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

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.