Moduli Spaces in Algebraic Geometry

代数几何提供了一个强有力的理论来定义模空间（几何问题的解决方案的空间）的有趣的数学对象。一旦它们被定义，关于它们的几何形状和拓扑结构自然会有一些令人信服的问题。他们长什么样？ 它们是不可约的，相连的，在什么维度上？它们光滑吗？如果不是，会产生什么样的奇点？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。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。 研究员还将继续与大量的学生在中学和大学阶段，吸引学生进入数学科学.研究员的工作在代数几何，虽然他的兴趣连接到其他领域的数学，包括拓扑学，组合数学，物理（弦理论），数论，辛几何和微分几何.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的基础“热带”几何，稳定的模空间中的Grothendieck环，研究K3表面通过椭圆纤维化，和拓扑结构的各种模空间的曲线。

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.