Chris Woodward

Non-abelian convexity by symplectic cuts

Abstract We give new proofs of the convexity and connectedness properties of the moment map using the technique of symplectic cutting and extend these results to the case of orbifolds.

On the quantum product of Schubert classes

We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Petersons quantum version of Chevalleys formula, also for general G/Ps.

Quilted Floer cohomology

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspondences, and transfer of displaceability under geometric composition.

Duistermaat-Heckman measures and moduli spaces of flat bundles over surfaces

Abstract. We introduce Liouville measures and Duistermaat—Heckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces.

The classification of transversal multiplicity-free group actions

Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes. This result generalizes a theorem of Delzant concerning torus actions.

Density-potential mapping in time-dependent density-functional theory

The key questions of uniqueness and existence in time-dependent density-functional theory are usually formulated only for potentials and densities that are analytic in time. Simple examples, standard in quantum mechanics, lead, however, to nonanalyticities. We reformulate these questions in terms of a nonlinear Schroedinger equation with a potential that depends nonlocally on the wave function.

MODULI SPACES OF FLAT CONNECTIONS ON 2-MANIFOLDS, COBORDISM, AND WITTEN'S VOLUME FORMULAS

According to Atiyah-Bott [ABA] the moduli space of flat connections on a compact oriented 2-manifold with prescribed holonomies around the boundary is a finite-dimensional symplectic manifold, possibly singular. A standard approach [W1W2] to computing invariants (symplectic volumes, Riemann-Roch numbers, etc.) of the moduli space is to study the “factorization” of invariants under gluing of 2-manifolds along boundary components. Given such a factorization result, any choice of a “pants decomposition” of the 2-manifold reduces the computation of invariants to the three-holed sphere.

Multiplicity-free Hamiltonian actions need not be Kähler

Abstract. Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible Kähler structures, and are therefore toric varieties. In this note we show that Delzants result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T3. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible Kähler structure.

Exact triangle for fibered Dehn twists

We use quilted Floer theory to generalize Seidel’s long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply the sequence to construct versions of the Floer and Khovanov–Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.

\(A_\infty \) functors for Lagrangian correspondences

We construct