New Directions in Gromov-Witten and Donaldson-Thomas theory

Gromov-Witten不变量在弦论中很重要，根据弦论的物理理论，基本粒子是微小的弦，而不是像牛顿那样认为的点。弦理论是所谓的万物理论的最佳候选者，万物理论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。 最近发现Gromov-Witten不变量与所谓的Donaldson-Thomas不变量有关。事实上，这种联系仍然是一个猜想，但它非常重要，因为唐纳森-托马斯不变量可以解释Gromov-Witten不变量所表现出的许多奇怪现象。 在我最近的研究中，我发现了一些关于唐纳森-托马斯不变量的令人惊讶的东西。它们表现得像欧拉特征线。形状的欧拉特征线是一个不变的数字，如果形状变形，就好像它是由橡胶制成的。例如，球面的欧拉特征线为2。圆环形状的表面具有欧拉特征0。唐纳森-托马斯不变量是某些类型的欧拉特征线，这一事实对格罗莫夫-威滕不变量以及弦论都有重要意义。 未来的目标是更深入地理解这些欧拉特征，并使其成为更灵活的工具。它们不应该只是数字，而应该是一些更复杂的结构。数字是用来数东西的，但在数的过程中，东西会丢失。目标是发现由欧拉特征“计算”的“事物”，这些特征会产生唐纳森-托马斯不变量。

Gromov-Witten invariants are important in String Theory, the physical theory according to which elementary particles are 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 the so-called Donaldson-Thomas invariants. This connection is, in fact, still a conjecture, but it is very important, because Donaldson-Thomas invariants can explain many 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 was 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. The goal for the future is to understand these Euler characteristics more deeply and make them a more flexible tool. They should not just be numbers, but some more complicated structure. Numbers are for counting things, but the things 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.