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Combinatorics Seminar: Laplacian Ideals, Arrangements, and Resolutions
September 11 at 5:00 PM to 6:00 PMUngar Room 402
Mathematics Lecture
Anton Dochtermann
University of Miami
will present
Laplacian Ideals, Arrangements, and Resolutions
Tuesday, September 11, 2012, 5:00pm
Ungar Room 402
Abstract: The lattice ideal of the Laplacian matrix of a graph G provides an algebraic perspective on the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The generators of this ideal form a Groebner bases with respect to a certain term order, and the associated initial ideals have nice connections to G-parking functions. We study resolutions of these initial ideals and show that, at least under certain conditions on G, a minimal free resolution is supported on the bounded subcomplex of a hyperplane section of the graphical arrangement of G. It is conjectured that these complexes also support resolutions for the Laplacian lattice ideal itself. This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles. Time permitting we will discuss some connections to the topology of generalized partition posets. This is joint work with Raman Sanyal.
University of Miami
will present
Laplacian Ideals, Arrangements, and Resolutions
Tuesday, September 11, 2012, 5:00pm
Ungar Room 402
Abstract: The lattice ideal of the Laplacian matrix of a graph G provides an algebraic perspective on the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The generators of this ideal form a Groebner bases with respect to a certain term order, and the associated initial ideals have nice connections to G-parking functions. We study resolutions of these initial ideals and show that, at least under certain conditions on G, a minimal free resolution is supported on the bounded subcomplex of a hyperplane section of the graphical arrangement of G. It is conjectured that these complexes also support resolutions for the Laplacian lattice ideal itself. This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles. Time permitting we will discuss some connections to the topology of generalized partition posets. This is joint work with Raman Sanyal.
