A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of these theories is Morse Theory. In this paper we present a combinatorial adaptation of Morse Theory, which we call discrete Morse Theory, that may be applied to any simplicial complex (or more general cell complex). The goal of this paper is to present an overview of the subject of discrete Morse Theory that is sucient both to understand the major applications of the theory to combinatorics, and to apply the the theory to new problems. We will not be presenting theorems in their most recent or most general form, and simple examples will often take the place of proofs. We hope to convey the fact that the theory is really very simple, and there is not much that one needs to know before one can become a "user".

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A user's guide to discrete morse theory

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Reflection Groups And Coxeter Groups

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology. These notes include introductory material, as well as recent developments and open problems. Some of the topics covered are: subspace arrangements, graph complexes, group actions on poset homology, shellability, recursive techniques, and fiber theorems.

/pdf/poset-topology-tools-and-applications-4hylfkbki6.pdf

Poset Topology: Tools and Applications

Noncrossing partitions

On optimizing discrete Morse functions

This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with O and I from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Mobius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset Π n /S λ of partitions of a set {1 λ 1,...,k λ k} with repetition is homotopy equivalent to a wedge of spheres of top dimension when A is a hook-shaped partition; it is likely that the proof may be extended to a larger class of A and perhaps to all A, despite a result of Ziegler (1986) which shows that Π n /S λ is not always Cohen-Macaulay.

/pdf/discrete-morse-functions-from-lexicographic-orders-309audnvix.pdf

Discrete Morse functions from lexicographic orders

Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximization of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the “100 Swiss Francs” problem, which we introduce and describe in detail.

/pdf/maximum-likelihood-estimation-in-latent-class-models-for-1rpuyitpsq.pdf

Maximum Likelihood Estimation in Latent Class Models For Contingency Table Data

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $\mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-representation on the $i(d-1)^{st}$ cohomology stabilizes sharply at $n=3i$ (resp. $n=3i+1$) when $d$ is odd (resp. even). 
The result comes from analyzing $S_n$-representations known to control the cohomology: the Whitney homology of set partition lattices for $d$ even, and the higher Lie representations for $d$ odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by $n\geq 4i$, where $i$ is the maximum rank selected. 
Further properties of the Whitney homology and more refined stability statements for $S_n$-isotypic components are also proven, including conjectures of J. Wiltshire-Gordon.

Representation stability for cohomology of configuration spaces in $\mathbf{R}^d$

Steingrimsson's coloring complex and Jonsson's unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ? n arrangements.

/pdf/coloring-complexes-and-arrangements-3vok4aks1x.pdf

Patricia Hersh

Papers

On optimizing discrete Morse functions

Discrete Morse functions from lexicographic orders

Maximum Likelihood Estimation in Latent Class Models For Contingency Table Data

Representation stability for cohomology of configuration spaces in $\mathbf{R}^d$

Coloring complexes and arrangements