Representation Stability for Cellular Resolutions

dc.contributorAalto-yliopistofi
dc.contributorAalto Universityen
dc.contributor.advisorEngström, Alexander, Assoc. Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
dc.contributor.authorJakobsson, Laura
dc.contributor.departmentMatematiikan ja systeemianalyysin laitosfi
dc.contributor.departmentDepartment of Mathematics and Systems Analysisen
dc.contributor.schoolPerustieteiden korkeakoulufi
dc.contributor.schoolSchool of Scienceen
dc.contributor.supervisorEngström, Alexander, Assoc. Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
dc.date.accessioned2021-03-10T10:00:10Z
dc.date.available2021-03-10T10:00:10Z
dc.date.defence2021-03-25
dc.date.issued2021
dc.description.abstractThis thesis is on cellular resolutions and the invariants of resolutions of monomial ideals. The general area of these topics 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.In Publication I, we define the category of cellular resolutions and establish the basic properties for it. Among these results are showing that homotopy colimit lifts from topology and that discrete and algebraic Morse maps are morphisms in this category. Having the category of cellular resolutions opens up cellular resolutions for applying tools of representations of categories, and we use these in Publication II to show that that specific families of cellular resolutions have finitely generated syzygies. The main tools used are defining a linear family that satisfies noetherianity properties of representation stability and viewing syzygies as a representation of the category of cellular resolutions. In particular, we show that the powers of maximal monomial ideals of a polynomial ring have finitely generated syzygies. We also touch upon the case of families with cellular resolutions over different rings and show that the finite generation of syzygies applies in this setting in special cases. The last publication covers combinatorial formulas for algebraic invariants of edge ideals of Booth-Lueker graphs.en
dc.format.extent88 + app. 116
dc.format.mimetypeapplication/pdfen
dc.identifier.isbn978-952-64-0264-2 (electronic)
dc.identifier.isbn978-952-64-0263-5 (printed)
dc.identifier.issn1799-4942 (electronic)
dc.identifier.issn1799-4934 (printed)
dc.identifier.issn1799-4934 (ISSN-L)
dc.identifier.urihttps://aaltodoc.aalto.fi/handle/123456789/103017
dc.identifier.urnURN:ISBN:978-952-64-0264-2
dc.language.isoenen
dc.opnDochtermann, Anton, Prof., Texas State University, USA
dc.publisherAalto Universityen
dc.publisherAalto-yliopistofi
dc.relation.haspart[Publication 1]: Laura Jakobsson. The category of cellular resolutions. Submitted to Journal of Commutative Algebra, submission date April 2019
dc.relation.haspart[Publication 2]: Laura Jakobsson. Families of cellular resolution, their syzygies, and stability. Submitted to Algebraic Combinatorics, submission date March 2020
dc.relation.haspart[Publication 3]: Alexander Engström, Laura Jakobsson, Milo Orlich. Explicit Boij-Söderberg theory of ideals from a graph isomorphism reduction. Accepted for publication in Journal of Pure and Applied Algebra, 25 pp., February 2020. DOI: 10.1016/j.jpaa.2020.106405
dc.relation.ispartofseriesAalto University publication series DOCTORAL DISSERTATIONSen
dc.relation.ispartofseries17/2021
dc.revBenedetti, Bruno, Prof., University of Miami, USA
dc.revDochtermann, Anton, Prof., Texas State University, USA
dc.subject.keywordcellular resolutionsen
dc.subject.keywordrepresentation stabilityen
dc.subject.otherMathematicsen
dc.titleRepresentation Stability for Cellular Resolutionsen
dc.typeG5 Artikkeliväitöskirjafi
dc.type.dcmitypetexten
dc.type.ontasotDoctoral dissertation (article-based)en
dc.type.ontasotVäitöskirja (artikkeli)fi
local.aalto.acrisexportstatuschecked 2021-03-29_1005
local.aalto.archiveyes
local.aalto.formfolder2021_03_10_klo_10_46
Files
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
Name:
isbn9789526402642.pdf
Size:
588.74 KB
Format:
Adobe Portable Document Format