Topology of Kaehler Manifolds, Surface Bundles, and Outer Automorphism Groups

凯勒流形、表面丛和外自同构群的拓扑

• 批准号: 2401403
• 负责人: Corey Bregman
• 金额: $ 12.63万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-12-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401403&HistoricalAwards=false
• 关键词: Topology; Kaehler; Manifolds; Surface; Bundles

The main subject of this project is geometric group theory. One of the guiding principles behind geometric group theory, as developed by Klein and more recently Gromov, is that one can understand a geometric object by studying its symmetries. The primary goal of this project is to utilize techniques from geometric group theory as a bridge to simplify and solve problems in other fields of mathematics. The first part of this project focuses on algebraic varieties, which are geometric spaces defined by polynomial equations. Algebraic varieties arise naturally in a wide-range of disciplines, including high-energy physics and cryptography. Although these objects have been studied for centuries, many of their geometric properties still remain unknown, and cannot be uncovered using traditional means. The PI proposes novel geometric group theory methods to develop restrictions on properties of algebraic varieties. The second part of this project studies the symmetries of right-angled Artin groups, which have important connections to low-dimensional topology, as well as robotics, phylogenetic trees, and computer science. In addition, the PI will advise undergraduate mathematics majors and mentor graduate students through organizing seminars and other mathematical activities.The study of mapping class groups and the moduli space of curves lies at the intersection of algebraic geometry, Riemannian geometry, and topology. The first part of this project studies the topology of surface and torus bundles admitting some extra structure such as a Kaehler metric, or which are formal in the sense of rational homotopy theory. The PI proposes techniques from geometric group theory and mapping class groups that can place restrictions on the fundamental group and monodromy of such bundles, but also connect questions about the geometry of complex projective surfaces to questions about mapping class groups. The second part of this project studies the automorphism groups of right-angled Artin groups (RAAGs), which comprise a large class of groups extending both free and free abelian groups. There is a fruitful analogy between the study of mapping class groups of surfaces, outer automorphism groups of free groups, and lattices in semisimple Lie groups. The role played by Teichmuller space, Culler-Vogtmann outer space, and symmetric spaces, respectively, is of fundamental importance in proving many key results about these groups. The PI proposes an analogous space for outer automorphisms of RAAGs, to provide a unified framework for studying automorphisms of free and free abelian groups.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.

该项目的主要课题是几何群论。由克莱因和最近的格罗莫夫提出的几何群论背后的指导原则之一是，人们可以通过研究几何对象的对称性来理解它。该项目的主要目标是利用几何群论的技术作为桥梁来简化和解决其他数学领域的问题。该项目的第一部分重点关注代数簇，即由多项式方程定义的几何空间。代数簇自然出现在许多学科中，包括高能物理和密码学。尽管这些物体已经被研究了几个世纪，但它们的许多几何特性仍然未知，并且无法使用传统手段来揭示。 PI 提出了新的几何群论方法来开发对代数簇性质的限制。该项目的第二部分研究直角 Artin 群的对称性，它与低维拓扑、机器人技术、系统发育树和计算机科学有着重要的联系。此外，PI还将通过组织研讨会和其他数学活动为本科生数学专业提供建议并指导研究生。映射类群和曲线模空间的研究属于代数几何、黎曼几何和拓扑的交叉点。 该项目的第一部分研究表面束和环面束的拓扑结构，允许一些额外的结构，例如凯勒度量，或者在有理同伦理论意义上是形式化的。 PI 提出了来自几何群论和映射类群的技术，这些技术可以对此类束的基本群和单性进行限制，但也可以将有关复杂射影曲面的几何问题与有关映射类群的问题联系起来。该项目的第二部分研究直角 Artin 群 (RAAG) 的自同构群，该群包含一大类延伸自由群和自由阿贝尔群的群。在曲面的映射类群、自由群的外自同构群以及半单李群中的格的研究之间存在着富有成效的类比。 Teichmuller 空间、Culler-Vogtmann 外层空间和对称空间分别发挥的作用对于证明这些群的许多关键结果至关重要。 PI 为 RAAG 的外自同构提出了一个类似的空间，为研究自由和自由阿贝尔群的自同构提供了一个统一的框架。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进行评估，被认为值得支持审查标准。