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

这个项目的主要课题是几何群论。几何群论背后的指导原则之一，正如克莱因和最近格罗莫夫所发展的那样，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几何群论的技术作为桥梁，简化和解决数学其他领域的问题。该项目的第一部分重点关注代数簇，代数簇是由多项式方程定义的几何空间。代数变种在广泛的学科中自然出现，包括高能物理学和密码学。虽然这些物体已经被研究了几个世纪，但它们的许多几何特性仍然未知，并且无法使用传统手段来发现。PI提出了新的几何群论方法，以发展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，这与低维拓扑学以及机器人技术，系统发育树和计算机科学有重要联系。此外，PI还将通过组织研讨会和其他数学活动为本科数学专业提供建议，并指导研究生。映射类群和曲线模空间的研究位于代数几何，黎曼几何和拓扑学的交叉点。 这个项目的第一部分研究拓扑的表面和环面丛承认一些额外的结构，如一个Kaehler度量，或正式的意义上的理性同伦理论。 PI提出了几何群论和映射类群的技术，这些技术可以限制基本群和这种丛的单值性，但也将复杂投影曲面的几何问题与映射类群的问题联系起来。这个项目的第二部分研究直角Artin群（RAAGs）的自同构群，它包括一个大类的自由和自由阿贝尔群的扩展。有一个富有成效的研究之间的类比映射类群的表面，外自同构群的自由群，格在半单李群。Teichmuller空间、Culler-Vogtmann外空间和对称空间分别扮演的角色，在证明这些群的许多关键结果中具有根本的重要性。PI为RAAGs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。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。

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.