The main goal of the research described in this proposal is to investigate the role of group actions in symplectic geometry and other closely related geometries. Some of the questions that Tolman and her collaborators plan to study are closely related to important problems in other fields; they hope that these projects will not only give insight into symplectic manifolds but also shed new light on the original problems. For example, R. Goldin and Tolman are working on extending the results of Schubert calculus to more general symplectic manifolds. L. Godinho and Tolman are attempting to prove the symplectic analog of the Petrie conjecture. Other questions that Tolman is working on are designed to explore the "geography" of symplectic manifolds with group actions by determining whether such spaces are always as well behaved as the natural examples which we usually consider. For example, Tolman is considering when symplectic actions are Hamiltonian, when such manifolds possess the hard Lefshetz property, and what restrictions their graphs must obey. She and Karshon are also classifying (n-1)-dimensional Hamiltonian torus actions on 2n-dimensional symplectic manifolds; this will provide a new source of examples. Many important manifolds arise most naturally as symplectic quotients. While this fact has been used extremely successfully to compute their rational cohomology rings, many of the theorems do not hold over the integers. Therefore, T. Holm and Tolman are studying how to compute the integral cohomology ring of such quotients. Generalizations of symplectic forms which allow some degeneration, such as near symplectic forms and folded symplectic forms, have recently played an important role in the study of four-manifolds. Tolman plans to work on understanding the role group actions in these geometries. Finally, Y. Lin and Tolman are using generalized Kaehler reduction to construct new examples of Bihermitian structures, which play an important role in string theory.
In classical physics, symmetries of a physical system give rise to conserved quantities.For example, the total angular momentum of the solar system is constant. From a mathematical perspective, studying these physical systems corresponds to studying Hamiltonian group actions on a special type of symplectic manifold - a cotangent bundle. The underlying goal of the proposed research is to gain a better understanding of what types of symplectic manifolds admit group actions, and how to calculate their invariants. Working with her collaborators, Tolman plans to attack this question on a number of fronts. She hopes that this will lead to a greater understanding of an area of increasing importance, both within mathematics and for physics.
