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

本课题的主要课题是几何群论。克莱因（Klein）和最近的格罗莫夫（Gromov）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物体的对称性来理解它。该项目的主要目标是利用几何群论的技术作为桥梁，简化和解决其他数学领域的问题。这个项目的第一部分集中在代数变量上，代数变量是由多项式方程定义的几何空间。代数变体自然出现在广泛的学科中，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，无法用传统的方法发现。PI提出了新的几何群论方法来发展代数变种性质的限制。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、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，为数学专业的本科生提供指导，并指导研究生。映射类群和曲线模空间的研究是代数几何、黎曼几何和拓扑学的交叉领域。本课题的第一部分研究了表面和环面束的拓扑结构，其中包含一些额外的结构，如Kaehler度量，或者在有理同伦理论的意义上是形式化的。PI提出了来自几何群论和映射类群的技术，这些技术可以对这些束的基本群和单性施加限制，但也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。本课题的第二部分研究了直角Artin群（RAAGs）的自同构群，它包含了一大类扩展自由阿贝尔群和自由阿贝尔群的群。在半单李群中的映射类群、自由群的外自同构群和格的研究之间有一个卓有成效的类比。Teichmuller空间、Culler-Vogtmann外空间和对称空间在证明这些群的许多关键结果中所起的作用是至关重要的。在此基础上，提出了rag外自同构的类似空间，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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.