Milena Pabiniak

The central theme of this work are Hamiltonian torus actions on symplectic manifolds. We investigate the invariants of the action, and use the action to answer questions about the invariants of the manifold itself.

In the first chapter we concentrate on equivariant cohomology ring, a topological invariant for a manifold equipped with a group action. We consider a Hamiltonian action of n-dimensional torus, Tⁿ, on a compact symplectic manifold (M, ω) with d isolated fixed points. There exists a basis {a_p} for H*_T (M ; Q) as an H* (BT ; Q) module indexed by the fixed points p ∈ M^T. The classes a_p are not uniquely determined. The map induced by inclusion,

ι* : H*_T (M ; Q) → H*_T (M^T ; Q) = ⊕_j=1^dQ[x₁,…,x_n]

is injective. We will use the basis {a_p} to give necessary and sufficient conditions for f = (f₁,…,f_d) in ⊕_j=1^dQ[x₁,…,x_n] to be in the image of ι*, i.e. to represent an equiviariant cohomology class on M. When the one skeleton is 2-dimensional, we recover the GKM Theorem. Moreover, our techniques give combinatorial description of H*_K (M ; Q), for a subgroup K of the torus T, even though we are then no longer in GKM case.

The second part of the thesis is devoted to a symplectic invariant called the Gromov width. Let G be a compact connected Lie group and T its maximal torus. The Thi orbit O_λ through λ ∈ t* is canonically a symplectic manifold. Therefore a natural question is to determine its Gromov width. In many cases the width is known to be exactly the minimum over the set

{〈α^∨_j, λ〉; α^∨_j a coroot, 〈α^∨_j, λ〉> 0 }.

We show that the lower bound for Gromov width of regular coadjoint orbits of the unitary group and of the special orthogonal group is given by the above minimum. To prove this result we will equip the (open dense subset of the) orbit with a Hamiltonian torus action, and use the action to construct explicit embeddings of symplectic balls. The proof uses the torus action coming from the Gelfand-Tsetlin system.