Cylinders and paths in simplicial categories

We prove the uniqueness, the functoriality and the naturality of cylinder objects and path objects in closed simplicial model categories

Cylinder, Tensor and Tensor-Closed Module

The purpose of this note is to show that, if $\mathcal{V}$ is a closed monoidal category, the following three notions are equivalent. (1) Category with $\mathcal{V}$-structure and cylinder. (2) Tensored $\mathcal{V}$-category. (3) Tensor-closed $\mathcal{V}$-module. As an application we will show that, if $\mathcal{V}$ is closed and symmetric, then given a category $\mathcal{S}$ there is an one-to-one correspondence between the set of $\mathcal{V}$-structures with cylinder and path on $\mathcal{S}$ introduced by Quillen and the set of closed $\mathcal{V}$-module structures on $\mathcal{S}$ introduced by Hovey

The asymptotic expansion of the heat kernel on a compact Lie group

Let $G$ be a compact connected Lie group equipped with a bi-invariant metric. We calculate the asymptotic expansion of the heat kernel of the laplacian on $G$ and the heat trace using Lie algebra methods. The Duflo isomorphism plays a key role

A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.Comment: 22 pages, final revision (mostly notation and presentation) for publicatio

Duflo Isomorphism and Chern-Weil Theory

We explain how the distributional index of a transversally elliptic operator on a principal G-manifold P that is obtained by lifting a Dirac operator on P/G can serve as a link between the Duflo isomorphism and Chern-Weil forms.Comment: 12 pages; removed restriction on the lifted operato

Borel-Weil-Bott Theorem via Equivariant McKean-Singer Formula

After reviewing how the Borel-Weil-Bott theorem can be interpreted as an index theorem, we present a proof using Kostant's cubic Dirac operator and the equivariant McKean-Singer formula

Inventory of Gravitational Microlenses toward the Galactic Bulge

We determine the microlensing event rate distribution, $\Gamma(t_{E})$, that is properly normalized by the first year MACHO group observations (Alcock et al. 1997). By comparing the determined $\Gamma(t_{E})$ with various MF models of lens populations to the observed distribution, we find that stars and WDs explain just $\sim 49$ of the total event rate and $\sim 37$ of the observationally determined optical depth even including very faint stars just above hydrogen-burning limit. Additionally, the expected time scale distribution of events caused by these known populations of lenses deviates significantly from the observed distribution especially in short time scale region. However, if the rest of the dynamically measured mass of the bulge ($\sim 2.1\times 10^{10} M_\odot$) is composed of brown dwarfs, both expected and observed event rate distribution matches very well.Comment: 15 pages including 1 tables and 4 figures, submitted to Ap

On a conjecture of Karasev

Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result

Orthogonal colorings of the sphere

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^2$, is a partition of $\mathbb{S}^2$ into parts such that no part contains a pair of orthogonal points, that is, a pair of points at spherical distance $\pi/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^2$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^2$ is such an octahedral coloring. In this paper we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given

The sandpile group of a trinity and a canonical definition for the planar Bernardi action

Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs. We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs. Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the $0$-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.Comment: Introduction rewritten, new figures adde