CAREER: Equivariant and Infinite-Dimensional Combinatorial Algebraic Geometry

对称性在我们对世界的数学理解中扮演着重要的角色。有离散的对称，如镜子中一张脸的反射或地砖的六角形图案；还有连续的对称组，如球体围绕不同轴的旋转。在连续群中，有涉及有限多个参数的群，也有涉及无限维参数的群。具有有限维对称群的对象通常允许显式的“组合”描述。无限维群自然要复杂得多。这个项目的一个关键目标是发展对无限维对称群及其作用空间的更具体的理解。作为该项目教育部分的一部分，PI将举办与研究相关的研讨会，并将开发一门将组合数学与社区应用联系起来的本科课程。研究计划涉及三个主要主题：舒伯特多项式，它给出了由矩阵的秩条件局部定义的代数簇的次数公式；牛顿-奥库科夫体，它是编码射影簇上线丛几何的凸体；以及量子K-理论，它包含关于簇上曲线的精细信息。主要目的是(1)完成一组Schubert多项式，将它们与无限维Grassmannians和FLAG簇的几何联系起来；(2)数值计算牛顿-奥库科夫体，并利用它们构造特殊簇的镜像对称对偶；(3)建立齐次簇的量子K-理论的基本结果。另一个目标是将简并轨迹技术应用于从非线性优化到代数曲线上的线性系统几何的各种问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Symmetry plays a fundamental role in our mathematical understanding of the world. There are discrete symmetries, like the reflection of a face in a mirror or the hexagonal pattern of floor tiles; and there are continuous groups of symmetries, like the rotations of a sphere about various axes. Among continuous groups, there are those that involve finitely many parameters, as well as infinite-dimensional ones. Objects with finite-dimensional groups of symmetries often admit explicit "combinatorial" descriptions. Infinite-dimensional groups are naturally more complicated. A key goal of this project is to develop a more concrete understanding of infinite-dimensional groups of symmetries and the spaces on which they act. As part of the educational component of the project, the PI will run workshops related to the research and will develop an undergraduate course linking combinatorial mathematics to applications in the community.The research program involves three main topics: Schubert polynomials, which give formulas for the degrees of algebraic varieties locally defined by rank conditions on matrices; Newton-Okounkov bodies, which are convex bodies encoding the geometry of line bundles on projective varieties; and quantum K-theory, which contains refined information about curves on varieties. The main objectives are (1) to complete a package of Schubert polynomials, tying them to the geometry of infinite-dimensional Grassmannians and flag varieties; (2) to numerically compute Newton-Okounkov bodies and use them to construct mirror-symmetric duals of special varieties; and (3) to establish foundational results about the quantum K-theory of homogeneous varieties. A further aim is to apply degeneracy locus techniques to questions ranging from nonlinear optimization to the geometry of linear systems on algebraic curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.