Enumeration Problems in Algebraic Geometry and Representation Theory

这项研究项目既涉及代数几何，它寻求用几何刻画代数方程的解，也涉及表示理论，它是对对称性的系统研究。尽管它们都是抽象的，但这两个科目中许多问题的答案都可以归结为能够计算某些数字。这项研究的目的是利用代数几何和交换代数的新技术，将这些数重新计算为组合量，使计算机更容易理解它们。通过利用与其他科学领域的概念联系，这项研究还将促进对数学物理和数学生物学中的几个问题的理解。本科生直接作为合作者参与该项目，为他们提供高等数学主题的培训和3D打印技术在数学研究中的应用。主丛和分枝变种的模空间的代数几何自然产生了两个有趣的计数问题：计算约化群映射的分枝多重数，以及从共形场论的Wess-Zumino-Novikov-Witten模型求共形块空间的维度。这项研究旨在利用牛顿-奥孔科夫体理论和迅速发展的伯克维奇几何学领域来进一步理解这些量。这些理论将被用来提供保形块和分支多面体的新的描述，以及进一步理解所考虑的空间的拓扑和辛几何。

This research project concerns both algebraic geometry, which seeks to characterize solutions to algebraic equations with geometry, and representation theory, which is a systematic investigation of symmetry. Despite their abstract nature, the answers to many questions in both of these subjects boil down to being able to compute certain numbers. The purpose of this research project is to use new techniques from algebraic geometry and commutative algebra to recast these numbers as combinatorial quantities, making them easier to understand with a computer. By utilizing conceptual connections to other scientific fields, this research will also advance the understanding of several questions in mathematical physics and mathematical biology. Undergraduate students are involved directly as collaborators in the project, providing them with training in advanced mathematical topics and the use of 3D printing techniques in mathematical research.The algebraic geometry of moduli spaces of principal bundles and branching varieties naturally produces two interesting enumeration problems: counting branching multiplicities of a map of reductive groups, and finding the dimension of the spaces of conformal blocks from the Wess-Zumino-Novikov-Witten model of conformal field theory. This research aims to further understanding of these quantities using the theory of Newton-Okounkov bodies and the quickly evolving field of Berkovich geometry. These theories will be used to provide new polyhedral descriptions of conformal blocks and branching multiplicities, as well as further the understanding of the topology and symplectic geometry of the spaces under consideration.