A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture

In this article we encode Hadwiger's covering conjecture and Borsuk's partition conjecture into continuous functions defined on the spaces of convex bodies, propose a four-step program to approach them, and obtain some partial results.Comment: 17 pages, four figure

Basic finite \'etale equivalence relations

We characterize quotient of a non-degenerate abelian fibration by a finite \'etale equivalence relation. We show that non-uniruled degenerations of each such quotient tend to be almost non-degenerate

On the translative packing densities of tetrahedra and cubooctahedra

In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.Comment: 37 pages, 9 figure

On Arnold's Problem on the Classifications of Convex Lattice Polytopes

In 1980, V.I. Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its analogues have been studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers of non-equivalent ddimensional convex lattice polytopes of given volume or cardinality have been achieved. In this paper, by introducing and studying the unimodular groups acting on convex lattice polytopes, we obtain lower bounds for the number of non-equivalent d-dimensional convex lattice polytopes of bounded volume or given cardinality, which are essentially tight.Comment: 15 pages, 3 figure

Minkowski Bisectors, Minkowski Cells, and Lattice Coverings

This article introduces and studies Minkowski Bisectors, Minkowski Cells, and Lattice Coverings.Comment: 9 figure

Classification of the sublattices of a lattice

In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $\Lambda$ of determinant ${\rm det}(\Lambda )$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(\Lambda )$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. This paper presents a systematic treatment for counting the sublattices and deduces a formula for the number of the sublattice classes of determinant $m\cdot {\rm det}(\Lambda )$.Comment: 8 page

A Formula of the One-leg Orbifold Gromov-Witten Vertex and Gromov-Witten Invariants of the Local \cB\bZ_m Gerbe

We give a formula of the framed one-leg orbifold Gromov-Witten vertex where the leg is gerby with isotropy group \bZ_m. Then we use this formula to compute the Gromov-Witten invariants of the local \cB\bZ_m gerbe. We will also compute some examples of the degree 1 and degree 2 \bZ_2-Hodge integrals.Comment: 32 page

Equidimensionality and regularity

The existence of an equidimensional morphism f with etale local sections from a regular algebraic space X to a locally noetherian normal algebraic space S of characteristic zero with excellent local rings implies that S is regular and f flat

Minimal models of L*

Let S be an algebraic space, A an S-abelian algebraic space, L an S-fiberwise numerically trivial invertible module on A, and L* the sheaf of regular sections of L considered as a G_m-torsor on A. We classify the S-minimal models of L* into two types

Generalized Mari\~no-Vafa Formula and Local Gromov-Witten Theory of Orbi-curves

We prove a generalized Mari\~{n}o-Vafa formula for Hodge integrals over \Mbar_{g, \gamma-\mu}(\cB G) with $G$ an arbitrary finite abelian group. Then we use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.Comment: 30 pages. arXiv admin note: text overlap with arXiv:math/0306434 by other author