The mod 2 cohomology of fixed point sets of anti-symplectic involutions

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $G=T^n$. Suppose that $\sigma$ is an anti-symplectic involution compatible with the $G$-action. The real locus of $M$ is $X$, the fixed point set of $\sigma$. Duistermaat uses Morse theory to give a description of the ordinary cohomology of $X$ in terms of the cohomology of $M$. There is a residual \G=(\Zt)^n action on $X$, and we can use Duistermaat's result, as well as some general facts about equivariant cohomology, to prove an equivariant analogue to Duistermaat's theorem. In some cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and Goldin-Holm to the real locus.Comment: 21 pages, 1 figur

Semiclassical spectral invariants for Schrodinger operators

Original manuscript September 23, 2009In this article we show how to compute the semiclassical spectral measure associated with the Schrodinger operator on R[superscript n], and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schrodinger operator on R[superscript 2] with a radially symmetric electric potential, V, and magnetic potential, B, both V and B are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schrodinger operator with its Birkhoff canonical form.National Science Foundation (U.S.) (Grant DMS-1005696

A semiclassical heat trace expansion for the perturbed harmonic oscillator

Original Manuscript September 1, 2011 (International Conference on Spectral Geometry held at Dartmouth College on July 19-23, 2010)In this paper we study the heat trace expansion of the perturbed harmonic oscillator by adapting to the semiclassical setting techniques developed by Hitrick-Polterovich in [HP]. We use the expansion to obtain certain inverse spectral results.National Science Foundation (U.S.) (Grant DMS-1005696

Semiclassical States Associated with Isotropic Submanifolds of Phase Space

We define classes of quantum states associated with isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a symbol calculus for them; the symbols are symplectic spinors. We outline various applications

Polynomial assignments

The concept of assignments was introduced in Ginzburg et al. (1999) as a method for extracting geometric information about group actions on manifolds from combinatorial data encoded in the infinitesimal orbit-type stratification. In this paper we answer a question posed in Ginzburg et al. (1999) by describing to what extent the equivariant cohomology ring of M is determined by this data

Balanced Fiber Bundles and GKM Theory

Let T be a torus and B a compact T-manifold. Goresky et al. show in [3] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H*[over]T(B) as a subring of H*[over]T(B[superscript 2]). In this paper, we prove an analog of this result for T-equivariant fiber bundles: we show that if M is a T-manifold and π:M→B a fiber bundle for which π intertwines the two T-actions, there is a simple combinatorial description of H*[over]T(M) as a subring of H*[over]T(π[superscript -1](B[superscript T])). Using this result, we obtain fiber bundle analogs of results of Guillemin et al. on GKM theory for homogeneous spaces

Stability Functions

In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to various types of asymptotic problems in the Kahler geometry of GIT quotients. We discuss several particular classes of examples, namely, toric varieties, spherical varieties and the symplectic version of quiver varieties.National Science Foundation (U.S.) (Grant DMS-0408993

Symplectic and Poisson geometry on b-manifolds

Let M2nM2n be a Poisson manifold with Poisson bivector field Π. We say that M is b -Poisson if the map Πn:M→Λ2n(TM)Πn:M→Λ2n(TM) intersects the zero section transversally on a codimension one submanifold Z⊂MZ⊂M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M,Π)(M,Π) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology

Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras

We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.Comment: 24 page