Thomas Westerbäck: Cyclic Flats for a Generalization of Matroids with Applications to Modules

Abstract:

Matroids were introduced independently in 1935–1936 by Hassler Whitney and Takeo Nakasawa, with the aim of capturing the essence of independence in an abstract, axiomatic framework encompassing objects such as vector spaces and graphs. Today, matroid theory is a rich field with deep connections to several areas of mathematics, including linear algebra, graph theory, hyperplane arrangements, Hodge theory, greedy algorithms, and information theory. One of the strengths of matroids is the many equivalent (cryptomorphic) definitions, which make the theory flexible by allowing one to choose the best representation, shift between viewpoints, and enable generalizations.

There are several generalizations of matroids. One of them allows us to examine modules with composition series, which ordinary matroids do not capture. This generalization is defined via a submodular function. In this talk, I will present how it can be cryptomorphically defined using cyclic flats, providing a more compact representation.