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Dissertation Defense-Nicholas Armenoff

Date:
-
Location:
745 Patterson Office Tower
Speaker(s) / Presenter(s):
Nicholas Armenoff, University of Kentucky

Title:  Free Resolutions Associated to Representable Matroids

 

Abstract:  As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution.  Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes.  Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid duals to finite projective geometries and discuss consequences for finite affine geometries.  Finally, we provide algorithms for computing such cellular resolutions explicitly.