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

Characteristic Classes

This story is part of a series entitled Leaving Mathematics. See also:
1. The Baltimore Snowstorm; 2. The Italian School; 3. Characteristic Classes

Rainer noticed the pattern halfway through July. He had constructed smooth surfaces in a certain four-dimensional smooth algebraic variety, using the Chern classes of vector bundles. He noticed that he could anticipate these surfaces’ Hodge numbers. “This seems to amount virtually to something like a non-existence result,” he wrote, later that day, in an email to a junior faculty member at another school.

Rainer soon sank into a deep depression. “I go full days without saying a single word,” he told Diego, over the phone. Continue reading

Ruminations

Jalāl ad-Dīn Rumi informs math and life.

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A page from Rumi’s Mathnawi.

“It’s very perverse, what you’re doing,” Professor Vyacheslav Shokurov said, one afternoon in his office chair, with a skepticism characteristic of the mathematician. “One usually begins with examples – examples not treatable by the existing theory. One then develops new theory.”

Shokurov peered at me for an instant, and then lifted his hands from his lap. “But you’ve developed the theory first. And now you go looking for examples.”

My mathematical research had been stalled for about two weeks, for exactly this reason. Continue reading

A Fragile Truth

My four-week psychiatry rotation at Western State Hospital landed smack in the middle of peak general election season. And, oddly enough, these two experiences have yielded remarkable similarities. In both cases, I have been forced to entertain various versions of the truth.

Many of the patients here at Western State are psychotic. Our known and stated goal, then, is to return these patients to reality-based thinking. Only then might they qualify for discharge. This exercise has presented philosophical challenges. Certainly, sometimes, our job is easy. One of our patients, who signs her forms as Michelle Obama Prince Harry Elizabeth Queen Zealand, communicates with Russia, Germany, Berlin, Jerusalem, East Germany, West Germany, South Germany, and Russia, by radio, television, and satellite, including the satellite in the backyard of her palace, which she built, and in which we currently reside. Another patient, though, gave me pause. Continue reading

Demons Within

If only Nicolas Winding Refn were as intent on my liking his films as I was.

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The first few scenes of Neon Demon offer the richness of experience that, as it seems, only Nicolas Winding Refn is capable of delivering. The protagonist (Elle Fanning), a gorgeous aspiring model from small-town Georgia, glides wide-eyed through the streets of LA in a red Mustang, driven by her new (and soon-to-be-ex-) boyfriend. America’s Next Top Model, after crossing America’s Former Top Model, comes home to a puma, prowling about her thrashed motel room. A haggard-looking Keanu Reeves strides out of the dingy, blue-lit motel lobby to smoke a cigarette in the red light of the Motel 6 sign. Indeed, this was quite something. Was Neon Demon going to be what Only God Forgives never was—the movie that could rival Drive?

Continue reading

The Italian School

This story is part of a series entitled Leaving Mathematics. See also:
1. The Baltimore Snowstorm; 2. The Italian School; 3. Characteristic Classes

“Of course you should go!” Professor Torino nodded honestly, smiling inexplicably. Torino hung like that for an instant, balanced in his chair. He relaxed suddenly, his smile vanishing. “Pieri is a good mathematician,” he continued. “We spent a summer together at the Institute in 1991.”

Professor Torino always seemed to like Josif, though Josif didn’t fully understand why. Continue reading

Taking Care to Take Care

The Genius of Drake. 

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In a previous post, I flagged Lil Wayne for his genius, largely because of his witty irreverence. In a world of wanna-be gangsters and braggarts, it somehow occurred to Lil Wayne that he didn’t have to act hard—he could be funny, instead.

Drake’s genius, then, becomes clear as well, for a similar but distinct reason. While his contemporaries extoll their kill counts, sexploits, and paychecks, Drake stands out in a sea of monotony for his ability to express his emotions. Just imagine: it’s 2009; Drake is rising to fame; and suddenly, it’s no longer uncool to feel. The reader will surely join me, then, in deeming Drake’s impact on the rap world revolutionary.

Continue reading