The main goal of Prof. Tolman's research is to investigate the role of group actions in symplectic geometry. She intends to focus on three related areas. First, she will explore the conditions that force a symplectic action to be Hamiltonian; in particular, she plans to construct a compact six-dimensional symplectic manifold with a non-Hamiltonian symplectic circle action with exactly two fixed points. Second, she will work on classifying symplectic manifolds with ``large" torus actions. More specifically, she will finish her project with Y. Karshon on n-1 dimensional torus actions on 2n dimensional manifolds, analyze Hamiltonian circle actions on six-manifolds, and find applications in both cases -- especially to Fano manifolds. Third, R. Goldin and S. Tolman will generalize ideas from algebraic combinatorics to the more general symplectic setting of Hamiltonian actions on symplectic manifolds. In addition to these three main areas, she will work with her collaborators on a number of other projects: calculating the cohomology of the symplectic quotients of Hamiltonian loop group actions, and computing the integer cohomology of symplectic manifolds and their quotients in the finite dimensional case. Taken together, these results will significantly advance our understanding of this field.
Consider a physical system, such as a planet orbiting around a star. We need to keep track of the position and momentum of each object: The set of all possible measurements is called phase space. For the two-body system, the phase space is twelve dimensional Euclidean space. Even for more complicated systems, the phase space looks locally like Euclidean space. Moreover, the rate of change over time of the momentum of an object is determined by the rate of change of the total energy of the system with respect to the position of that object; the converse also holds. Finally, many physical systems have symmetries. For example, if we ignore any external gravitational field, the solar system has three-dimensional rotational symmetry. Prof. Tolman studies a mathematical generalization of phase space, called "symplectic manifolds." Her research focuses on understanding the role of symmetries on these spaces. For example, she is studying when symplectic manifolds with symmetry are as well behaved as algebraic varieties, which are the spaces cut out by solutions to polynomial equations. Working with Prof. Karshon, she is trying to describe all symplectic manifolds with extremely large amounts of symmetry.
