Enumeration Problems in Algebraic Geometry and Representation Theory

这个研究项目涉及代数几何，它试图用几何来描述代数方程的解，以及表示理论，这是对对称性的系统研究。尽管它们具有抽象的性质，但这两门学科中许多问题的答案都归结为能够计算某些数字。本研究项目的目的是利用代数几何和交换代数的新技术将这些数字重新转换为组合量，使它们更容易用计算机理解。通过利用与其他科学领域的概念联系，本研究还将促进对数学物理和数学生物学中几个问题的理解。本科生直接作为合作者参与项目，为他们提供高等数学主题和3D打印技术在数学研究中的应用方面的培训。主束和分支变体的模空间的代数几何自然产生了两个有趣的枚举问题：计算约化群映射的分支多重性，以及从共形场理论的wessw - zumino - novikov - witten模型中求出共形块空间的维数。本研究旨在利用牛顿-奥孔科夫体理论和快速发展的伯科维奇几何领域进一步理解这些量。这些理论将用于提供新的多面体描述的共形块和分支多样性，以及进一步的理解拓扑和辛几何的空间所考虑的。

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.