The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 2:20-3:35PM in Room 164 of the Adel Mathematics Building. If you are interested in giving a talk, please contact Mike Falk or Dana C. Ernst, ACGT coordinators.
Speaker: Nandor Sieben
Title: Introduction to graph pebbling
Abstract: Graph pebbling is a model for the transport of consumable resources like fuel. Consider a country where some of the towns have fuel reserves. To transfer a barrel from a town to a neighboring town requires the use of one barrel. We will consider questions like: How many barrels will guarantee that a barrel can be transported to any town having an emergency? What is the best placement of the fuel repositories? The talk will start with the basics and will progress slowly. It will be accessible to undergraduates.
Speaker: Michael Falk
Title: Hypergraphs, matroids, and graded algebras
Abstract: Given a simple graph, or more generally a matroid, there is an associated graded algebra, the Orlik-Solomon (OS) algebra, and its quadratic closure, the quadratic OS algebra, that carry a lot of combinatorial information about the original graph or matroid, as well as topological and geometric information about associated topological spaces and algebraic varieties. It is an unsolved problem to determine necessary and sufficient conditions for two matroids, even two graphs, to have isomorphic OS algebras. We will first sketch the proof that two matroids have isomorphic quadratic OS algebras if and only if the hypergraphs formed by their nontrivial ranktwo flats are 2-isomorphic. The proof uses a result of Vertigan and Whittle concerning Dilworth truncation of hypergraphic polymatroids, generalizing the well-known Whitney 2-isomorphism theorem. We will proceed to describe how the argument generalizes to classify OS algebras using filtered 2-isomorphisms, using a construction of Lovasz, and observe connections with exterior StanleyReisner rings and small covers, as time permits. This is a report on work in progress with Geoff Whittle. All notions will be explained and illustrated; no familiarity with any of the above will be assumed.
Speaker: Dana C. Ernst
Title: Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes
Abstract: Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. We say that two reduced expressions are braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes (i.e., isometric to an induced subgraph of a hypercube). For a special class of links, called Fibonacci links, we explicitly describe the isometry from the corresponding braid graph to a Fibonacci cube graph. Time permitting, we will also summarize current research with Jillian Barnes that attempts to prove that braid graphs in simply-laced triangle-free Coxeter systems are median and/or the covering graph for distributive lattices.