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Full-Text Articles in Algebra
Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok
Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok
Theses and Dissertations--Mathematics
This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.
We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' …
Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj
Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj
Theses and Dissertations--Mathematics
In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.