Puzzles, Quantum K-Theory, and Other Topics in Schubert Calculus

这个研究项目涉及到枚举几何一般领域的几个开放问题。本课题的激励目标是确定满足一系列条件的固定类型几何图形的数量。这些条件可能涉及图形包含给定的点，满足其他图形，或具有具有指定属性的切线。令人惊讶的是，虽然确定满足条件的解的精确列表是极其困难的，但通常可以预测这样的解的数量。对枚举问题的解的数量的公式的搜索已经揭示了许多领域感兴趣的深刻而美丽的组合结构，包括几何、组合学、表示理论、计算机科学中的复杂性理论和物理学中的镜像对称。PI将吸引研究生和本科生参与他的研究，目的是在数学方面，特别是在枚举几何方面招募新的人才。该PI目前是一名博士生的论文指导老师，另有两名学生最近在他的指导下完成了博士学位。在过去的8个夏季中，PI在罗格斯大学的本科生研究经验（REU）项目中监督了7个项目。获得的研究经费将使他能够继续这项活动。PI还将开发计算机软件以促进他所在领域的研究，他还将帮助组织会议和研讨会。关于标志流形中子变种的枚举几何的许多信息被编码在它的上同调环中，以及更一般的上同调理论中。舒伯特微积分的一个主要目标是得到这个环关于其舒伯特类基的乘法结构常数的公式。PI将尝试证明一个正组合公式，该公式将任何三步旗的上同调结构常数表示为可以使用拼图块列表创建的拼图数量。标志流形X的Gromov-Witten不变量计算在X上满足一般Schubert变分的有理曲线的个数。当无限多条曲线满足Schubert变分的一般构型时，这些曲线的集合形成一个称为Gromov-Witten变分的模空间。反过来，Gromov-Witten变型定义了k -理论Gromov-Witten不变量，这些不变量编码在x的量子k -理论环中。该环的研究为我们理解Gromov-Witten变型及其相关空间的奇异性和合理性提供了一个有用的基准。PI将试图回答几个关于量子k理论的正定性和有限性的开放性问题。本项目使用的方法来自几何学和组合学，以计算机实验为中心工具。

This research project concerns several open questions in the general area of enumerative geometry. The motivating goal in this subject is to determine the number of geometric figures of a fixed type that satisfy a list of conditions. These conditions could involve that the figures contain given points, meet other figures, or have tangents with specified properties. Surprisingly, while it can be extremely difficult to determine the precise list of solution figures that satisfy the conditions, it is often possible to predict the number of such solutions. The search for formulas for the number of solutions to enumerative problems has uncovered deep and beautiful combinatorial structures that are of interest in numerous fields, including geometry, combinatorics, representation theory, complexity theory in computer science, and mirror symmetry in physics. The PI will engage both graduate and undergraduate students in his research with the aim of recruiting new talent to mathematics in general and to enumerative geometry in particular. The PI is currently the thesis advisor of one Ph.D student, and two additional students have recently completed their Ph.D degrees under his supervision. The PI has supervised projects in the Research Experience for Undergraduates (REU) program at Rutgers University in 7 out of the past 8 summers. The awarded research grant will enable him to continue this activity. The PI will also develop computer software to facilitate research in his area, and he will help organizing conferences and workshops.Much information about the enumerative geometry of subvarieties in a flag manifold is encoded in its cohomology ring, as well as in more general cohomology theories. A major goal in Schubert calculus is to obtain formulas for the multiplicative structure constants of this ring with respect to its basis of Schubert classes. The PI will attempt to prove a positive combinatorial formula that expresses the cohomological structure constants of any three-step flag variety as the number of puzzles that can be created using a list of puzzle pieces. The Gromov-Witten invariants of a flag manifold X count the number of rational curves that meet general Schubert varieties in X when this number is finite. When infinitely many curves meet a general configuration of Schubert varieties, the collection of these curves form a moduli space called a Gromov-Witten variety. Gromov-Witten varieties in turn define K-theoretic Gromov-Witten invariants that are encoded in the quantum K-theory ring of X. The study of this ring provides a useful benchmark for our understanding of the singularities and rationality properties of Gromov-Witten varieties and related spaces. The PI will attempt to answer several open questions about positivity and finiteness properties of quantum K-theory. The methods to be used in this project come from geometry and combinatorics, with computer experiments as a central tool.