Moduli Spaces in Algebraic Geometry

尽管他的兴趣与其他数学领域有关，包括拓扑学、组合学、物理学（弦理论）、数论、辛几何和微分几何，但他的研究方向是代数几何。该方案延续了研究者工作的各个方面，处理了模空间和各种设置中的相关概念。特别是，该建议涉及一些基本问题，关于曲线的模空间，几何不变理论，模空间的拓扑结构，以及来自理论物理的推测性思想。代数几何提供了一个强大的理论来定义有趣对象的模空间。一旦它们被定义，自然就会出现令人信服的问题：它们是什么样子的？它们不可约吗？连接?什么维度的？它们光滑吗？如果不是，会出现什么奇点？它们的上同环表现出什么结构？为什么它在几何上是预期的？它们的方程是什么？研究者打算在一些情况下解决许多这些基本问题。研究者在与各级学生（高中、本科和研究生）的接触以及建立代数几何可以发展的机构方面都有持续和认真的努力。研究者将继续吸引斯坦福大学和其他地方的研究生研究代数几何，并继续培养研究生、博士后和年轻研究人员的职业生涯。研究者将继续与大量的中学生和大学生合作，吸引学生进入数学科学领域。

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, Geometric Invariant Theory, topology of moduli spaces, and speculative ideas coming out of theoretical physics.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.