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Dowling lattices arose in connection with the theory of linear codes and as q-analogs of partition lattices, and they were quickly seen to be of use for studying arrangements of hyperplanes. These geometric lattices also provided the motivation for biased graphs, and, like projective spaces and free matroids, they are the universal models for one of the five types of varieties of combinatorial geometries. This talk will provide an introduction to these lattices and will outline several recent advances in the theory.
The first class of results to be treated brings out similarities between Dowling lattices and projective spaces. This includes an axiom scheme for Dowling lattices, analogs of Desargues' and Pappus' theorems, and a coordinatization theory analogous to that for projective spaces. We will also present an application of the automorphism theory for Dowling lattices.
The second collection of results to be discussed involves the Tutte polynomial, the matroid counterpart of the dichromatic polynomial of graph theory. The Tutte polynomial of any matroid contains much information about the matroid, and this is especially true for Dowling lattices. We will explore exactly how much information about a Dowling lattice is reflected in its Tutte polynomial.