The absence of efficient dual pairs of spanning trees in planar graphs

A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T* are both within a uniform multiplicative constant (independent of G) of the diameters of their ambient graphs.Comment: 7 pages, 3 figure

A Note on Real Tunneling Geometries

In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in three spacetime dimensions, such a transition is ``probable,'' in the sense that the required Riemannian geometry yields a genuine maximum of the semiclassical wave function.Comment: 5 page

Homologous non-isotopic symplectic tori in a K3-surface

For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n>2.Comment: 15 pages, 9 figures; v2: extended the main theorem, gave a second construction of symplectic tori, added a figure, added/updated references, minor changes in figure

Limits of corporal punishment in public schools

A review of the U.S. Supreme Court decision in Ingraham v. Wright: The Limits of Corporal Punishment in Public Schools

3-manifolds with(out) metrics of nonpositive curvature

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.Comment: 16 page

Compassion : the inward journey to love

https://place.asburyseminary.edu/ecommonsatsdissertations/1893/thumbnail.jp

Reconstructing the global topology of the universe from the cosmic microwave background

If the universe is multiply-connected and sufficiently small, then the last scattering surface wraps around the universe and intersects itself. Each circle of intersection appears as two distinct circles on the microwave sky. The present article shows how to use the matched circles to explicitly reconstruct the global topology of space.Comment: 6 pages, 2 figures, IOP format. To be published in the proceedings of the Cleveland Cosmology and Topology Workshop 17-19 Oct 1997. Submitted to Class. Quant. Gra

Spherical structures on torus knots and links

The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot ${\rm t}(2n+1, 2)$ or the torus link ${\rm t}(2n, 2)$. Domains of existence for a spherical metric are found in terms of cone angles and volume formul{\ae} are presented.Comment: 17 pages, 5 figures; typo

All flat manifolds are cusps of hyperbolic orbifolds

We show that all closed flat n-manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)-orbifold.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-13.abs.htm

Closed surface bundles of least volume

Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus-g bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, we prove that this is indeed the case when g is large. The proof combines a soft geometric limit argument with a detailed Neumann-Zagier asymptotic formula for the volumes of Dehn fillings. Our examples are all Dehn fillings on the sibling of the Whitehead manifold, and we also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations.Comment: 22 pages, 4 figures. V2: Corrected Table 1.9; V3: Added Table 1.10; V4: Minor edits; V5: Corrected Figure 2.1. To appear in AG&