Moduli and quantization of Poisson varieties

日常生活中的物体都是按照“经典”物理学的定律运动的，这些定律在本质上是几何学的。 例如，一个球被抛向空中，在空间中划出一条曲线，通过描述这条曲线的形状，就可以预测球的运动。 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（1781-1840）的开创性贡献。 相比之下，像电子这样的微小物体遵循的是“量子”物理学的奇怪定律，这些定律是代数的而不是几何的。其中一个定律，称为海森堡的不确定性原理，指出我们不能同时知道粒子的位置和动量。如果我们试图测量一个，我们会影响另一个的测量，所以测量的顺序会影响它们的结果。在数学术语中，量子物理学的代数是“非对易的”，这意味着“位置乘以动量”不同于“动量乘以位置”。 20世纪90年代，著名数学家Kontsevich发现了一个深奥而神秘的公式，从经典的Poisson几何中导出了这个量子非交换代数。除了阐明经典物理和量子物理之间的关系外，它还为解决纯代数问题提供了革命性的工具。 然而，在实践中使用它是非常困难的：它涉及无限多的条款，很长一段时间没有人甚至可以计算个别条款。 不过最近，我的团队取得了突破性进展：我们证明了这些项可以用称为“多重zeta值”的特殊数字重写，并利用这种纯粹的数学陈述产生了有史以来第一个计算公式的计算机程序。 基于这一成功，拟议的研究计划有一个雄心勃勃的目标，使Kontsevich的公式易于处理的一类广泛的经典几何形状，称为泊松法诺品种，其中包括许多重要的系统研究数学物理。 这将通过一些平行的项目，在我的研究小组将继续我们的研究的数字Kontsevich的公式，回答长期以来关于他们的属性的问题;证明定理的泊松范诺品种，阐明他们的结构和他们的分类到家庭;和制定一个具体的食谱应用Kontsevich的公式，许多这些家庭。 这项研究涉及广泛的纯数学领域的尖端技术的融合，并代表了我在过去几年中在这些领域所做的工作的新颖综合。 这项工作在很大程度上属于“纯”数学范畴，因此，它对加拿大人的主要好处是通过提高基本科学知识，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.