Moduli spaces in algebraic geometry

研究工作在代数几何，虽然他的兴趣连接到其他领域的数学，包括拓扑学，组合学，物理学（弦理论），数论，辛和微分几何。这个建议，继续各种股的调查工作，涉及模空间和相关的概念，在各种设置。 特别是，该建议涉及一些基本问题，关于模空间的曲线，变形理论，舒伯特演算，几何不变理论，格罗莫夫-威滕理论，和fundamental groups. Algecompletics提供了一个强大的理论，以确定模空间的有趣的对象。 一旦它们被定义，自然令人信服的问题是：它们看起来像什么？ 它们是不可约的吗？ 有关系？ 什么维度？ 它们光滑吗？如果不是，会出现什么样的奇点？它们的上同调环表现出什么样的结构，为什么在几何学上应该这样？ 他们的方程式是什么？ 调查员打算在一些案件中解决许多这些基本问题。 调查员有一个跟踪记录的持续和认真的努力都在推广到学生在各级（高中，本科和研究生），并在建设机构，其中代数几何可以增长。调查员将继续吸引研究生进入代数几何，无论是在斯坦福大学和其他地方，并继续培养研究生，博士后和年轻的研究人员的职业生涯。 研究人员将继续与大量的学生在中学和本科水平，吸引学生进入数学科学。

The investigator works in algebraic geometry, although his interests connect to other areas of mathematics, including topology, combinatorics, physics (string theory), number theory, and symplectic and differential geometry. This proposal, continuing various strands of the investigator's work, deals with moduli spaces and related notions in a variety of settings. In particular, the proposal deals with a number of fundamental questions regarding the moduli space of curves, deformation theory, the Schubert calculus, Geometric Invariant Theory, Gromov-Witten theory, and fundamental groups.Algebraic geometry provides a powerful theory with which to define moduli spaces of interesting objects. Once they are defined, natural compelling questions are: what do they look like? Are they irreducible? Connected? Of what dimension? Are they smooth? If not, what singularities arise? What structure is exhibited by their cohomology rings, and why should it be geometrically expected? What are their equations? The investigator intends to address many of these fundamental problems in a number of cases. The investigator has a track record of sustained and serious effort both in outreach to students at all levels (high school, undergraduate, and graduate), and in building institutions in which algebraic geometry can grow. The investigator will continue to attract graduate students into algebraic geometry, both at Stanford and elsewhere, and to continue to nurture the careers of graduate students, post-docs, and young researchers. The investigator will continue to work with large numbers of students at the secondary and undergraduate levels, attracting students into the mathematical sciences.