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，和表示论，这是对对称性的系统研究。尽管这两门学科都很抽象，但它们的许多问题的答案都归结为能够计算某些数字。这个研究项目的目的是使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，这项研究也将促进对数学物理和数学生物学中几个问题的理解。本科生作为合作者直接参与该项目，为他们提供高级数学主题的培训以及在数学研究中使用3D打印技术。主丛和分支簇的模空间的代数几何自然产生两个有趣的枚举问题：计算约化群映射的分支重数，并从共形场论的Wess-Zumino-Novikov-Witten模型中找到共形块空间的维数。本研究的目的是利用牛顿-奥肯科夫体理论和快速发展的布氏几何领域来进一步理解这些量。这些理论将被用来提供新的多面体描述的共形块和分支的多重性，以及进一步了解的拓扑和辛几何的空间正在考虑。

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.