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Collector's Edition. Premium Edition. Special Edition. Ultimate Edition. Complete Edition. Genre see all. Platform see all. Sub-Genre see all. Rating see all. Guaranteed Delivery see all. No Preference. Some of the biggest leaps in understanding occur when someone extends a well-established theory in one area to seemingly unrelated phenomena in another.
Think, for example, about gravitation. People have always understood that objects fall to the ground when released from a height; the heavens became far more intelligible when Newton realized the same dynamic explained the motion of the planets.
In math the same kind of transplantation occurs all the time. But the derivative appears in other guises: as the slope of a line tangent to the graph of a function, or as the instantaneous speed given by a function at a specific time. Hodge theory was developed in the s by the Scottish mathematician William Hodge. The cohomology rings in Hodge theory arise from smooth functions that come packaged with a concept of the infinite.
By contrast, combinatorial objects like graphs and matroids are purely discrete objects — assemblages of dots and sticks. To ask what Hodge theory means in the context of matroids is a little like asking how to take the square root of a sphere. Yet there was good reason to ask. In the more than 60 years since Hodge theory was proposed, mathematicians have found a number of instances of Hodge-type structures appearing in settings far removed from their original algebraic context.
Some of these new settings have been combinatorial, which encouraged Huh to wonder whether relationships from Hodge theory might underlie these log concave patterns. Searching for a familiar math concept in a foreign land is not easy, though. In recent years Huh has done much of his most important work with two collaborators — Eric Katz, a mathematician at Ohio State University, and Karim Adiprasito, a mathematician at the Hebrew University of Jerusalem. They make an unusual trio.
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Katz has a frenetic mind and obsessively detailed knowledge of indie rock bands, developed in his earlier years as a college radio DJ. Of the three collaborators, he is the closest to having a typical math pedigree, and he views himself as a kind of interpreter between the creative ideas of the would-be poet and the would-be chef. But with nonrealizable matroids, there was no obvious way to instantiate a cohomology ring — they were like a language without a word for that concept.
For four years Huh and Katz tried and failed to find a way to define what a Hodge structure would mean in the context of nonrealizable matroids. In he traveled to IAS and visited Huh. I think they asked the question and I provided the technique.
This technique provided a full proof of the Rota conjecture. The trio posted their work online in November , and since then it has rippled through the math world. Their work provides a fully combinatorial vision of Hodge theory, which in turn provides a whole new way to approach open problems in combinatorics.
In addition to his new position at IAS, Huh is frequently mentioned as a strong contender for the Fields Medal, which is given every four years to the most accomplished mathematicians under the age of Hironaka was in the audience, and he recalls being surprised to learn that singularity theory had applications to graphs. Afterward, he asked Huh if this new work marked a change in his research interests. Huh remembers that conversation, too. It took place at a time when he was indeed setting out in a whole new direction in mathematics.
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Huh and Hironaka have not seen each other since. Hironaka is now In March he posted a long paper to his old faculty webpage at Harvard University that he says provides a proof. Other mathematicians, including Huh, have taken a preliminary look at the work but have not yet verified that the proof holds. He thought for a moment, then said he feels guilty. This has left a sort of long-term homework in me. At the same time, Huh regards the distance he has traveled from his mathematical roots as a good and maybe necessary step in the development of his work. Correction June 27, The original version of this article included an error in the calculation of the chromatic polynomial of a rectangle.
In fact, Hodge theory was developed in the s and Rota made his conjecture in This article was reprinted on Wired. Get highlights of the most important news delivered to your email inbox. It is customary to bow when greeting a person in Japan. When friends meet they usually bow to the same degree as their friends.
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