impulsively accepted Misha Glouberman’s invitations to be on Zoom calls with groups of Canadians I barely knew. I called old friends on the phone without warning them I was going to call, and talked to them. People I usually talk to about every five years I talked to every three months.

Writing a sonnet in class doesn’t mean you go around talking in sonnets afterwards. Maybe you never write a sonnet again. But the things I did when my social life ran through this weird channel are things I’m glad I did.

I made a playlist for Shape, which is coming out in just five days!

All these songs have something to do with geometry or with the book. A few notes:

“Shape,” BaLonely — a great band from Spokane, a guy who plays guitar and sings and his mom who plays bass.“Pythagorean Theorem,” The Invisible Cities. A terrific kinda-dormant-now band from San Francisco. The bass player dated somebody I knew a long time ago and one time I ran into the band at the Ferry Terminal Market and they invited me to a party at their apartment where there was a giant whole roasted pig on the table that everybody ate out of with a fork, and I talked to the bass player about how much we both admired the bassline on “Radio Free Europe.”“The True Wheel,” Brian Eno. Sometimes I feel this to be the greatest rock song ever made. The lyric “looking for a certain ratio” appears in Shape as a section title and Eno shows up a few other times too. “Let’s get it understood” might be a kind of motto for math itself. The topologist Benson Farb introduced me to this song.“You Make Me Feel (Mighty Real)” Sylvester. Because James Joseph Sylvester appears several times in the book. Also this song can be read as a commentary on the Platonist view of geometric entities. (Is it for or against?)“Feed The Tree,” Belly. There’s a whole chapter about trees, and a tree on the cover. “I know all this and more.”“The Distance,” Cake. I don’t even like Cake that much but their ridiculous shtick worked perfectly this one time. The idea that you have to pay attention to what you mean by “distance” is central to Shape.“Circles,” Post Malone. Once you know what distance is you know what a circle is.“The Globe,” Big Audio Dynamite II. And you also know what a sphere is, and what a ball is. (“gonna have a ball tonight / down at the Globe.” “Axis spins so round and round we go” might have something to do with the quaternions. “Headache,” Frank Black. A lot of this song is somehow about the book. Starts out “This wrinkle in time, I can’t give it no credit / I thought about my space and it really got me down,” goes on to “I was counting the trees” as if he’s about to invoke Kirchhoff’s theorem.“Spiraling Shape,” They Might Be Giants. Unsurprisingly a band that has a lot of geometry songs (like the one about Triangle Man) but this is the one that’s specifically about a shape.“Diagonals,” Stereolab. From an album called Dots and Loops.“Circle,” Miles Davis. Starting a run of shapes in the plane.“Triangles & Rhombuses,” Boards of Canada. More shapes in the plane.“Meet Me In St. Louis,” Judy Garland. Written for the St. Louis Exposition of 1904, where a lot of action in the book takes place, and where Ronald Ross, Ludwig Boltzmann, and Henri Poincare all speak (but don’t all meet.)“Perfect Circle,” R.E.M. More shapes in the plane. “A perfect circle of acquaintances and friends” seems to refer to the social networks I talk about in chapter 13.“Shape of Somethings,” Moving Targets. A little punk rock right before the end of a playlist cleanses the palate.“Once In a Lifetime,” Talking Heads. Co-written by Eno. Quoted in the book as a depiction of gradient descent. I listened to Stop Making Sense pretty much non-stop during the training program for Math Olympiad in 1987 and it still feels joyously like math to me. Maybe only me.

For the second year in a row, Seder was virtual, with Dr. Mrs. Q’s family the first night and mine the second, so for the second year in a row I cooked our own Passover meal. After a whole cycle of chagim I’m pretty OK at making brisket by now. I like matzah balls, really like them, and so does everybody else in the house, so I went nuts and tripled the matzah ball recipe. What I envisioned: matzah ball soup absolutely brimming with matzah balls cheek by unleavened jowl. What I did not take into account: the matzah balls absorb soup as they cook, which meant that what was left in the pot when they were done was a kind of matzah ball slurry with no soup at all (pictured, top right). The matzah balls themselves were moist and delicious! But it didn’t really feel right. The next night I made a whole new soup and plopped the leftover matzah balls in there (you make triple, you have leftovers) and that was much more like it.

What we usually do on Passover is visit Dr. Mrs. Q’s mom in Columbus, and get a big helping of deli from Katzinger’s ; missing that, I put in a mail order to Katz’s, which arrived today. So now, the brisket leftovers all gone, I am happily eating pastrami and tongue and matzah-chopped liver sandwiches.

(photo by CJ who never ceases to clown me about how much better his phone’s camera is than my phone’s camera.)

Spring arrived right on schedule, just a little snow left in the shady places, sunny out and windy in the high 60’s. AB and I did our first real bike ride of the year, going out about 15 miles to the very agreeable Riley Tavern where you eat outside on picnic tables. A lot of people are watching Wisconsin’s basketball season slowly sputter out as the Badgers fail to mount a comeback against the much higher-seeded team from Baylor. Riley Tavern serves amazing ice cream sandwiches with two chocolate chip cookies instead of the rectangular brown things; they’re not made there, they’re from Mullen’s Dairy Bar in Watertown. The thing about an ice cream sandwich is, they use the rectangular brown things which are soft and not very interesting because you can bite right through them without messing up the ice cream. Any cookie with a little more of a resistance to the tooth tends to smoosh the ice cream out the side when you bite down. That’s unacceptable. Mullen’s has somehow found a way to use a cookie with a real bite but give the ice cream itself enough structural integrity to hold itself in place while you eat it. Extraordinary!

I’d figured it had been warm enough long enough for the bike trail to be dry, and that was sort of true, but in many places it was badly rutted from the people who’d ridden on it when it was muddy, and even though it wasn’t really muddy anymore, it was soft for a couple of miles, so that your weight pushed your back wheel down into the dirt, which clutched your tire so that you were perpetually in a kind of low-grade partially submerged wheelie. We fought our way through at about 5mph for the whole stretch. So a more strenuous 30 miles than the usual. But the last 5 miles home, on pavement, felt like absolute gliding.

Like a lot of people I’m watching WandaVision, the latest Marvel show. CJ is an MCU fanatic and this show, well-acted, imaginatively shot, and legible without extreme knowledge of Marvel lore, is a good one for us to watch together.

It has settled, on the surface, into being a more “normal” MCU show after doing a lot of really interesting stuff in the first half of the season. But weirdness remains, under the surface. For example (and now the rest of this is spoilers) — the scene where Wanda magically blasts a new rendition of her dead husband Vision out of her own abdomen is clearly shot as a childbirth scene, which makes Vision both her son and her husband, so the whole thing has suddenly taken on a Freudian cast which I don’t think is from the comics. And this explains the shock of the old expert witch Agatha Harkness, who tells Wanda she’s something that isn’t supposed to exist; she is “chaos magic,” a witch with the power to spontaneously create. Witches, traditionally, are supposed to be infertile, but Wanda is not. (This is complicated, I guess, by the fact that Harkness herself apparently has a son in comics continuity but she’s presented as married and childless here.)

Isn’t the Mind Stone placed in the middle of Vision’s forehead a little like a third eye? And isn’t death by getting that eye ripped out kind of Vision’s thing?

I know, I know, sometimes a synthezoid is just a synthezoid.

A couple of days ago I started and ended my workday attending very interesting seminars. At 9am I was watching Zeev Dvir talk about his new paper with Manik Dhar proving the Kakeya conjecture over Z/mZ for m squarefree. At 5, as the sun set, Bjorn Poonen was talking at IAS (ok, at his office at MIT, but virtually at IAS) about his work with Kedlaya, Kolpakov and Rubinstein on space vectors forming rational angles. Those two things sound pretty different, but to my surprise, a very similar question appeared at the heart of both papers! Here it is, phrased a bit vaguely:

Describe the N x N matrices with rank much less than N whose entries are all 0 or roots of unity.

Such matrices exist, of course; A could be 0, or A could be the rank 1 matrix where v and w have root-of-unity entries. Or you could just make a block-diagonal matrix with each block of rank 1 as above. But are there more interesting cases? And what does this have to do with the results above? (Update: it occurs to me that you could pithily phrase the question as: what can we say about matrices of low rank and low height?)

Let’s start with Poonen, where the connection is a little more direct. He and his collaborators are interested in tetrahedra whose face angles are all rational. The unit vectors normal to those faces will thus have the property that

for every pair of vectors u,v. Bjorn & co. decide to go further, asking for a classification of all sets of unit vectors in R^3 such that every pair forms a rational angle. If you have a set of k such vectors, you can put them together into a kx3 matrix U, and the condition above says that the matrix UU^T (the so-called Gram matrix) has all entries of the form . But more is true: U U^T visibly has rank at most 3! Now UU^T doesn’t have entries that are roots of unity, but it does have roots drawn from a very similar countable subset. Conversely, if you have a matrix with 1’s on the diagonal, all other entries rational cosines, and rank at most r, then you have found a subset of S^{r-1} which forms only rational angles. So this has the same flavor as the question above. And indeed one way you could obtain such a matrix is by finding a matrix A of rank < r/2 with all entries roots of unity and then averaging A with its conjugate.

Now what about Zeev? He’s working on one of my favorite problems, Kakeya over rings. Of course it was Zeev himself who famously proved the Kakeya conjecture over finite fields, showing that if S is a subset of F_q^n containing a line in every direction, then |S| must be at least c_n q^n. This is meant to be an analogue of the original Kakeya problem from harmonic analysis, which asks whether a subset of R^n containing a unite line segment in every direction has to have full Hausdorff dimension. Dvir’s theorem proves this over finite fields; but actually, it proves more — that the Kakeya set has positive measure (namely, measure at least c_n, normalizing the whole space to have measure 1.) That’s a problem if you like analogies, because the analysis analogue of that statement isn’t true! A famous example of Besicovitch shows that a set can have a unit line segment in every direction but have measure 0. (It’s hard to draw. Very spiky.) R and F_p are different!

One reason they’re different is that R has scales and F_p does not. Two distinct real numbers can be very close together or very far apart. In a finite field, two numbers are either the same or they’re not. A better analogue for R (of course I’d think this, I’m a number theorist) is the ring of p-adic integers, Z_p, which metrically looks a lot more like the reals. Even the finite ring Z/p^k Z has scales, though only k, not infinitely many. (For more about this, see this old post.)

Dhar and Dvir don’t prove Kakeya over Z/p^k Z, a long-time goal of mine, but they do prove it over Z/mZ for m squarefree, which is very cool, and they lay out a very appealing strategy for proving it over R = Z/p^k Z, which I’ll now explain. Suppose S is a subset of R^n which contains a line in every direction. (By a “direction” let’s just mean a nonzero vector y in R^n; we could identify vectors that differ by a scalar but I don’t think that makes a real difference.) For each direction y, choose a line L_y in that direction whose p^k points are all contained in S. Now we have a linear map F_S from the C-vector space spanned by directions to the C-vector space spanned by S, defined by

.

And we have a map G from C[S] to the space spanned by R^n, which is just the discrete Fourier transform

with a primitive p^k-th root of unity in C.

What happens if we compose F with G? The points of L_y are just x+ty for some x in R^n, with t ranging over R. So

When is anything but 0, the sum over t cancels everything and you get a coefficient of 0. When , on the other hand, the dependence on t in the summand disappears and you just get . Write M for the matrix expressing the linear transformation .

To sum up: this composition has rank at most |S|, because it factors through C[S]. On the other hand, expressed as a matrix, M has y,v entry 0 whenever y and v are not orthogonal, and some p^k-th root of unity whenever y and v are orthogonal. Dhar and Dvir ask, in the spirit of the question we started with: how small can the rank of such a matrix actually be? Any lower bound on the rank of M is a lower bound for the size of the Kakeya set S.

Is that in the spirit of the question we started with? The fact that the order of the root of unity is known makes it feel a little different from the question arising for Bjorn, where bounding the order is part of the game. Your allowable entries are drawn from a finite set instead of an infinite one. And of course in this case you’re specifying which matrix entries are roots of unity and which are zero.

Still, I found it an enjoyably thematic day of math!

This is just to record what a Saturday during what we hope are the late stages of the pandemic looks like here.

Slept well but had complicated dreams; the only part I remember is that I ran into Mike Sonnenschein in Pittsburgh while eating a gigantic meatball I’d bought at a hipster bookstore, and he invited me over, but when I got there, it wasn’t Mike’s house anymore, it was Craig Westerland’s. Akshay Venkatesh was there too. We were going to work on something but nobody really knew how to start and Craig and Akshay were absently flipping through their phones. The thing was, Craig had a tiger for a pet and the tiger got out of its cage and seemed really threatening. It was a bad scene.

A cold wave from the arctic settled in here overnight and it was 7 Fahrenheit this morning. AB and I made French toast with the challah that was left over from last night and watched Kids Baking Challenge on Netflix. Then I had to go out into it and scrape the car, remembering, as I do every time I scrape the car, that I broke the head off the scraper so I have to use the jagged plastic edge of what used to be the head, which works well at breaking up the big chunks of ice but is pretty bad at getting the window fully clean. I’ve lived here long enough to not find 7 Fahrenheit that bad, for the fifteen minutes it takes to scrape off the car. I wore the voluminous sweater that’s so ugly I wear it only on the coldest days. I’m not even sure it’s that warm, but psychologically the body feels it wouldn’t be clad in such an ugly sweater unless the sweater was warm, and that creates the right sensation.

Quiet afternoon. CJ had a mock trial competition against teams from Oregon and Brookfield. AB and I worked on some fractions homework. I posted an early-term course questionnaire for the real analysis course I’m teaching for the first time in my life, and I went through another 50 pages of page proofs of Shape. How there can still be so many typos and small verbal infelicities, after I and others have gone over it so many times, I don’t really know. And there will still be some I miss, and which will appear on paper in thousands of printed books. I wrote a math email to Aaron Landesman, about something related to my work with Westerland and Venkatesh (no tigers.) In honor of Dr. Mrs. Q’s half-birthday we got takeout from Graze for dinner. They had the patty melt special, which I’ve only seen there once before, and which is superb, certainly the best patty melt in the city. I got it with Impossible since we don’t eat milk and meat together in the house.

After dinner, we did what we’ve been doing a lot of weekends, play online games at Jackbox with my sister’s family and my parents. Then we all retreated into our zones. AB is doing some homework. CJ is talking to friends on the phone. I washed dishes while I watched a movie, Fort Tilden, about people being out in the city, in the summer, coming in and out of contact with other people. It was funny.

I’m going to put AB to bed and then think, just a little bit, about a cohomology group whose contribution I don’t understand.

When I was younger I cared about sports a lot. If the Orioles lost a big game — especially to the hated Yankees — it ruined my day, or more than one day. I remember when Dr. Mrs. Q. first found out about this she thought I was kidding; it made no sense to her that somebody could actually care enough to let it turn your whole ship of mood.

CJ is different. It has been an emotionally complicated last few years for Wisconsin sports fans, with all the local teams being good, really good, but never good enough to win the title. The Badgers losing the NCAA final to (the hated) Duke. The Brewers getting rolled out of the NLCS by the Dodgers. Of course, the Bucks, the team with the best record in the league and the two-time MVP, getting knocked out of the playoffs. And today, the 14-3 Packers losing the NFC championship to the Buccaneers. And I gotta say — CJ, while watching a game, is as intensely into his team as I have ever been. But after it’s over? It’s over. He doesn’t stew. I don’t know where he got this equanimity. Not from me, maybe from Dr. Mrs. Q. But I think I’m starting to get it from him. Maybe it just comes with age — or maybe I’m actually learning something.