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Topological and Shifting Theoretic Methods in Combinatorics and AlgebraPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

KTH Royal Institute of Technology, 2016. , 152 p.
##### Series

TRITA-MAT-A, 2016:02
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-186136ISBN: 978-91-7595-899-6 (print)OAI: oai:DiVA.org:kth-186136DiVA: diva2:925608
##### Public defence

2016-06-07, F3, Lindstedtsvägen 26, Stockholm, 12:30 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

##### List of papers

This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.

QC 20160516

Available from: 2016-05-16 Created: 2016-05-02 Last updated: 2016-05-16Bibliographically approved1. Cellular structure for the Herzog–Takayama resolution$(function(){PrimeFaces.cw("OverlayPanel","overlay715774",{id:"formSmash:j_idt488:0:j_idt492",widgetVar:"overlay715774",target:"formSmash:j_idt488:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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4. Dimension filtration, sequential Cohen-Macaulayness and a new polynomial invariant of graded algebras$(function(){PrimeFaces.cw("OverlayPanel","overlay925571",{id:"formSmash:j_idt488:3:j_idt492",widgetVar:"overlay925571",target:"formSmash:j_idt488:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals$(function(){PrimeFaces.cw("OverlayPanel","overlay925575",{id:"formSmash:j_idt488:4:j_idt492",widgetVar:"overlay925575",target:"formSmash:j_idt488:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Connectivity of pseudomanifold graphs from an algebraic point of view$(function(){PrimeFaces.cw("OverlayPanel","overlay925574",{id:"formSmash:j_idt488:5:j_idt492",widgetVar:"overlay925574",target:"formSmash:j_idt488:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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