Symplectic Geometry and Schubert Calculus

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DMS-0305128Rebecca GoldinSymplectic geometry occurs at a crossroad of combinatorics, representation theory, physics, geometry and topology. One of the most important examples of a Hamiltonian T-space is the coadjoint orbit of a compact Lie group. Coadjoint orbits arise in physics, as they are the set of all matrices with specified spectra. They also occur in representation theory, as all irreducible representations of a complex reductive Lie group occur as holomorphic sections of a line bundle over a coadjoint orbit. They occur in algebraic geometry as projective varieties. The intersection theory of naturally arising subvarieties, called Schubert varieties, appears in questions of linear algebra, valuation rings, and representation theory. Schubert calculus is important as it is the linear case for (and hence the first step towards) understanding complex algebraic intersections more generally. Algebraically, Schubert calculus analyses the multiplicative structure of the cohomology of flag varieties. The main open question is to find a totally positive formula for the (equivariant) structure constants. In joint work with A. Knutson, Goldin has discovered new techniques that give hope for a positive formula in certain cases. Goldin is also interested in topological invariants that distinguish reduced spaces for any compact Hamiltonian T-space, such as certain ideals found in its equivariant cohomology ring or generalized equivariant Euler classes. Goldin is additionally developing techniques to calculate integrals on certain symplectic reduced spaces using small amounts of information about how the torus acts on the original symplectic manifold.Understanding Schubert calculus is a small step towards the larger question of how to study intersections of algebraic varieties. Such intersections arise outside of mathematics, as scientists ask questions that involve the concurrence of several physical constraints, and want to understand what possible solutions there may be. Symplectic geometry is a natural setting to describe physical systems. The space of the position and the momentum of a particle, for example, is called "phase space" and has a natural symplectic structure on it. In many cases, there is a lot of symmetry; physical observations of the Hydrogen atom, for example, are unaffected by the position of the observer a fixed distance away from the particle. The study of "reduced spaces" is a way of studying the physics while simplifying the space by ignoring (in fact, dividing out) these symmetries. Much of Goldin's work involves techniques to do computations on these simplified systems, which may shed light on questions that physicists and chemists are posing about molecules and particles. Goldin is also writing a book (joint with G. Goldin) on myths in mathematics education.