Child’s Play

I hadn’t seen an exercise in silliness of this magnitude in a while. The Wall Street Journal blared, on its front page, that “A CHESS NOVICE CHALLENGED MAGNUS CARLSEN. HE HAD ONE MONTH TO TRAIN.” My eyes were already rolling. “You fucking serious?” was the first question I asked. The second one was, “How badly did he lose?”

Badly, it turns out. Self-styled speed-learner Max Deutsch blundered a piece on move 12. It’s not quite a move someone who’s never played chess before would make—but it’s close. In fact, it’s just about the type of move someone who’s played for 30 days would make. By move 14, the game was essentially lost.

Board

On first glance, Max’s 12. Qf3 appears merely useless. But further study reveals that it’s problematic.12….Qh4 threatens a bad attack, which is addressed with 13. h3. But the queen on h4 also looks at d4, a threat which is discovered after 13…Nxe4. To make matters worse, Max recaptures with 14. Qxe3 instead of fxe3, putting him down a whole piece, instead of just a pawn, after 14…Bxd4.

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Alien Languages

The recent movie Arrival treats an imagined arrival on earth by alien beings. The United States government, at a loss to understand the visitors’ intentions, conscripts the film’s hero—unusually for Hollywood, a linguist—to help understand the aliens’ language, and in turn, their purpose.

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The aliens’ language’s “freedom from time” evokes the functional programming language Haskell.

The linguist, Louise Banks, soon makes headway. She discovers that the aliens’ language “has no forward or backward direction” and “is free of time”. Moreover, in a nod to the (unfortunately, all-but discredited) Sapir–Whorf hypothesis—according to which, as Banks suggests, “the language you speak determines how you think and… affects how you see everything”—Banks soon finds her own cognition shifting:

If you learn it, when you really learn it, you begin to perceive time the way that they do, so you can see what’s to come. But time, it isn’t the same for them. It’s non-linear.

Far from inducing an reaction of incredulity and awe, these descriptions of the visitors’ language provoked in me just one persistent response: “This is just like the programming language Haskell.” Continue reading

Rabbinic Mathematics

יַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ ׳וּקְוֵה׳ ״וְקָו֙״ שְׁלֹשִׁ֣ים בָּֽאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃
מלכים א 7:23

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
I Kings 7:23

This Hebrew Bible passage from I Kings—along with a similar one from II Chronicles—forms the biblical basis for Talmudic scholar Matityahu Hacohen Munk’s suggestion that “some of the geometrical rules did not hold in King Solomon’s temple,” a heavenly ‘‘world of truth’’ beyond our own, mathematical historians Tsaban and Garber write [1].

What’s so heavenly about the Molten Sea, a putative basin created by King Solomon in the ancient Temple of Jerusalem for ritual ablution? And why do the Rabbis Johanan and Papa discuss it extensively in the Babylonian Talmud, bickering in particular about its brim—“[as thin as] the flower of a lily… a handbreadth thick… wrought like the brim of a cup” [2, Eruvin 14a:29-31]?

The simple answer is that this particular snippet of the Word of God contains an oddity, asserting that this circular basin’s circumference is thrice its diameter—or that the geometrical constant π, rather than an irrational number, with an infinite and unpredictable decimal expansion, is in fact rational, and indeed an integer—the number 3, to be exact. Continue reading

A Mathematical Daydream

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“Those limits in certain functor categories that cannot be computed pointwise? They don’t actually exist,” one of us declared authoritatively. We, all math grad students, sat on an apartment floor, in a circle, with cards scattered over the floor.

Blank looks abounded. “Meaning, there don’t actually exist monic natural transformations some of whose components are not monomorphisms,” he clarified.

We had invented a variant of a popular card game called “spies vs. revolutionaries” – we called it “students vs. professors” – in which, in particular, each new round was heralded by its leader’s presentation of an established mathematical truth that he or she had decided we were to overturn for good. “The Banach-Tarski paradox is still true, but requires using at best six pieces, not five,” another student later suggested. “2 isn’t actually a prime,” one student blustered, citing the integer’s pathological character in many number-theoretic environments.

Soon it was my turn. “The Hodge Conjecture is false,” I fibbed, “and a counter-example is provided in my paper.”

A good laugh and a general readiness to proceed with the game cut my monologue short. Ridiculous as it was, though, I was ready to continue. The matter was one to which I had given some thought. Indeed, my work presents an interesting testing ground for a few of the ideas surrounding the Hodge Conjecture, and in particular seems to invite a heuristic argument whereby it could be used to furnish a counter-example to the conjecture. I’ll explain this mathematical daydream, and how it can be ultimately debunked. Continue reading

The Valley

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“I see a massive valley before me. The only question is… Should I cross it?”

This is how, about three years ago, and living in Moscow, I posed to a friend the dilemma of whether I should attend graduate school for math.

The analogy held up well. I often envisioned my progress through the program in terms of the same imposing valley. “The brambles are getting so thick, at this depth, that the light which once flowed liberally from the surface is becoming damp and attenuated,” I once thought, as I entered the later stages of the notorious first-year program in algebra and analysis. “I’m so far from charted land that I’m encountering species wholly unknown to the outside world,” I thought later, during my second year – as I first began to encounter Hodge Theory – envisioning passing into a dark forest where marvelous, strange creatures chirped and glowed.

Once in while I’d perceive myself on an unexpected trail towards a promising discovery. “It’s as if I’ve glimpsed light on the other side much earlier than I had anticipated,” I’d think.

The analogy seems to have lost some of its power now. I’ve made my way to the other side, sure. But much of the valley remains unexplored – by me or by anyone else. Continue reading

The Hardest Conjecture

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

Among the so-called Millenium Prize Problems – seven notoriously difficult mathematical problems, each open for decades, and each now carrying, courtesy of the Clay Mathematics Institute, a million-dollar prize – Kieth J. Devlin places the Hodge Conjecture last. “[A]n author should delay as long as possible introducing anything is likely to make his reader give up in despair,” Devlin writes in his book, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of our Time. “There is no… path even to the problem’s front door.”

The Hodge Conjecture fascinated me even before I began studying mathematics. Its sheer inscrutability surely played a role. I sensed, behind the incomprehensible words and symbols of its Wikipedia page, a bafflingly deep, and coherent, world. I had to understand.

The problem’s formidability was only partly to blame. This world – which I envisioned – was not just expansive, but beautiful. I perceived, there, something like Dante’s “music of the heavenly spheres”.

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French artist Gustave Doré’s depictions of Dante’s Paradise.

My graduate study has taken me into a field of math relatively close to that which the Hodge Conjecture occupies. As I’ve explored the rich foothills of this towering mountain, its peak has become even more stunning and mysterious.

I will try to explain the conjecture to this blog’s lay readers. Continue reading

Game of Theorems

This article is part of a series on The Structure of Theorems. See also:
1. Theorems’ Almanack; 2. The Greatest Theorem; 3. Game of Theorems

Few (currently) practicing mathematicians – in my experience – deign to concern themselves with issues surrounding set theory and the foundations of mathematics. In these areas reside the very definitions upon which the rest of our discipline rests; in any case, our discipline proceeds nonetheless, despite its practitioners’ regrettable ignorance. Mathematicians are pragmatic people. In 1949, Bourbaki – perhaps apprehending a subtle need to defend itself – titled a paper Foundations of Mathematics for the Working Mathematician [1].

This state of affairs, unfortunate as it is, explains the surprise and intrigue I often feel when I take time to explore foundational issues. The definition of the so-called Axiom of Determinacy particularly struck me. This set-theoretic axiom is formulated in terms of a certain type of two-player game – an infinite sequential game, in fact, in which two players take turns playing integers, leading ultimately to a sequence of integers of infinite length. The axiom (indeed, it’s something we might choose to suppose) states that every such game – that is, every choice of a victory set, a distinguished collection of possible infinite sequences whose members define the winning outcomes for the first player – is determined, in the sense that one player or the other in the game has a dominant strategy.

I’ll explain these terms below. The important thing, here, is that this mathematical property is defined by the existence of dominant strategies for a certain class of two-player games. This intrusion of an apparently economic, or game-theoretic, notion – that of the two-player game – into mathematics surprised me.

This intrusion, in retrospect, should have been less than surprising. Continue reading