The Integer Valued SU(3) Casson Invariant for Brieskorn spheres

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculate the invariant for all Brieskorn homology spheres.Comment: 50 pages, 3 figure

Structural dynamic interaction with solar tracking control for evolutionary Space Station concepts

The sun tracking control system design of the Solar Alpha Rotary Joint (SARJ) and the interaction of the control system with the flexible structure of Space Station Freedom (SSF) evolutionary concepts are addressed. The significant components of the space station pertaining to the SARJ control are described and the tracking control system design is presented. Finite element models representing two evolutionary concepts, enhanced operations capability (EOC) and extended operations capability (XOC), are employed to evaluate the influence of low frequency flexible structure on the control system design and performance. The design variables of the control system are synthesized using a constrained optimization technique to meet design requirements, to provide a given level of control system stability margin, and to achieve the most responsive tracking performance. The resulting SARJ control system design and performance of the EOC and XOC configurations are presented and compared to those of the SSF configuration. Performance limitations caused by the low frequency of the dominant flexible mode are discussed

Relativistic Shock Acceleration: A Hartree-Fock Approach

We examine the problem of particle acceleration at a relativistic shocks assuming pitch-angle scattering and using a Hartree-Fock method to approximate the associated eigenfunctions. This leads to a simple transcendental equation determining the power-law index, $s$, given the up and downstream velocities. We compare our results with accurate numerical solutions obtained using the eigenfunction method. In addition to the power-law index this method yields the angular and spatial distributions upstream of the shock.Comment: 4 pages, 2 figures, proceedings of the "4th Heidelberg International Symposium on High Energy Gamma-Ray Astronomy" July 7-11, 2008, Heidelberg, German

Gauge Theoretic Invariants of, Dehn Surgeries on Knots

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k Dehn surgery on (2,q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff. Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7 and 9.Comment: Version 3: minor corrections from version 2. Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper6.abs.htm

A symplectic manifold homeomorphic but not diffeomorphic to CP\u3csup\u3e2\u3c/sup\u3e # 3CP\u3csup\u3e2\u3c/sup\u3e

In this article we construct a minimal symplectic 4-manifold and prove it is homeomorphic but not diffeomorphic to CP # 3CP . © 2008 Mathematical Sciences Publishers. 2

On the rho invariant for manifolds with boundary

This article is a follow up of the previous article of the authors on the analytic surgery of eta- and rho-invariants. We investigate in detail the (Atiyah-Patodi-Singer)-rho-invariant for manifolds with boundary. First we generalize the cut-and-paste formula to arbitrary boundary conditions. A priori the rho-invariant is an invariant of the Riemannian structure and a representation of the fundamental group. We show, however, that the dependence on the metric is only very mild: it is independent of the metric in the interior and the dependence on the metric on the boundary is only up to its pseudo--isotopy class. Furthermore, we show that this cannot be improved: we give explicit examples and a theoretical argument that different metrics on the boundary in general give rise to different rho-invariants. Theoretically, this follows from an interpretation of the exponentiated rho-invariant as a covariantly constant section of a determinant bundle over a certain moduli space of flat connections and Riemannian metrics on the boundary. Finally we extend to manifolds with boundary the results of Farber-Levine-Weinberger concerning the homotopy invariance of the rho-invariant and spectral flow of the odd signature operator.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-22.abs.htm