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

We propose a comprehensive Bayesian approach for graphical model determination in observational studies that can accommodate binary, ordinal or continuous variables simultaneously. Our new models are called copula Gaussian graphical models (CGGMs) and embed graphical model selection inside a semiparametric Gaussian copula. The domain of applicability of our methods is very broad and encompasses many studies from social science and economics. We illustrate the use of the copula Gaussian graphical models in the analysis of a 16-dimensional functional disability contingency table.

Copula Gaussian graphical models and their application to modeling functional disability data

/pdf/copula-gaussian-graphical-models-and-their-application-to-2eoexzm6jq.pdf

Algebraic statistics advocates polynomial algebra as a tool for addressing problems in statistics and its applications. This connection is based on the fact that most statistical models are defined either parametrically or implicitly via polynomial equations. The idea is summarized by the phrase "Statistical models are semialgebraic sets". My tutorial will consist of a detailed study of two examples where the algebra/statistics connection has proven especially useful: in the study of phylogenetic models and in the analysis of contingency tables.

Algebraic Statistics

In Providence, R. I.

On optimizing discrete Morse functions

The Mathematical Legacy of Richard P. Stanley

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 $\mathbf{R}^d$

llReiner and Webb (preprint, 2002) compute the S n -module structure for the complex of injective words This paper refines their formula by providing a Hodge type decomposition Along the way, this paper proves that the simplicial boundary map interacts in a nice fashion with the Eulerian idempotents The Laplacian acting on the top chain group in the complex of injective words is also shown to equal the signed random to random shuffle operator Uyemura-Reyes, 2002, conjectured that the (unsigned) random to random shuffle operator has integral spectrum We prove that this conjecture would imply that the Laplacian on (each chain group in) the complex of injective words has integral spectrum

https://msp.org/pjm/2004/214-1/pjm-v214-n1-p07-s.pdf

A Hodge decomposition for the complex of injective words

Representation Stability for Cohomology of Configuration Spaces in ${\mathbb{R}}^d$

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