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

Advertisements

Great Expectations and Supermen

Dickens, Nietzsche, and the science of a better life

Churchyard.jpg

One of the most memorable scenes of Great Expectations, as illustrated in the original 1861 text

Pip, of Dickens’s Great Expectations, was set to have a normal childhood and to lead a happy existence, albeit a humble one, until Estella came along. He had a job ready for him in Joe’s forge; he had a father-figure, a mentor, and a friend, in Joe; he had a faithful friend, even a prospective romantic companion, in Biddy. He took happiness even from—indeed, only from—the simplest of things.

For example, in their nightly eating of bread and butter by the hearth, Pip and Joe shared in an amusing ritual.

In our already-mentioned freemasonry as fellow-sufferers, and in his good-natured companionship with me, it was our evening habit to compare the way we bit through our slices, by silently holding them up to each other’s admiration now and then,—which stimulated us to new exertions.

They did all this while trying to avoid the wrath of Pip’s tyrannical older sister. The reader comes to look back with fondness on a time when avoiding Mrs. Joe’s temper was the greatest of Pip’s troubles.

Everything changes when Pip meets Estella, the gorgeous but ice-hearted daughter of Miss. Havisham, a reclusive, mysterious old rich woman, at a mansion in the nice part of town.

Over a game of beggar my neighbor: “He calls the knaves Jacks, this boy!” said Estella with disdain, before our first game was out. “And what coarse hands he has! And what thick boots!”

Back home, Pip broods over his hands and his boots.

I took the opportunity of being alone in the courtyard to look at my coarse hands and my common boots. My opinion of those accessories was not favorable. They had never troubled me before, but they troubled me now, as vulgar appendages. I determined to ask Joe why he had ever taught me to call those picture-cards Jacks, which ought to be called knaves. I wished Joe had been rather more genteelly brought up, and then I should have been so too.

Thus begins Pip’s obsession with becoming a “gentleman”. He’d like to wear the finest clothes; to become literate and read the best books; to associate with the most refined of people. Life in the kitchen and forge was good enough for Pip, until it wasn’t.

Yet when Pip comes into a fortune and starts a new life in London, all is not always splendid. Worse than his daily troubles and trifles is the fact that his once-easygoing relationship between Joe becomes stilted and forced. Back in rural Kent, now-gentlemanly Pip has dinner with his uncultivated companions:

Soon afterwards, Biddy, Joe, and I, had a cold dinner together; but we dined in the best parlor, not in the old kitchen, and Joe was so exceedingly particular what he did with his knife and fork and the saltcellar and what not, that there was great restraint upon us.

And their easygoing relationship of the past seems distant and inaccessible.

“Mr. and Mrs. Hubble might like to see you in your new gen-teel figure too, Pip,” said Joe, industriously cutting his bread, with his cheese on it, in the palm of his left hand, and glancing at my untasted supper as if he thought of the time when we used to compare slices.

Despite all the frills of life with London’s upper crust, Pip can’t help but wonder at times, “with a weariness on my spirits, that I should have been happier and better if I had never seen Miss Havisham’s face, and had risen to manhood content to be partners with Joe in the honest old forge.”

Great Expectations gives rise to a question I myself have considered quite a bit: is it necessarily better to eat better food, to see better plays, and to dine with people who have better manners? Or might one rationally opt to gain pleasure from baser sources? If a college student gains as much pleasure from Burger King as a chef does from French haute cuisine, is not the student better off in this regard? Is Pip wise to pursue a life of excellence, at the expense of the simple things which once gave him pleasure? 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”.

Screen Shot 2016-12-16 at 11.19.07 AM.png

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