CAREER: Equivariant and Infinite-Dimensional Combinatorial Algebraic Geometry

职业：等变和无限维组合代数几何

• 批准号: 1945212
• 负责人: David Anderson
• 金额: $ 40万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945212&HistoricalAwards=false
• 关键词: CAREER; Equivariant; Infinite; Dimensional; Combinatorial

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.

对称性在我们对世界的数学理解中起着基础性的作用。 有离散的对称性，如镜子中的脸的反射或地板砖的六边形图案;也有连续的对称性组，如球体围绕不同轴的旋转。 在连续群中，有那些涉及许多参数的群，也有那些涉及无限维参数的群。 具有有限维对称群的对象通常允许明确的“组合”描述。 无限维群自然更加复杂。 这个项目的一个关键目标是发展一个更具体的理解无限维群的对称性和空间上，他们的行为。 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。研究计划包括三个主要主题：舒伯特多项式，它给出了由矩阵秩条件局部定义的代数簇的次数公式;牛顿-奥肯科夫体，这是凸体编码的几何形状的线丛上的投影品种;和量子K理论，其中包含了细化信息的曲线品种。 主要目标是（1）完成舒伯特多项式的包，将它们与无限维格拉斯曼和旗簇的几何联系起来;（2）数值计算Newton-Okounkov体，并使用它们来构造特殊簇的镜像对称群;（3）建立齐次簇的量子K理论的基础结果。 另一个目标是将退化轨迹技术应用于从非线性优化到代数曲线上线性系统的几何形状等问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

K-theoretic balancing conditions and the Grothendieck group of a toric variety

K 理论平衡条件和复曲面簇的 Grothendieck 群

• DOI: 10.1016/j.jalgebra.2022.07.038
• 发表时间: 2022
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Shah, Aniket

Motivic classes of degeneracy loci and pointed Brill‐Noether varieties

简并位点和尖头布里尔-诺特变体的动机类别

• DOI: 10.1112/jlms.12547
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Anderson, Dave;Chen, Linda;Tarasca, Nicola
• 通讯作者: Tarasca, Nicola

?-classes of Brill–Noether Loci and a Determinantal Formula

Brill–Noether 轨迹的 ? 类和行列式

• DOI: 10.1093/imrn/rnab025
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Anderson, Dave;Chen, Linda;Tarasca, Nicola
• 通讯作者: Tarasca, Nicola