Papers and Preprints
7.
On the mass center of the tent map, (with Kuo-chang Chen), (PDF file, PostScript file, DVI file), preprint, last
updated on 2/19/2008.
Abstract: It is well known that the time average or the
center of mass for generic orbits of the standard tent map is $0.5$. In this
paper we show some interesting properties of the exceptional orbits, including
periodic orbits, orbits without mass center, and orbits with mass centers
different from $0.5$. We prove that for any positive integer $n$, there exist
$n$ distinct periodic orbits for the standard tent map with the same center of
mass, and the set of mass centers of periodic orbits is a dense subset of
$[0,2/3]$. Considering all possible orbits, then the set of mass centers is the
interval $[0,2/3]$. Moreover, for every $x$ in $[0,2/3]$, there are uncountably many
orbits with mass center $x$. We also show that there are uncountably
many orbits without mass center.
6.
The bounded complex of a uniform affine oriented matroid
is a ball, (PDF file, PostScript file, DVI file), updated version (this
is the final version that will appear in J.
Combin. Theory Ser. A).
Abstract: Zaslavsky conjectures
that the bounded complex of a simple hyperplane
arrangement is homeomorphic to a ball. We prove this
conjecture for the more general uniform affine oriented matroids.
5. On the bounded complex of an affine
oriented matroid, (PDF
file, PostScript
file, DVI
file), Discrete Comput. Geom.
35 (2006), no. 3, 457—471.
Abstract: We prove
that the bounded complex of an affine oriented matroid
is pure and collapsible. We also generalize Zaslavsky's
Central Decomposition Theorem for hyperplane
arrangements to affine oriented matroids.
4. Topology of bounded-degree graph complexes, J. of Algebra 262( 2003), no. 2, 287--312.
Abstract: The bounded degree graph complexes were first introduced by Reiner and Roberts. They arise from the finite free resolution of quadratic Veronese rings and modules. We prove various results about the homotopy types of these complexes, and deduce corresponding characteristic-free results about the quadratic Veronese resolutions. In particular, we characterize the set of multidegrees which support at least one higher syzygy in this resolution. The answer turns out to be independent of the field characteristic.
3. Alexander duality for projections of polytopes, Topology 41 (2002), no. 6, 1109--1121.
Abstract: An affine projection
$\pi: P^p \rightarrow Q^q$ of convex polytopes induces
an inclusion map of the face posets $i: \F(Q) \rightarrow
\F(P)$. We define an order-preserving map of posets
$h: \F(P) \rightarrow \Susp^{p-q}\F(Q)$
such that for any filter $J$ of $\Susp^{p-q}\F(Q)$,
the map $h$ restricts to a homotopy equivalence
between the order complexes of $h^{-1}(J)$ and $J$. As applications we prove:
(1) A conjecture of
2. Combinatorial Laplacian of the matching complex, (with Michelle Wachs), Electron. J. Combin. 9 (2002), no. 1, R17, 11pp.
Abstract: A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral.
1. Canonical
modules of semigroup rings and a conjecture of Reiner,
Geometric combinatorics (
Abstract: We prove that the homotopy types of two naturally defined simplicial complexes are related in the following way: one is homotopy equivalent to a multiple suspension of the canonical Alexander dual of the other. These simplicial complexes arise in free resolutions of semigroup rings and modules. The relation between their homotopy types was conjectured by Reiner, and was suggested by a homological consequence of a result due independently to Danilov and Stanley on canonical modules of normal semigroup rings. Our proof is purely topological, and gives an alternative proof of their result. We also prove a generalization of result of Hochster saying that these rings are Cohen-Macaulay, and indicate a new proof of Ehrhart's reciprocity law for rational polytopes.
My Ph.D. Thesis
The
topology of bounded degree graph complexes and finite free resolutions,
Ph.D. Thesis, University of