Hypertoric manifolds of infinite topological type

We analyse properties of hypertoric manifolds of infinite topological type, including their topology and complex structures. We show that our manifolds have the homotopy type of an infinite union of compact toric varieties. We also discuss hypertoric analogues of the periodic Ooguri-Vafa spaces

Are hygiene standards useful in assessing infection risk?

We monitored the surface level cleanliness of a five-bedded surgical intensive care unit (SICU) over a ten-week period in order to evaluate proposed hygiene standards.Ten environmental sites within SICU were sampled twice weekly along with collection of clinical and patient activity data. The standards designate aerobic colony counts (ACCs) >2.5cfu/cm2 from hand-touch sites and the presence of Staphylococcus aureus as hygiene failures. Nearly a quarter of 200 samples failed the standards, mostly from hand-touch sites on curtains, beds and medical equipment. The total number of fails each week was associated with bed occupancy (p=0.04), trending towards association with SICU-acquired infections (p=0.11). Environmental S.aureus was associated with the proportion of beds occupied (p = 0.02). Indistinguishable genotypes were found between patient and environmental staphylococci, with timescales supporting staphylococcal transmission in both directions. Hygiene standards based on microbial growth levels and the presence of S.aureus reflect patient activity and provide a means to risk manage infection. They also exposed a staphylococcal reservoir that could represent a more tangible risk to patients. Standards for surface level cleanliness deserve further evaluation

A multiplicative analogue of complex symplectic implosion

We introduce a multiplicative version of complex-symplectic implosion in the case of SL(n, \C). The universal multiplicative implosion for SL(n, \C) is an affine variety and can be viewed as a nonreductive geometric invariant theory quotient. It carries a torus action. and reductions by this action give the Steinberg fibres of SL(n, \C). We also explain how the real symplectic group-valued universal implosion introduced by Hurtubise, Jeffrey and Sjamaar may be identified inside this space.Comment: To appear in European Journal of Mathematic

Toric Hypersymplectic Quotients

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in R^{3n}. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on R^{4n} and metrics on complements of hypersurfaces in compact manifolds.Comment: 26 pages, 6 figures, small linguistic correction

On Spin(7) holonomy metric based on SU(3)/U(1)

We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest $\Sigma_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space ${\bf CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric.Comment: 21 pages, Latex, Introduction slightly expanded, an error in section 6 corrected and references added, (v3) minor correction

On the nonexistence of quasi-Einstein metrics

We study complete Riemannian manifolds satisfying the equation $Ric+\nabla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $\Delta_f f + m\mu e^{2f/m}=0$ for $\mu\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers which have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity $R+|\nabla f|^2$ is a positive constant.Comment: Final version: Improved exposition of Section 2, corrected minor typo