Intelligent Computer Mathematics vs. Intelligent Mathematics, II
Intelligent Computer Mathematics vs. Intelligent Mathematics, II
Part II: A review of a dramatic and suspenseful lecture by Dick Gross
"Literature is the place of the contingent. It is not tied to the necessity of problem-solving." (Gayatri Chakravorty Spivak, November 4, 2020)
This week and last week I compare two perspectives on the "the surplus of meaning" and "the essence of mathematical thought," both implicit in texts about mathematics: the first by a prominent exponent and prophet of "intelligent computer mathematics," the second by a prominent number theorist. It would be too easy to say that the first text takes the position that there is no surplus and that the "essence of mathematical thought" resides in formalization and nothing more, while the second text exemplifies the surplus of meaning as well as what one reviewer of Karen Olsson's book The Weil Conjectures (to which we return below) called "the poetry and precision of a theorem." But it would be impossible not to say that.
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Last week I reported on my reading of Christian Szegedy's invited paper to the 2020 International Conference on Intelligent Computer Mathematics. But in my rush to complete the entry before heading to the train station I forgot to comment on an especially peculiar passage.
The first technical issue concerns the input format of informal mathematical content (i.e. mathematical papers and text books). A textual representation could work well for use cases that do not require the understanding of formulas, diagrams and graphs. However, mathematical content often uses a lot of formulas and diagrams. Geometric illustrations play a role in informing the reader as well. The safest path seems to rely on images instead of textual representation. While this puts more burden on the machine learning part of the system, it can reduce the engineering effort significantly.
If I understand this correctly, Szegedy is proposing to train the algorithm on paired sets of "informal mathematical content" in the form of "formulas, diagrams and graphs" together with their formal translations. Later he writes that the "informal input statement" is "given as a picture." It sounds like he is proposing to pixellate things like this (randomly chosen) figure from William P. Thurston's 1986 article "Hyperbolic structures on 3-manifolds I":
— or this illustration from an 1879 article by Felix Klein, that I copy from the slides of a July lecture, to which I turn in a moment, by Dick Gross:
— or maybe a picture of a blackboard. The resulting file, in "raw image format," to quote Szegedy, is then realized somehow as a subset of a Euclidean space, with the hope that the algorithm generates an approximate formal translation with the same content. I can't begin to associate a mental image to such a process, but it does form a kind of bridge between automatic language translation, on the one hand, and training a pattern recognition algorithm to distinguish dogs from cats, on the other hand. I would be curious to exchange impressions with a consciousness that experiences all three kinds of tasks on a continuum.
But now I want to turn to today's assignment, which is a close reading of a lecture Dick Gross gave this past July, entitled The Fricke-Macbeath curve and triple product L-functions, at a conference in honor of the 70th birthday of Steve Kudla, a professor at the University of Toronto with whom Gross and I have both collaborated. The audience was evenly divided between experts and students who aspire to being experts, and I do not expect readers of this newsletter to be able to follow the details of Gross's presentation, nor for that matter to care about the subject matter, which lies at the border of algebraic geometry and number theory. But Gross, in addition to being an extremely distinguished and influential number theorist, is also an uncommonly skilled and versatile lecturer.1 You needn't take my word for it.2 In a book that one reviewer (not the one quoted above) called "her conjectural, juxtapositional, and occasional dreamlike zigzags down the corridors of the inner lives of André and Simone Weil," and what a historian of mathematics called "One of the most insightful meditations on modern mathematics I have ever read," the novelist Karen Olsson describes how she revisited abstract algebra, years after her last undergraduate mathematics class, by watching lectures by Gross on YouTube.
In his lectures, the Professor Gross of 2003 manages to impart something like suspense and drama to a subject without much inherent narrative tension.
What, I would like to ask, can be like suspense and drama without actually being suspense and drama? Perhaps only a novelist like Olsson can offer such an original reading of Gross's lecture, but I would amend it. Narrative tension is absolutely inherent to the experience of doing mathematics, that as a rule can only be detected by those who are acquainted with the experience. Dick Gross has made a long list of lasting marks on number theory and related fields,3 but the special talent that Karen Olsson recognized is his rare ability to impart the authentic suspense and drama of the mathematics that he values, when speaking to audiences with widely varying experience and degrees of familiarity with his material.
My close reading of Gross's lecture is informed by my experience with narrative tension as an organizing principle in teaching as well as in lecturing to experts, but I owe the reader a warning. A reliance on narrative alone does not suffice to pinpoint "the essence" of mathematics and its "surplus of meaning" relative to formalism. The unreliable narrator, to name just one recurrent literary archetype, is an unwelcome figure in a mathematical text. The mathematical narrator is constrained by rules and tradition; but occasionally the narrator is sufficiently forceful to innovate within the tradition. In watching his lecture last July and in rereading his slides, I had the impression that Gross is such a narrator.
His opening slide already displays the internal dialogue characteristic of his narrative style. The lecture's first theme is the symmetries of algebraic curves,4 traditionally a topic in elementary algebraic geometry. He reminds the audience of the classic theorem about these symmetries, due to Adolf Hurwitz, that his listeners all already know. But he immediately focuses on the maximally symmetric curves, called Hurwitz curves, or H-curves, and explains a less familiar fact: that these curves have a different description in terms of hyperbolic geometry; this geometry is illustrated on the following slide by the image due to Felix Klein copied above.
An initial dialogue has now been established between the H-curves, denoted X, and their symmetry groups, denoted G. Slide 3 introduces a new symmetry involving G and asserts that it can be measured by a precise formula. Instead of interrupting the narrative to prove this formula, he establishes complicity with his listeners by implicitly inviting them to prove it at their own leisure, indicating simply that it is a consequence of the Lefschetz fixed point formula, which he is confident they all know how to manipulate. The alert listener, who is familiar with Gross's primary interests, will accept this invitation as evidence of the correctness of the formula and will instead be more intrigued by the slide's foreshadowing of the eventual appearance of the theory of modular forms, not mentioned by name but clearly visible in the new symmetry5 mentioned in this slide.
The topic of the talk's title makes its first appearance in Slide 4, at the beginning of a list of numbers — 3, 7, 14, 17, 118. The principal invariant of an algebraic curve is its genus, and this is the list of the first few genera of H-curves. Gross is well aware that the listener who is encountering H-curves for the first time will find this enchanting (I certainly did), in the way of cabinets of curiosities: the rules established by Hurwitz force H-curves to conform to this curious list. After identifying the H-curve with genus 3 as the Klein quartic
Gross moves on to identify the Fricke-Macbeath curve of his title with the second genus on his list, and in the space of a few lines recapitulates the themes he has introduced thus far: the relation to hyperbolic geometry, the symmetry group G, and the new symmetry discussed in Slide 3 that will tie in later with modular forms.
I briefly digress with another observation by Olsson regarding Gross's lecturing style:
He has a fondness for historical digressions (as do I, obviously), and during the first week of class, he presents one as though it were a theorem: "If you go to Paris and if you take the ligne de sud to the Parisian suburb of Bourg-la-Reine, an absolutely disgusting suburb,6 and you go to the main intersection of Bourg-la-Reine, an absolutely disgusting intersection, you will find a plaque that says, Ici est née Évariste Galois, mathematicien."
Before I turn to Macbeath, who appears on the very next slide, let me say a few words about Fricke. Like any good story-teller, Gross can count on shared knowledge as the basis of his complicity with his audience. In this case, the audience knows that Fricke and Klein are pioneers of the branch of hyperbolic geometry related to modular forms, and their role in the founding of the subject is marked by the publication of two massive two-volume studies between 1890 and 1912.7 As pioneers they were entitled to choose where to direct their attention; they were not constrained by what Spivak calls "the necessity of problem-solving." Or, since the interaction of the trained reader with the grammar of most mathematical texts leads the former to read the latter as structured around "problem-solving" as such, maybe it's more accurate to say that Fricke and Klein were not constrained necessarily by pre-existing problems, but that they chose — contingently — to define new problems, which they then proceeded, freely, to solve. Fricke chose in 1899 to study the H-curve with genus 7, but the contemporary audience is entitled to ask whether the curve illustrates the kind of general principles to which mathematics aspires, or whether it rather illustrates the pleasure that can be derived when these general principles are illustrated in a singular item picked out from the cabinet of mathematics' historical curiosities, like the collection of mathematical models at Klein's department in Göttingen.
I ask these questions with the humility and urgency of the recovering rationalist who even now endures bouts of impatience with contingent mathematics. The Fricke-Macbeath curve frustrates any attempt to enforce a rigid separation between mathematical necessity and contingency. The theorems announced on the first four slides declare that an H-curve of genus 7 is necessarily the one studied by Fricke and then by Macbeath; that is part of what it means to say that they are theorems. At the same time, the coalescence of the disparate properties mentioned thus far — symmetry of algebraic curves, hyperbolic geometry, the very special modular form that the final slides will reveal to be the Fricke-Macbeath curve's alter ego, and more still to be encountered — is not necessarily present in the concept of any of these properties taken separately. For a philosopher this may be taken as the starting point for an examination of the intractable antinomy between seeing mathematics as a field of discovery or of invention. To my mind Dick Gross resolves the antinomy between the necessity and contingency of the multiple identities of the Fricke-Macbeath curve by endowing it as a subject of literature, in precisely the sense intended by Spivak in today's epigraph.
Here I defer to Olsson, describing her impressions of a Dick Gross lecture:
There's an urgency to his presentation, a vigor born of logic itself. "Now I claim," he says, raising his voice and drawing out the I and the claim, like a magician announcing his next trick - "Now I CLAIM ... that r=0 ."
Macbeath, a more shadowy figure than Fricke for a contemporary audience, appears on Slide 5 in 1965 where he builds a bridge from hyperbolic geometry to algebraic geometry. This provides the opportunity to introduce a new character, the elliptic curve E, which arises from X by discarding some of the latter's symmetries. Gross devotes a slide to mentioning some of the stories that could be (but will not be) told about E before withdrawing the elliptic curve temporarily, and instead pivots to a second main theme — a Shimura curve, whose immediate role is to enrich the hyperbolic geometry we have already seen with familiar content from number theory. Over the next few slides we learn that the H-curves of the beginning of the talk are all Shimura curves, and from here to the end of the lecture the dominant idiom is that of number theory rather than geometry. The end of this sequence underlines the successful synthesis of the lecture's first themes by displaying the equations that characterize the curve E within this idiom.
Now we are ready for the third theme of automorphic forms, whose own idiom admits a reformulation of the original theme of the title. The main trunk of mathematics is conventionally divided between the branches of geometry, algebra, and analysis, where the latter includes the techniques of differential and integral calculus — computer scientists and philosophers will look in vain for logic in that list. So it is gratifying to see that Gross is now ready to unify mathematics around the Fricke-Macbeath curve as a topic within the third branch, symbolized by the first appearance of the integral sign
as well as the first two branches whose characteristic images the reader of this essay has already seen in figures 2 and 3, respectively. After a (personally gratifying) allusion to my own collaboration with Kudla, which represents one early intervention in one of the main plot lines of Gross's mathematical career, Gross ties together the themes of the lecture in one line:
Specialists, and in particular those who are familiar with Gross's own collaboration with Kudla, will recognize the significance of this simple but deeply meaningful formula, which is precisely analogous to the scene in a traditional novel where all the (surviving) main characters appear and where all the plot lines converge. Olsson knows best how to capture what Gross's audience is feeling at this point:
A quality of both good literature and good mathematics is that they may lead you to a result that is wholly surprising yet seems inevitable once you’ve been shown the way, so that—aha!—you become newly aware of connections you didn’t see before.
Interestingly, Gross doesn't mention whether he considers the (two-sentence) verification of the formula to be contingent, based on a numerical coincidence, or a necessary feature of the Fricke-Macbeath curve — of its essence, if you like. It's generally expected that expressions like those on the left side of Figure 5, whatever they could possibly mean, equal 0 roughly half the time, and do not equal zero the rest of the time, and in either case the circle of ideas to which Gross has contributed as much as anyone includes an explanation of what it means either way. His very last paragraph returns to the theme, as will we, but first he pauses to acknowledge the audience's anticipated reaction, with a rhetorical question:
Why is this computation so simple?
His answer lifts the curtain on his choice to focus on the Fricke-Macbeath curve: it is explained not only by its historic significance but also by the highly contingent fact of its relation to an unusually simple computation. So he reveals that it is possible to exemplify a difficult general principle by a story with a rare historical resonance.
The last few slides bring the Fricke-Macbeath curve into the mainstream of contemporary research, with the introduction of one last theme, that of algebraic cycles. Conjectures about algebraic cycles were already central in when Gross and Kudla and I all became professional mathematicians, and they have proliferated since then, but they remain among the most intractable and mysterious questions in all of mathematics. Gross's last slide reminds his listeners what one of these conjectures predicts with regard to the Fricke-Macbeath curve — that somewhere there is an algebraic surface,8 that explains the formula in Figure 5 in a precise way.
Gross sustains suspense and drama and narrative tension in his lecture by the successive revelations of new properties of the object under study and the successive introduction of new plot lines, and are resolved when these plot lines, and with them the three main branches of mathematics — algebra (algebraic curves), geometry (hyperbolic geometry), and analysis (automorphic forms) — converge in the formula in Figure 5. A quick scan of Vladimir Propp's list of the 31 functions of Russian folk tales suggests features shared with "The Fricke-Macbeath Curve…", notably functions 22 (Unrecognized Arrival), 25 (Difficult Task), 26 (Solution), and especially 27-29 (Recognition, Exposure, Transfiguration). A new narrative tension emerges in the final lines, however, absent from Propp’s list9, when Gross’s lecture reminds the mathematicians in the audience that the questions on the other side of the formula lead directly to an enormous mystery, and warns everyone else that mathematics is not, or not primarily, concerned with folk tales.
The components of the narrative are all familiar enough. Narratologists should not expect to find radical structural innovations in a lecture by Dick Gross or any other skilled expositor of mathematics. But this is not the point. At the end of Gross's talk in July I felt the distinct and vivid urge to explain to anyone who would listen that the "essence of mathematical thought" consists in making the world a place where such a lecture is possible. I can attest, on the basis of my experience while preparing this essay, that the pleasure occasioned by Gross's narrative does not diminish upon successive rereadings of his slides. It would be futile to pretend to know that no "Intelligent Computer" will ever be capable of creating such an effect in mathematics. But the bar is very high.
The subtitle describes this lecture as "dramatic and suspenseful." I was tempted to use the word "extraordinary," but that would suggest that it was unusually good for a Gross lecture. Since all his lectures are extraordinary in that sense the word loses its meaning. There was something special, however, about the way he introduced successive themes from different branches of mathematics and different historical periods in a way that I suspect none of his listeners would have anticipated, but that retrospectively feels absolutely natural. In that way it qualifies as literature in the sense expressed by Spivak's epigraph, and my goal today is to try to understand just what kind of literature it was, and why it exemplifies the kind of Intelligent Mathematics of which Intelligent Computer Mathematics can only dream.
In fact, you can watch Gross’s July 2021 talk and decide for yourself.
It would take too long to list and explain his accomplishments in number theory and allied areas of algebraic geometry and representation theory for readers who are not mathematicians; those who are have no need of such a list. For the purposes of this essay, I will just mention that his most famous contribution was undoubtedly his 1986 paper with Don Zagier on elliptic curves, which is the starting point for most subsequent developments on the Birch-Swinnerton-Dyer (BSD) Conjecture, a key focus of attention of number theorists around the world for nearly 60 years, and also the inspiration for several new areas of research that were largely created by Gross and his students and collaborators. Elliptic curves will reappear several times during this essay and there will be a long discussion of BSD in a future installment.
Roughly, the theory of solutions to equations in two variables.
Specifically, the action of G on the holomorphic differentials on X
I have to protest, on behalf of my colleagues who live there, that Gross was being unfair to Bourg-la-Reine.
Vorlesungen über die Theorie der elliptischen Modulfunctionen (1890/1892) and Vorlesungen über die Theorie der automorphen Functionen s1897/1912).
(very) roughly, the solutions to equations in three variables.
The final item on Propp’s list is Wedding.