Topology of the Moduli Spaces of Stable Curves and Stable Maps

小行星9803553 在这个项目中，研究人员正在研究稳定曲线的模空间及其近亲，稳定映射的模空间。 这些空间在数学的各个领域都扮演着重要的角色，包括拓扑学、代数几何、几何分析和量子场论。 例如，在代数几何中，这些空间在解决枚举问题时已经得到了富有成效的探索。 在这个项目中，研究者正在研究稳定曲线和稳定映射的模空间上的相交数。 他打算利用这项研究，以了解自然变形的某一类上同调场论。 研究者还研究了稳定映射的模空间的拓扑，这本身就很重要。 即使是一个特殊的情况，稳定曲线的模空间的情况下，已经产生了很多有趣的数学。 在过去的十年里，数学和物理之间的相互作用提供了许多美丽的，有时是意想不到的结果和启示。 特别是，几何学吸收了物理学中弦理论的许多思想和直觉。 弦理论将粒子视为一个小环而不是一个点。 当一个粒子在时间中运动时，它会描绘出一个表面。 因此，理解曲面上的几何，这一直是数学家感兴趣的，在物理学中也变得很重要。 在这个项目中，研究人员正在研究包含给定表面上所有几何信息的空间。 他还研究了生活在给定空间内的表面上的几何形状。 例如，可以计算在给定空间内满足某些约束的曲面的数量。 令人惊讶的是，这个主题在物理学中的起源是拓扑引力。 ***

9803553 Kabanov In this project the investigator is studying the moduli spaces of stable curves and their close cousins, the moduli spaces of stable maps. These spaces play an important role in various areas of mathematics, including topology, algebraic geometry, geometric analysis, and quantum field theory. For example, in algebraic geometry these spaces have been fruitfully explored in solving enumerative problems. In this project, the investigator is studying intersection numbers on the moduli spaces of stable curves and stable maps. He intends to use this study in order to understand natural deformations of a certain class of cohomological field theories. The investigator is also studying the topology of moduli spaces of stable maps, which are important in their own right. Even a particular case, the case of the moduli spaces of stable curves, has generated a lot of interesting mathematics. For the past decade, the interaction between mathematics and physics has provided many beautiful, and sometimes unexpected, results and conjectures. In particular, geometry has incorporated many ideas and much intuition from string theory in physics. String theory treats a particle as a small loop rather than a point. When a particle moves in time it traces out a surface. Therefore understanding geometries on surfaces, which has always been of interest to mathematicians, has also become important in physics. In this project, the investigator is studying the spaces that incorporate information about all geometries on a given surface. He is also studying geometries on surfaces living inside a given space. For example, one can count the number of surfaces satisfying certain constraints inside the given space. Amazingly, the origin in physics of this subject is topological gravity. ***