Moduli and quantization of Poisson varieties

日常生活中的物体是按照“经典”物理定律运动的，这些定律在本质上是几何的。例如，一个抛在空中的球在空间中划出一条曲线，通过描述这条曲线的形状可以预测球的运动。用来描述这些定律的现代抽象数学语言被称为泊松几何，以纪念西蒙·德·泊松(Siméon de Poisson，1781-1840)的开创性贡献。相比之下，像电子这样的微小物体遵循“量子”物理学的奇怪定律，这些定律是代数的，而不是几何的。其中一条被称为海森堡测不准原理的定律指出，我们不可能同时知道一个粒子的位置和动量。如果我们试图测量一个，我们就会影响另一个的测量，所以测量的顺序会影响他们的结果。用数学术语来说，量子物理的代数是“非对易的”，意思是“位置乘动量”不同于“动量乘位置”。20世纪90年代，著名数学家康采维奇发现了一个深刻而神秘的公式，它从经典的泊松几何中推导出了这个量子非对易代数。除了揭示经典物理和量子物理之间的关系外，它还为解决纯代数中的问题提供了革命性的工具。然而，在实践中使用它是极其困难的：它涉及无限多的项，而且在很长一段时间里，甚至没有人能计算出单独的项。然而，最近，我的团队取得了突破：我们证明了这些项可以用称为“多个Zeta值”的特殊数字来重写，并使用这个纯粹的数学陈述产生了有史以来第一个计算该公式的计算机程序。在这一成功的基础上，拟议中的研究计划有一个雄心勃勃的目标，即使康采维奇公式适用于一大类被称为泊松-法诺族的经典几何，其中包括数学物理中研究的许多重要系统。这将通过几个平行的项目来实现，在这些项目中，我的研究小组将继续我们对康采维奇公式中的数字的研究，回答关于它们性质的长期猜测；证明关于Poisson Fano变种的定理，这些定理阐明了它们的结构和它们对家族的分类；并开发出一个具体的配方，将Kontsevich公式应用于许多这样的家族。这项研究涉及跨越广泛的纯数学的尖端技术的混合，并代表了我在过去几年中在这些领域所做的工作的新颖的综合。这项工作在很大程度上属于“纯”数学的范畴，因此它对加拿大人的主要好处是通过促进基本的科学知识，并通过培训下一代数学家，他们将继续通过教学、研究和私营部门的工作为加拿大社会服务。

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.