New directions in Gromov-Witten theory

Gromov-Witten不变量在弦理论中很重要，根据弦理论，基本粒子是微小的弦，而不是自牛顿以来被认为的点。弦理论是所谓万物理论的最佳候选者，它可以统一所有的物理理论，特别是爱因斯坦的相对论和量子场论。最近人们发现格罗莫夫-威滕不变量与所谓的唐纳森-托马斯不变量有关。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格罗莫夫-托马斯不变量所表现出的许多奇怪现象。在我最近的研究中，我发现了关于唐纳森-托马斯不变量的一些令人惊讶的事情。它们的行为类似于欧拉特性。例如，一个形状的欧拉特性是一个不变的数，如果该形状被变形，就好像它是由橡胶制成的。例如，一个球体的表面具有欧拉特性2。或者一个甜甜圈形状的表面具有欧拉特性0。Donaldson-Thomas不变量是欧拉特性的某些类型，这一事实对Gromov-Witten不变量也有重要的影响，因此对弦理论也是如此。未来的目标是更深入地理解这些Euler特性，并使其成为更灵活的工具。它们不应该只是数字，而是一些更复杂的结构。这些数字是用来计算事物的，但东西在这个过程中丢失了。我将发现由欧拉特性“计数”的“东西”，这些东西导致了Donaldson-Thomas不变量的产生。

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 the quantum field theory.Recently it was discovered that Gromov-Witten invariants are related to the so-called Donaldson-Thomas invariants.  This is 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.  I will discover the "things" which are "counted" by the Euler characteristics which give rise to Donaldson-Thomas invariants.