Tara S. Holm

Orbifold cohomology of torus quotients

We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to NH^*_T(Y) the graphical GKM calculus (as detailed in e.g. [Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman, Goldin]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with \Q coefficients, in [Borisov-Chen-Smith]); symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods), and extend them to \Z coefficients in certain cases, including weighted projective spaces.

Real loci of symplectic reductions

Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus T. Suppose that M is equipped with an anti-symplectic involution a compatible with the T-action. The real locus of M is the fixed point set M σ of σ. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction M//T has the same ordinary cohomology as its real locus (M//T) σred , with degrees halved. This extends Duistermaats original result on real loci to a case in which there is not a natural Hamiltonian torus action.

GKM theory for torus actions with nonisolated fixed Points

Let M 2d be a compact symplectic manifold and T a compact n-dimensional torus. A Hamiltonian action, τ, of T on M is a GKM action if, for every p ∈ M T , the isotropy representation of T on T p M has pairwise linearly independent weights. For such an action, the image of the set of zero- and one-dimensional orbits under the projection onto M/T is aregular d-valent graph; Goresky, Kottwitz, and MacPherson have proved that the equivariant cohomology of M can be computed from the combinatorics of this graph. In this paper we define a “GKM action with nonisolated fixed points” to be an action, τ, of T on M with the property that for every connected component, F of M T and p ∈ F, the isotropy representation of T on the normal space to F at p has pairwise linearly independent weights. For such an action, we show that all components of M T are diffeomorphic and prove an analogue of the theorem above.

USING A CARD TRICK TO TEACH DISCRETE MATHEMATICS

ABSTRACT We present a card trick that can be used to review or teach a variety of topics in discrete mathematics. We address many subjects, including permutations, combinations, functions, graphs, depth first search, the pigeonhole principle, greedy algorithms, and concepts from number theory. Moreover, the trick motivates the use of computers in mathematical research. The ultimate solution to the card trick makes use of Halls Distinct Representative Theorem.

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

We study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman and Weitsman’s proof of the GKM theorem [TW] in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S ⊂ T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen–Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen–Ruan cohomology in terms of orbifold fixed point data.

The fundamental group and Betti numbers of toric origami manifolds

Toric origami manifolds are characterized by origami templates, which are combinatorial models built by gluing polytopes together along facets. In this paper, we examine the topology of orientable toric oigami manifolds with coorientable folding hypersurface. We determine the fundamental group. In our previous paper [HP], we studied the ordinary and equivariant cohomology rings of simply connected toric origami manifolds. We conclude this paper by computing some Betti numbers in the non-simply connected case.

Connectivity properties of moment maps on based loop groups

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of

On majority domination in graphs

LG

Conjugation spaces and edges of compatible torus actions

and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.

The Morse–Bott–Kirwan condition is local

Abstract A majority dominating function on the vertex set of a graph G=(V,E) is a function g:V→{1,−1} such that g(N[v])⩾1 for at least half of the vertices v in V. The weight of a majority dominating function is denoted as g(V) and is ∑ g(v) over all v in V. The majority domination number of a graph is the minimum possible weight of a majority dominating function, and is denoted as γmaj(G). We determine the majority domination numbers of certain families of graphs. Moreover, we show that the decision problem corresponding to computing the majority domination number of an arbitrary disjoint union of complete graphs is NP-complete.