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. ***
