96 research outputs found
A Lower Estimate for the Modified Steiner Functional
We prove inequality (1) for the modified Steiner functional A(M), which
extends the notion of the integral of mean curvature for convex surfaces.We
also establish an exression for A(M) in terms of an integral over all
hyperplanes intersecting the polyhedralral surface M.Comment: 6 pages, Late
Convex Hulls in the Hyperbolic Space
We show that there exists a universal constant C>0 such that the convex hull
of any N points in the hyperbolic space H^n is of volume smaller than C N, and
that for any dimension n there exists a constant C_n > 0 such that for any
subset A of H^n,
Vol(Conv(A_1)) < C_n Vol(A_1) where A_1 is the set of points of hyperbolic
distance to A smaller than 1.Comment: 7 page
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
On the Obstructions to non-Cliffordian Pin Structures
We derive the topological obstructions to the existence of non-Cliffordian
pin structures on four-dimensional spacetimes. We apply these obstructions to
the study of non-Cliffordian pin-Lorentz cobordism. We note that our method of
derivation applies equally well in any dimension and in any signature, and we
present a general format for calculating obstructions in these situations.
Finally, we interpret the breakdown of pin structure and discuss the relevance
of this to aspects of physics.Comment: 31 pages, latex, published in Comm. Math. Phys. 164, No. 1, pages
65-87 (1994
On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus
The width of a convex curve in the plane is the minimal distance between a
pair of parallel supporting lines of the curve. In this paper we study the
width of nodal lines of eigenfunctions of the Laplacian on the standard flat
torus. We prove a variety of results on the width, some having stronger
versions assuming a conjecture of Cilleruelo and Granville asserting a uniform
bound for the number of lattice points on the circle lying in short arcs.Comment: 4 figures. Added some comments about total curvature and other
detail
Topological transversals to a family of convex sets
Let be a family of compact convex sets in . We say
that has a \emph{topological -transversal of index }
(, ) if there are, homologically, as many transversal
-planes to as -planes containing a fixed -plane in
.
Clearly, if has a -transversal plane, then
has a topological -transversal of index for and . The converse is not true in general.
We prove that for a family of compact convex sets in
a topological -transversal of index implies an
ordinary -transversal. We use this result, together with the
multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann
category of the Grassmannian, and different versions of the colorful Helly
theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences
On three-manifolds dominated by circle bundles
We determine which three-manifolds are dominated by products. The result is
that a closed, oriented, connected three-manifold is dominated by a product if
and only if it is finitely covered either by a product or by a connected sum of
copies of the product of the two-sphere and the circle. This characterization
can also be formulated in terms of Thurston geometries, or in terms of purely
algebraic properties of the fundamental group. We also determine which
three-manifolds are dominated by non-trivial circle bundles, and which
three-manifold groups are presentable by products.Comment: 12 pages; to appear in Math. Zeitschrift; ISSN 1103-467
Knots, Braids and BPS States in M-Theory
In previous work we considered M-theory five branes wrapped on elliptic
Calabi-Yau threefold near the smooth part of the discriminant curve. In this
paper, we extend that work to compute the light states on the worldvolume of
five-branes wrapped on fibers near certain singular loci of the discriminant.
We regulate the singular behavior near these loci by deforming the discriminant
curve and expressing the singularity in terms of knots and their associated
braids. There braids allow us to compute the appropriate string junction
lattice for the singularity and,hence to determine the spectrum of light BPS
states. We find that these techniques are valid near singular points with N=2
supersymmetry.Comment: 38 page
Dilogarithm Identities in Conformal Field Theory and Group Homology
Recently, Rogers' dilogarithm identities have attracted much attention in the
setting of conformal field theory as well as lattice model calculations. One of
the connecting threads is an identity of Richmond-Szekeres that appeared in the
computation of central charges in conformal field theory. We show that the
Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be
interpreted as a lift of a generator of the third integral homology of a finite
cyclic subgroup sitting inside the projective special linear group of all real matrices viewed as a {\it discrete} group. This connection
allows us to clarify a few of the assertions and conjectures stated in the work
of Nahm-Recknagel-Terhoven concerning the role of algebraic -theory and
Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related
to hyperbolic 3-manifolds as suggested but is more appropriately related to the
group manifold of the universal covering group of the projective special linear
group of all real matrices viewed as a topological group. This
also resolves the weaker version of the conjecture as formulated by Kirillov.
We end with the summary of a number of open conjectures on the mathematical
side.Comment: 20 pages, 2 figures not include
A geometric discretisation scheme applied to the Abelian Chern-Simons theory
We give a detailed general description of a recent geometrical discretisation
scheme and illustrate, by explicit numerical calculation, the scheme's ability
to capture topological features. The scheme is applied to the Abelian
Chern-Simons theory and leads, after a necessary field doubling, to an
expression for the discrete partition function in terms of untwisted
Reidemeister torsion and of various triangulation dependent factors. The
discrete partition function is evaluated computationally for various
triangulations of and of lens spaces. The results confirm that the
discretisation scheme is triangulation independent and coincides with the
continuum partition functionComment: 27 pages, 5 figures, 6 tables. in late
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