New Directions in the Theory of Moduli

格罗莫夫-威滕不变量在弦理论中很重要，弦理论是一种物理理论，根据该理论，基本粒子的行为像微小的弦，而不是像牛顿以来所认为的那样像点。弦理论是所谓万物理论的最佳候选，万物理论统一了所有的物理理论，尤其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不变量有关联。这种联系仍然是推测性的，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，我发现了一些关于Donaldson-Thomas不变量的令人惊讶的事情。它们表现得像欧拉特征。形状的欧拉特性是一个数字，如果形状像橡胶一样变形，它也不会改变。例如，球面具有欧拉特性2。甜甜圈形状的表面具有欧拉特性0。Donaldson-Thomas不变量是某种欧拉特征这一事实对Gromov-Witten不变量以及弦理论都有重要的影响。******本研究的一个目标是更深入地理解这些欧拉特征，并使它们成为更灵活的工具。它们不应该只是数字，而是从一些更复杂的结构中抽象出来的数字。数字是用来计算东西的，但在计算的过程中，东西本身就丢失了。我们的目标是发现那些被欧拉特征所计数的东西这些特征产生了Donaldson-Thomas不变量。******模空间是多维几何形状，每个点对应一个几何对象。例如，有一个三角形的模空间达到相似度。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被分类到模空间中。实际上，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。这揭示了数值不变量，并加深了我们对物理理论（如弦理论）背后的数学的理解。******本研究的一个特定目标是将我们所学到的模空间应用于数论。数论和几何之间有一个令人信服的类比，例如，质数对应于结。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.