This month’s topics:

### Alexander Grothendieck, *topoi*, and Huawei.

“‘He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI—or a lonely madman?” This is the headline for Phil Hoad’s long article about the life, work and legacy of the renowned French mathematician Alexander Grothendieck, published in *The Guardian* on August 31. Hoad concentrates on Grothendieck’s life after 1970, when he quit the Institut des Hautes Études Scientifiques and moved first to Provence in southern France and then, from 1991 until his death in 2014, to Lassalle in the remote foothills of the Pyrenées. Hoad spoke with three of Grothendieck’s children and visited at least two of them; from their stories, it is clear that at the end Grothendieck was a tormented and somewhat delusional man. But when he began his exile, he was still completely in command of his thoughts and his words. His 1000-page *Récoltes et Semailles* (*Harvests and Sowings*, partly translated into English), published in 1984, is the work of a confident and persuasive writer.

Hoad quotes one passage in particular, where Grothendieck describes metaphorically how he feels himself different from most of his colleagues: “They are like the inheritors of a large and beautiful house all ready-built, with its living rooms and kitchens and workshops, and its kitchen utensils and tools for all and sundry, with which there is indeed everything to cook and tinker.” They don’t worry about who built the house or why; they take it for granted. Sometimes they’ll fix a bench or sharpen a tool; the ambitious among them may construct a piece of furniture, or, rarely, a new tool, Grothendieck writes. But the windows stay closed and the furniture accumulates. Grothendieck, however, builds new houses entirely.

Hoad tells us about one of the houses Grothendieck built, the *topos* (Greek for “place,” plural *topoi*, but also *toposes*): “toposes were his furthest step in his quest to identify the deeper algebraic values at the heart of mathematical space, and in doing so generate a geometry without fixed points”—my impression is that a topos is supposed to be no more and no less than the minimal structure on which one can build a mathematical theory of any kind. John Baez has a breezy (“Okay, you wanna know what a topos is?”) introductory webpage on the topic.

Topoi come up in Hoad’s presentation because of their recently discovered potential to enhance artificial intelligence by helping machines to think as we do, with concepts and not just with words. Hoad ascribes to the Italian Grothendieck expert Olivia Caramello their characterization “as a mathematical incarnation of the idea of vision; an integration of all the possible points of view on a given mathematical situation that reveals its most essential features.” This sounds something like the formation of a concept and is prompting real-world investment. The Chinese information and communications technology giant Huawei has opened a research institute, as Hoad tells us, at an elegant location in Paris; one of their recruits is the Fields medalist and topos specialist Laurent Lafforgue, who was brought on to apply topoi to “a number of domains, including telecoms and AI.” Some details are available on YouTube in a seminar talk Lafforgue gave in June 2023.

### AI and the topology of molecules.

The website *Physics World* posted “Spot the knot: using AI to untangle the topology of molecules” by Davide Michieletto on August 13. Michieletto is one of the four Edinburgh-based authors of the research being covered: “Geometric learning of knot topology,” published in *Soft Matter* earlier this year. This is the latest in a sequence of papers using the *local writhe*, a new characterization appropriate for the knots appearing in long organic molecules. Local writhe turns out to be especially suitable for machine learning.

Mathematicians have been busy classifying knots since the mid-1800s. The method has been to find *invariants*—algebraic tags that can be calculated from a representation of a knot and that do not change if the knot is deformed without tearing. One example of an invariant is the <em>crossing number</em>. When projected onto a plane, a knot will cross itself; the crossing number is the smallest possible number of crossings in a projection.

The ideal would be to find a *complete invariant*. Just as the topology of any closed surface in 3-space can be characterized by its *genus* (number of holes): sphere (genus 0), torus (genus 1), 2-holed torus (genus 2), etc., a complete invariant would characterize a knot completely: if it was the same for two knots, then they would be topologically the same, and one could be smoothly moved to overlap the other. The crossing number is not enough, since there are two knots with 5 crossings, three with 6, seven with 7, etc. (Michieletto reproduces an 1884 knot table made by the Scottish scientist Peter Guthrie Tate; it lists all the knots with up to 9 crossings; the website Knot Atlas links to more recent and more extensive tables). During the 20th century, more powerful invariants were developed, culminating in the Jones polynomial (1984) and the Vassiliev invariants (around 1990), but a complete invariant is still out of reach. For every new invariant, topologists have come up with two different knots which the invariant cannot distinguish.

As Michieletto explains, he and his co-authors reasoned that a more appropriate method for identifying knots in proteins would focus on their most intricately twisted parts. This is what the local writhe is trying to measure. The writhe itself has more than one definition, but they all relate to how twisty the knot is. The simplest to explain is sometimes called the “2D-writhe.” You choose a direction along the knot, project the knot onto a plane and calculate the algebraic sum of the crossing numbers, where a right-handed crossing counts as $+1$, and a left-handed crossing as $-1$. This writhe is always an integer, but it depends on the projection plane. Another definition is an integral, going back to Gauss’s calculation of the *linking number* for two curves. This integral turns out to equal the average of the 2D-writhe taken over all possible planar projections and is usually not an integer. Note that the writhe itself, however defined, is not a knot invariant (see figure below). Nevertheless, the authors found that machines trained to distinguish knots using a derived quantity, the local writhe, were able to make better distinctions than some of the best of the known invariants.

**Local writhe.** Think of a knot as an interval wrapped around in space so that its two ends go to the same point. The segment-to-all or StA (local) writhe of the knot is a function defined on that interval. For each point in the interval, take a small fixed-length segment around that point, and average, for all the other segments of that length, a number measuring how closely the corresponding knot segment is twisted about the original one. (Averaging this function in turn would give the Gauss-inspired writhe integral).

Michieletto recounts how he and his collaborators used machine learning techniques to train a computer to distinguish knots based on their local writhe curves. To test their approach, they repeated the same training regime, but using the Cartesian coordinates of the knot instead. In this particular test, the machine was repeatedly presented data ($10^5$ statistically uncorrelated conformations) from an unknotted circle, the Kinoshita–Terasaka knot and the Conway knot.

The Cartesian-coordinate training led to substantial confusion between the two 11-crossing knots, whereas the local-writhe training allowed them to be distinguished with 99.6% accuracy. It is significant that (as remarked above) those two knots are similar enough to have the same Jones polynomial. How does this work? The authors remark: “Somewhat unsatisfactorily, we cannot fully pinpoint why StA writhe is so powerful at identifying different topologies,” especially since the writhe itself, which it is based on, is not a topological invariant. They conclude by writing: “We hope that our results will also inspire mathematicians and topologists to formulate new topological invariants based on the geometrical embeddings of knotted curves.”

*—Tony Phillips, Stony Brook University*