New Directions in the Theory of Moduli

Gromov-Witten不变量在弦理论中很重要，这是一种物理理论，根据该理论，基本粒子的行为像微小的弦，而不是自牛顿以来被认为的点。弦理论是所谓的万物理论的最佳候选者，它统一了所有的物理理论，特别是爱因斯坦的相对论和量子场论。*最近，人们发现Gromov-Witten不变量与Donaldson-Thomas不变量有关。这种联系仍然是猜测，但它非常重要，因为Donaldson-Thomas不变量可以解释Gromov-Witten不变量所表现出的许多奇怪现象。在我最近的研究中，我发现了关于Donaldson-Thomas不变量的一些令人惊讶的事情。他们的行为就像是欧拉的特征。形状的欧拉特征是一个数字，如果该形状被变形，就像它是由橡胶制成的一样，该数字不会改变。例如，球体的表面具有欧拉特性2。甜甜圈形状的表面具有欧拉特性0。Donaldson-Thomas不变量是某些类型的欧拉特性，这一事实对Gromov-Witten不变量也有重要的影响，因此也适用于弦理论。*本研究的目的之一是更深入地了解这些欧拉特性，使其成为更灵活的工具。它们不应该仅仅是数字，而应该是从某种更复杂的结构中抽象出来的数字。数字是用来计算事物的，但事物本身在计算过程中丢失了。其目的是发现由产生Donaldson-Thomas不变量的欧拉特征所计数的事物。*模空间是多维几何形状，其每个点对应于一个几何对象。例如，有一个由三角形组成的模空间，直到相似为止。这个模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成模空间。事实上，Gromov-Witten不变量和Donaldson-Thomas不变量是与各种模空间相关的数。模空间反映了被分类对象的对称性和变形行为等属性。*本研究的主要目的是研究模空间的几何。这阐明了数值不变量，并加深了我们对诸如弦理论等物理理论背后的数学的理解。*这项研究的一个特别目标是将我们所学到的关于模空间的知识应用于数论。例如，在数论和几何之间有一个令人信服的类比，素数对应于纽结。利用这个类比将阐明数论中的微妙问题。

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.