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.

该项目的主要主题是几何群体理论。克莱因（Klein）和最近格罗莫夫（Gromov）开发的几何群体理论背后的指导原理之一是，人们可以通过研究其对称性来理解几何对象。该项目的主要目标是利用从几何群体理论中的技术来简化和解决其他数学领域的问题的桥梁。该项目的第一部分着重于代数品种，这是由多项式方程定义的几何空间。代数品种自然出现在广泛的学科中，包括高能物理学和密码学。尽管已经研究了这些物体已经存在了几个世纪，但它们的许多几何特性仍然未知，并且无法使用传统手段进行发现。 PI提出了新的几何组理论方法，以开发对代数品种特性的限制。该项目的第二部分研究了与低维拓扑以及机器人技术，系统发育和计算机科学的重要连接的对称性。此外，PI将通过组织研讨会和其他数学活动来为本科数学专业和导师研究生提供建议。对映射课程组和曲线模量空间的研究在于代数几何学，riemannian几何形状和拓扑结构。 该项目的第一部分研究表面和圆环捆绑包的拓扑结构，以承认一些额外的结构，例如kaehler指标，或者在理性同义理论的意义上是正式的。 PI提出了从几何群体理论和映射课程组中提出的技术，这些技术可以对此类捆绑包的基本组和单片进行限制，但也将有关复杂投影表面的几何形状的问题与有关映射课程组的问题联系起来。该项目的第二部分研究了右角Artin组（RAAGS）的自动形态群体，该群体组成了一大批群体，这些群体均扩展了自由和自由的Abelian群体。在映射表面的阶级群，自由组的外态群体和半圣像谎言组的格子的研究之间，有一个富有成果的比喻。 Teichmuller空间，Culler-Vogtmann外太空和对称空间的作用对于证明这些群体的许多关键结果至关重要。 PI为RAAG的外部自动形态提供了一个类似的空间，以提供一个统一的框架来研究自由和免费的Abelian群体的自动形态。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查审查的审查标准来通过评估来通过评估来支持的。