New Directions in the Theory of Moduli

模理论的新方向

• 批准号: RGPIN-2017-04156
• 负责人: Behrend, Kai
• 金额: $ 3.35万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=644992
• 关键词: New; Directions; Theory; Moduli

Gromov-Witten invariants are important in String Theory, the physical theory according to which elementary particles behave like tiny strings, instead of points, as was believed since Newton. String Theory is the best candidate for the so-called Theory of Everything, which unifies all physical theories, in particular Einstein's theory of relativity and quantum field theory.******Recently, it was discovered that Gromov-Witten invariants are related to Donaldson-Thomas invariants. This connection is still conjectural, but it is very important, because Donaldson-Thomas invariants can explain many of the strange phenomena exhibited by Gromov-Witten invariants.******In my recent research I discovered something surprising about Donaldson-Thomas invariants. They behave like Euler characteristics. The Euler characteristic of a shape is a number which does not change if the shape is deformed as if it were made of rubber. The surface of a sphere, for example, has Euler characteristic 2. The surface of a donut shape has Euler characteristic 0. The fact that Donaldson-Thomas invariants are certain kinds of Euler characteristics has important consequences, also for Gromov-Witten invariants and hence for String Theory.******One goal of this research is to understand these Euler characteristics more deeply and make them a more flexible tool. They should not just be numbers, but numbers abstracted from some more complicated structure. Numbers are for counting things, but the things themselves are lost in the process of counting. The goal is to discover the things which are counted by the Euler characteristics which give rise to Donaldson-Thomas invariants.******Moduli spaces are multidimensional geometric shapes, each of whose points corresponds to a geometric object. For example, there is a moduli space of triangles up to similarity. Each point of this moduli space represents one triangle shape. Many mathematical or physical objects can be sorted into moduli spaces. In fact, Gromov-Witten invariants and Donaldson-Thomas invariants are numbers associated to various moduli spaces. The moduli space reflects properties such as symmetries and deformation behaviour of the objects being classified. ******The main purpose of this research is to study the geometry of moduli spaces. This sheds light on the numerical invariants, and deepens our understanding of the mathematics underlying physical theories such as String Theory. ******A particular goal of this research is to apply what we have learned about moduli spaces to number theory. There is a compelling analogy between number theory and geometry, in which prime numbers correspond to knots, for example. Exploiting this analogy will illuminate subtle questions in number theory.

Gromov-witten的不变性在弦理论中很重要，这是物理理论，根据牛顿以来，基本粒子的行为像小字符串一样，而不是点，而不是点。 弦理论是所有事物的所谓理论的最佳候选者，它统一了所有物理理论，尤其是爱因斯坦的相对论和量子场理论。 这种联系仍然是猜想的，但非常重要，因为唐纳森 - 托马斯不变的人可以解释格罗莫夫 - 织机不变的许多奇怪现象。 他们的行为就像欧拉的特征。形状的欧拉特征是一个数字，如果形状变形，就好像是由橡胶制成的。 例如，球的表面具有欧拉的特征2。甜甜圈形状的表面具有欧拉的特征0。唐纳森 - 托马斯不变性是某些类型的欧拉特征具有重要的后果，对gromov-witter不变性剂也具有重要的后果。 它们不仅应该是数字，而且还应该从一些更复杂的结构中抽象出来。 数字是用于计算事物的，但是在计数过程中，事物本身就丢失了。目的是发现由欧拉（Euler）特征计算出来的事物，这些特征引起了唐纳森（Donaldson） - 托马斯（Donaldson） - 托马斯（Thomas）的不变。 例如，三角形的模量空间达到相似性。 该模量空间的每个点代表一个三角形。 许多数学或物理对象可以分类为模量空间。 实际上，格罗莫夫（Gromov）的不变性和唐纳森（Donaldson） - 托马斯（Donaldson-Thomas）不变性是与各种模量空间相关的数字。 模量空间反映了要分类的对象的对称性和变形行为等属性。 ******这项研究的主要目的是研究模量空间的几何形状。 这阐明了数字不变性，并加深了我们对诸如弦理论之类的物理理论基础数学的理解。 ******这项研究的一个特定目标是将我们了解的模量空间应用到数字理论。 数字理论和几何形状之间有一个令人信服的类比，例如，质数与结相对应。 利用此类比将在数字理论中阐明微妙的问题。