# An algebraic approach to the German noun phrase

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Linguistic Analysis, Volume 31, p.291-300 (2004)# An algebraic approach to Arabic sentence structure

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Linguistic Analysis, Volume 31, p.301-315 (2004)# A computational algebraic approach to English grammar

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Syntax, Volume 7, Issue 2, p.128-147 (2004)# What is the world of mathematics?

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Annals of Pure and Applied Logic, Volume 126, p.149-158 (2004)# Subgroups of fully residually free groups: algorithmic problems

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Contemporary Mathematical Series of the AMS. Group theory, Statistics and Cryptography, Volume 360, p.63-101 (2004)# Supersymmetry and algebraic Darboux transformations

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Journal of Physics A, Mathematical and General, Volume 37, p.10065-10078 (2004)# The Darboux transformation and algebraic deformations of shape-invariant potentials

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Journal of Physics A, Mathematical and General, Volume 37, p.1789-1804 (2004)# Geometry and Dynamics of Surface Group Representations.

The space of representations of the fundamental group of a surface in a Lie group is a rich geometric object. Examples include symplectic vector spaces, Jacobi varieties and Teichmueller spaces. The topological symmetries of the surface acts on this space preserving a natural Poisson geometry. This action of the mapping class group closely relates to Hamiltonian flows on these moduli spaces. When the Lie group is compact, the action is chaotic. For uniformization representations corresponding to geometric structures, the action is properly discontinuous. In general the dynamics falls between these two extremes.

# Isbell duality II

An Isbell duality arises when the "same" object lives in two different categories. Examples abound, often when the object is 2. We attempt to flesh out this concept of an object in two categories. We give one new example that leads to a duality between totally disconnected Tychonoff (completely regular hausdorff) spaces and a class of subrings of powers of Z. In this case, Z is the object in question. We show how this applies to the known dualities between all Tychonoff spaces and a class of subrings of powers of R as well as to the classical Isbell duality between sober spaces and powers of the Sierpinski space.

Joint work with John Kennison and Robert Raphael.