Degenerations and moduli

Alexeev博士将致力于代数几何中的一系列问题，这些问题的中心是退化、稳定簇和对的紧致模空间以及与最小模型程序的联系。稳定簇和稳定对是Deligne-Mumford-Knudsen稳定曲线的高维推广。这个项目的一些特殊主题是加权稳定超平面排列的模，一般类型的稳定曲面，以及交换变种和K3曲面的模空间的各种紧化的几何意义。代数几何是数学中最核心的分支之一，其目的是在实践和概念上理解多元多项式方程组的解。它在其他数学领域也有重要的应用，如数论、拓扑学、分析，以及物理、生物学、密码学和工程学。Alexeev博士将研究的特殊问题涉及模空间，这是一个可以追溯到Riemann的经典主题，特别是稳定曲线和映射的模空间，以及它们的高维类似物。其中一些起源于物理学，并可能反过来在那里找到应用。这笔赠款将用于研究生的培训，并将支持佐治亚大学一个活跃的代数几何项目。

Dr. Alexeev will work on a range of problems in algebraic geometry centering around degenerations, compact moduli spaces of stable varieties and pairs, and connections to the Minimal Model Program. Stable varieties and pairs are higher-dimensional generalizations of Deligne-Mumford-Knudsen's stable curves. Some particular subjects of this project are moduli of weighted stable hyperplane arrangements, stable surfaces of general type, and geometric meaning of various compactifications of moduli spaces of abelian varieties and K3 surfaces.Algebraic geometry is one of most central branches of mathematics which aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. It has important applications to other fields of mathematics, such as number theory, topology, analysis, as well as to physics, biology, cryptography, and engineering. The particular problems that Dr Alexeev will study involve moduli spaces, a classical subject going back to Riemann, and in particular moduli spaces of stable curves and maps, and their higher-dimensional analogues. Some of these originated in physics, and may in turn find application there. The grant will contribute to training of graduate students, and will support an active program in algebraic geometry at the University of Georgia.