Defense of dissertation in the field of mathematics, M.Sc. Laura Jakobsson
The thesis studies cellular resolutions and the invariants of resolutions of monomial ideals. The area of the thesis is combinatorial commutative algebra, and as much of pure mathematics, the studied questions in the thesis are motivated mainly by fascination towards these combinatorial mathematical objects and applying new tools to study them. The questions on resolutions and their invariants have been around for a long time, and over the years they have become a rich topic, with a variety of directions including cellular resolutions. We look at cellular resolutions from a category-theoretic point of view and apply tools from representation stability to study them. Representation stability is a relatively new area of mathematics and uses abstract methods to study stability in families of mathematical objects.
The first part of the thesis focuses on cellular resolutions and categorical representation stability. Among the main results is the definition of the category of cellular resolutions and establishing the basic properties for it. Furthermore, it is shown that important topological constructions, like the homotopy colimit, lift to this category and that discrete and algebraic Morse maps are morphisms in this category. The category of cellular resolutions opens up cellular resolutions for applying tools of representations of categories, and the application these tools provides the remaining main results of the thesis. The most important result being that the particular families of cellular resolutions have finite generating sets for their syzygies. The second part of the thesis is on the combinatorial formulas for algebraic invariants of ideals coming from specific graphs, known as Booth—Lueker graphs. The invariants in question come from free resolutions and keep in the theme of the topics studied in the other parts of the thesis.
Opponent is Professor Anton Dochtermann, Texas State University, USA
Custos is Professor Alexander Engström, Aalto University School of Science, Department of Mathematics and Systems Analysis
The public defence will be organised via Zoom. Link to the event
The dissertation is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University