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

本课题的主要课题是几何群论。由克莱因和最近的格罗莫夫发展起来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。这个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。这个项目的第一部分集中在代数变体上，它是由多项式方程定义的几何空间。代数变种自然而然地出现在广泛的学科中，包括高能物理和密码学。虽然这些物体已经被研究了几个世纪，但它们的许多几何性质仍然未知，无法用传统的方法来发现。PI提出了新的几何群论方法来发展对代数簇性质的限制。这个项目的第二部分研究直角Artin群的对称性，它与低维拓扑以及机器人学、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和研究生导师提供建议。映射班组和曲线的模空间是代数几何、黎曼几何和拓扑学的交叉学科。这个项目的第一部分研究了曲面和环丛的拓扑，这些丛允许一些额外的结构，如Kaehler度量，或者在有理同伦理论意义下是形式的。PI从几何群论和映射类群中提出了一些技巧，这些技巧不仅可以限制这类丛的基本群和单一性，而且还可以将复杂射影曲面的几何问题与映射类群的问题联系起来。本项目的第二部分研究直角Artin群(RAGs)的自同构群，RAGs包括一大类扩展自由和自由阿贝尔群的群。曲面的映射类群、自由群的外自同构群和半单李群中的格之间有着丰富的相似之处。Teichmuller空间、Culler-Vogtmann外空间和对称空间分别在证明这些群的许多关键结果方面起着至关重要的作用。PI为RAG的外部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。这个奖项反映了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。

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.