Moduli Spaces in Algebraic Geometry

代数几何为有趣的数学对象定义模空间（几何问题解的空间）提供了一个强大的理论。一旦它们被定义，自然就会出现关于它们的几何和拓扑的令人信服的问题。它们长什么样子？它们是不可约的吗，是相连的吗，在哪个维度上？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。研究者将继续吸引研究生进入代数几何领域，并继续培养研究生，博士后和年轻研究人员的职业生涯。研究者还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。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，该提案涉及一些关于“热带”几何基础的基本问题，格罗滕迪克环中模空间的稳定性，通过椭圆纤维的K3表面的研究，以及曲线的各种模空间的拓扑结构。

Algebraic geometry provides a powerful theory with which to define moduli spaces (spaces of solutions of geometric problems) for interesting mathematical objects. Once they are defined, there are natural compelling questions about their geometry and topology. 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 and continue to nurture the careers of graduate students, post-docs, and young researchers. The investigator will also continue to work with large numbers of students at the secondary and undergraduate levels, attracting students into the mathematical sciences.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 foundations of "tropical" geometry, the stabilization of moduli spaces in the Grothendieck ring, the study of K3 surfaces through elliptic fibrations, and the topology of various moduli spaces of curves.