Moduli Spaces in Algebraic Geometry

代数几何提供了一个强大的理论，用它来定义有趣的数学对象的模空间(几何问题的解空间)。一旦定义了它们，就会有关于它们的几何和拓扑的自然令人信服的问题。他们是什么样子的？它们是不可约的，是相互联系的，是什么维度的？它们光滑吗？如果不是，会出现什么奇点呢？它们的上同调环展示了什么结构，为什么要在几何上预期呢？他们的方程式是什么？调查员打算在一些案件中解决这些基本问题中的许多。研究人员在接触各级学生(高中生、本科生和研究生)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记录。研究人员将继续吸引研究生进入代数几何，并继续培养研究生、博士后和年轻研究人员的职业生涯。调查员还将继续与中学和本科的大量学生合作，吸引学生进入数学科学。调查员从事代数几何方面的工作，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(弦论)、数论以及辛几何和微分几何。这一建议继续了研究者的各种工作，在各种背景下处理了模空间和相关概念。特别是，该建议涉及一些基本问题，涉及“热带”几何的基础，Grothendieck环中的模空间的稳定性，通过椭圆纤维研究K3曲面，以及各种曲线的模空间的拓扑。

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.