Tony’s Take May 2022

This month’s topics:

Alexander Grothendieck in The New Yorker

“The Grothendieck Mystery” is the title of Rivka Galchen’s article in the issue for May 16, 2022; the subhead is “Alexander Grothendieck revolutionized mathematics—then he disappeared.” Galchen takes us from Grothendieck’s birth (Berlin, 1928; Grothendieck is his mother’s last name) and childhood as a refugee in France, through a brilliant mathematical career that essentially ended in 1970, and through the last half of his life—which he spent in self-imposed exile, farther and farther from Paris, finally as a kind of hermit in a village in foothills of the Pyrenees. He died in 2014.

Galchen was not going to tax the New Yorker readership with mathematical details, but manages to convey, through metaphors Grothendieck used, some understanding of his approach to mathematics. The first is thinking of a problem as a hard nut. One way to get to the meat is to go at it with “sharp tools and a hammer.” But a better way would be “to put the nut in liquid, let it soak, even to walk away from it, until eventually it opened.” You build a theory, step by simple step, and by the time you get around to your problem, the solution is obvious. The second metaphor is “the rising sea.” You have to get your boat across a rocky reef. Instead of attacking the reef, you can “wait for the sea to rise, providing a smooth surface to cross effortlessly.”

The metaphors pay off when Galchen speculates on why Grothendieck dropped out from what seemed the top of the mathematical world. One of his main motivations had been the solution of the Weil Conjectures. These had been formulated during the war by André Weil, another towering figure of 20th-century mathematics. What makes the conjectures striking is that they relate numerical properties of the set of solutions to a polynomial equation over a finite field—a totally discrete gadget—to topological invariants of a geometric object: the complex variety corresponding to that same equation. There were four conjectures. Bernard Dwork had proved the first in 1960; Grothendieck and collaborators proved two of the others by 1965; but the last one just would not succumb to his methods. Instead, after he quit, his student Pierre Deligne found a different approach and scored the goal. Galchen quotes Ravi Vakil: “It was as if, in order to get from one peak to another, Deligne shot an arrow across the valley and made a high wire and then crossed on it.” Whereas, Galchen tells us, “Grothendieck wanted the problem to be solved by filling in the entire valley with stones.”

DNA topology in Popular Science

The digital magazine Popular Science ran “What tangled headphones can teach us about DNA” on April 18, 2022. The story, by Ryan F. Mandelbaum, is about the mathematical biologist Mariel Vazquez, a professor of mathematics and of microbiology and molecular genetics at UC Davis, and her research into topological and geometric features of DNA molecues. Here’s how it starts: “It’s a truth universally acknowledged that if you shove wired headphones into your pocket, they’ll eventually emerge in a jumble of knots.” Then he reminds us that each of our cells [average diameter $30 \mu\text{m}$] contains about 6 linear feet of DNA. A lot of stuffing, a lot of tangles. And tangles in DNA are Vazquez’s field of expertise.

Mandelbaum gives a brief definition of topology, with a useful link to Quanta magazine. A sphere may be equivalent to a cube, but “Doughnuts are a different beast […] : Turning an orb into a ring requires slicing a hole in it or sticking its ends together, making them two fundamentally different shapes.” We learn how Vazquez, as a math major at the National Autonomous University in Mexico, discovered that topology was just what she needed to link her interest in math with her curiosity about the natural world: It was a key to begin understanding how living creatures handle the hugely complicated task of copying and interpreting the genetic information encoded in their DNA.

The image below is not in Mandelbaum’s report, but it gives an elementary example of how topological structures occur in Vazquez’s earlier work on DNA. It comes from “DNA knots reveal a chiral organization of DNA in phage capsids” by Vazquez and five co-authors in Proceedings of the National Academy of Sciences, June 28, 2005. They were studying DNA from a virus, the bacteriophage P4 (the phage). The DNA extracted from the head (capsid) of this virus presents a large proportion of highly knotted DNA circles. One conclusion of their research is that those knots capture information about the original packing of the DNA.

Diagram showing mathematical knots and knots in phage DNA
On the right (“Phage Knots”) is the result of two consecutive electrophoresis analyses of a phage DNA sample, the second one at right angles to the first, and at higher voltage. The second run separates the torus knots $5_1, 7_1$ from the twist knots $5_2, 7_2$ with the same number of crossings. (The trefoil $3_1$ counts as both torus and twist). The column in the center (“Twist Knots”), shows how a sample of exclusively twist knots disperses under electrophoresis. Image from PNAS 102 9165-9169, © 2005 National Academy of Sciences.

Möbius strip nanobelt

Blue and orange Mobius strip made from a chain of hexagons
A Möbius strip nanobelt. Image from an Open Access article in Nature Synthesis.

Ellen Phidian centributed “Molecular Möbius strip: chemists make a geometric anomaly from atoms” to the web science magazine Cosmos (May 20, 2022). As Phidian explains it, “sometimes, chemists want to make molecules that are simply geometrically interesting.” These adventurous experiments can have important real-life consequences—Phidian gives the example of carbon nanotubes. In fact the Nagoya University team (Yasumoto Segawa, Kosuke Itawa and collaborators) that synthesized the Möbius strip carbon nanobelt (MCNB) had made an untwisted carbon nanobelt (CNB), which is an ultra-short nanotube, back in 2017.

Both CNBs and MCNBs are assembled from hexagonal benzene rings (schematically represented in the figure above); what makes both syntheses hard is wrestling a strip of rings into the proper shape. As Phidian describes it: “Unlike a paper Möbius strip, the molecule took more than some scissors and tape to create. In fact, because of its unusual shape, placing strain on the carbon atoms in its belt, making the molecule was deeply complicated.” Segawa et al. report the synthesis of the MCNB to have taken 14 steps. Interestingly, they remark “a CNB can be generated when the number of repeat units is even, whereas an MCNB can be obtained when the number is odd.”

Once synthesized, the MCNB is a very lively molecule: Segawa and collaborators report that, as predicted in 2009, the twist travels rapidly around the belt. The period is on the order of several picoseconds ($10^{-12}$ seconds) at room temperature.

Equiangular lines in Popular Mechanics

“How to Slice a Pie in Four Dimensions, According to Math” ran in Popular Mechanics on May 17, 2022. The author, Juandre, tells us: “If you’ve ever drooled over a pie, cut into eight beautifully equal slices, you’re already a little bit familiar with the concept of equiangular lines—those that intersect at a single point, with each pair of lines forming the same angle.” This is not quite right: with eight equal angles, some lines intersect at $45^{\circ}$ and some at $90^{\circ}$. (We take the angle between two lines to be the smaller of the two angles they form). But changing the eight to six gets us out of trouble.

Left: hexagon with oppostive corners connected by colored lines. Right: Icosahedron with opposite corners connected by colored lines.
Equiangular lines in 2 dimensions (the lines connecting opposite vertices of a hexagon) and in 3 (same with an icosahedron).

Notice that slicing a pie in four equal slices, or taking the three coordinate axes in 3-space, also produces a set of equiangular lines. The examples in the figure above represent the largest possible collections of equiangular lines in 2 and 3 dimensions. What happens in higher dimensions? Juandre seems to imply that researchers recently solved all the higher dimensional cases of equiangular lines, but in fact the maximum number $N(d)$ of equiangular lines in $d$-dimensional space is not known exactly in general. Some recent contributions to the problem are in Equiangular Lines in Low Dimensional Euclidean Spaces published last year (ArXiv version here), including $N(14)=28$ and $N(16)=40$.

The work that motivates Juandre’s posting addresses a modification of the question. That is, given a fixed angle, what is the maximum number of lines in $d$-space which are pairwise separated by that angle? This is the problem that was recently solved (for values of $d$ sufficiently large) by Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao, all of them at MIT at the time. Yao and Zhang (both 2018 Putnam Fellows) were undergraduates, Tidor (MIT ’17) was a graduate student, Jiang was a postdoc, and Zhao (MIT ’10) was an assistant professor. Their work was published in the prestigious Annals of Mathematics on November 2, 2021.