Moduli and quantization of Poisson varieties

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Everyday objects move according to the laws of "classical" physics, which are geometric in nature. For instance, a ball thrown in the air traces out a curve in space, and the ball's motion can be predicted by describing the shape of this curve. The modern abstract mathematical language used to describe these laws is called Poisson geometry, in homage to the pioneering contributions of Siméon de Poisson (1781-1840). In contrast, tiny objects like electrons obey the strange laws of "quantum" physics, which are algebraic rather than geometric. One of these laws, called Heisenberg's uncertainty principle, states that we cannot know both the position and the momentum of a particle simultaneously. If we try to measure one, we influence the measurement of the other, so the order of the measurements affects their results. In mathematical terms, the algebra of quantum physics is "noncommutative", meaning that "position times momentum" is different from "momentum times position". In the 1990s, the famous mathematician Kontsevich discovered a deep and mysterious formula that derives this quantum noncommutative algebra from the classical Poisson geometry. In addition to shedding new light on the relationship between classical and quantum physics, it provides a revolutionary tool for solving problems in pure algebra. However, it is extremely difficult to use in practice: it involves infinitely many terms, and for a long time nobody could even calculate the individual terms. Recently, though, my team made a breakthrough: we proved that the terms could be rewritten using special numbers called "multiple zeta values", and used this purely mathematical statement to produce the first-ever computer program for calculating the formula. Building on this success, the proposed research program has the ambitious aim of rendering Kontsevich's formula tractable for a wide class of classical geometries, called Poisson Fano varieties, which includes many important systems studied in mathematical physics. This will be approached through a number of parallel projects, in which my research group will continue our study of the numbers in Kontsevich's formula, answering long-standing conjectures about their properties; prove theorems about Poisson Fano varieties that elucidate their structure and their classification into families; and develop a concrete recipe for applying Kontsevich's formula to many of these families. This research involves a blend of cutting-edge techniques across a wide spectrum of pure mathematics, and represents a novel synthesis of the work that I have been doing in each of these areas during the past several years. This work is very much in the category of "pure" mathematics, and as such its primary benefit to Canadians is through the advancement of basic scientific knowledge, and through the training of the next generation of mathematicians, who will go on to serve Canadian society through teaching, research and private-sector work.