One of the most exciting moments in mathematical research occurs when two seemingly unrelated theories come into contact and suddenly it seems as if someone has turned on the light and everything is much clearer. The main contribution of June Huh (1983, Stanford, California, United States), one of the four recipients of the 2022 Fields Medals, is one of these stellar moments in mathematics. In the words of the Fields Medals committee, the prize has been awarded for, among other notable achievements, “contributing the ideas of Hodge’s theory [una rama de la llamada geometría algebraica] to combinatorics”.
Algebraic geometry was precisely what captivated Huh to the point of making him abandon his career as a science journalist to pursue mathematics. Huh showed no interest in the latter discipline during his childhood and early youth. His father tried to teach him math with an exercise book, but Huh just copied the solutions at the end of the text. During high school, his dream was to become a great poet. Not getting the recognition he expected in this field, he began his university studies in Seoul (South Korea) with the intention of becoming a science journalist, specializing in physics and astronomy.
It took him six years to finish his degree, fortunately, because in his last year, Heisuke Hironaka (who received the Fields medal in 1970) was invited to teach a course in algebraic geometry at Seoul National University. The course raised great expectations at the university and more than 200 students enrolled. However, Hironaka, instead of giving a conventional course —where all the details are worked out and fit perfectly—, a product already packaged —and, we could say, a product that is already dead—, gave a course live about the research I was conducting at the moment, with all the inconsistencies, setbacks, twists and turns that this entails. After a week, of the 200 registered, only five remained. One of them, Huh, was fascinated by the freedom and mystery of true mathematics, in which he discovered the absolute poetry, liberated from the ego, that he had sought so much.
The following year, Huh decided to pursue a Ph.D. in mathematics in the US. With an unremarkable resume, he was rejected by most of the universities to which he applied. Only the University of Illinois at Urbana-Champaign accepted him for his master’s studies. During the first two years, Huh solved the Hoggar Conjecture, which had been open for 40 years.
This conjecture is about the so-called chromatic polynomialwhich describes of how many ways can a graph be colored with a given number of colors. In the 1960s, mathematician Donald Read observed that coefficients of this polynomial they always seemed to fulfill a curious property: first they increased, until they reached a maximum, and then they decreased. This characteristic receives the name of unimodal; Read surmisedtherefore, that all chromatic polynomials are unimodal.
Shortly after, Stuart G. Hoggar refined this conjecture, stating that the coefficients of the chromatic polynomial were not only unimodal, but also fulfilled a more demanding property, called logarithmically concave. This means that the square of any coefficient is always greater than or equal to the multiplication of its two adjacent terms..
Huh proved this conjecture with a new and totally unexpected method, based on tools from Hironaka’s course. That job opened the doors for him to the University of Michigan (USA), where he received his doctorate under the direction of Mircea Mustaţădelving into the relationship between combinatorics and algebraic geometry.
Huh’s great contribution to mathematics is not a concrete result, but is precisely a new way of looking at combinatorics, using ideas from algebraic geometry. In particular, it uses a structure that is born in algebraic geometry, called the Hodge-Riemann bilinear relations. Huh discovered a new beginning which can be summed up in the following sentence: behind every family of logarithmically concave numbers (such as the coefficients of the chromatic polynomial) there is a variant of the bilinear Hodge-Riemann relations that explains this property and the job is to find it.
This unexpected bridge between disciplines has guided his research from then on and has allowed him to prove important conjectures such as the Dowling-Wilson conjecture for geometric lattices, and the Heron-Rota-Welsh conjecture for matroids.
Working on a problem related to this last conjecture, Huh, together with Federico Ardila, Graham Denham, wrote a scientific article of more than 50 pages. Everyone seemed satisfied except Huh, who proposed to “find a more elegant argument.” They discarded the work done, started over and two years later they found the key. found that beauty that Huh was looking for and that, without a doubt, determines his life trajectory.
Jose Ignacio Burgos He is a researcher at the Higher Council for Scientific Research (CSIC) at the Institute of Mathematical Sciences (ICMAT)
Timon G Longoria Agate is coordinator of the Mathematical Culture Unit of the ICMAT.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
Edition and coordination: Agate A. Timón G Longoria (ICMAT).
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