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Topological CombinatoricsPrimeFaces.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}); 2009 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Stockholm: KTH , 2009. , vii, 16 p.
##### Series

Trita-MAT. MA, ISSN 1401-2278
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-10383ISBN: 978-91-7415-256-2OAI: oai:DiVA.org:kth-10383DiVA: diva2:216411
##### Public defence

2009-05-08, Sal E2, KTH, Lindstedtsvägen 5, Stockholm, 13:00 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

QC 20100712Available from: 2009-05-08 Created: 2009-05-08 Last updated: 2010-07-12Bibliographically approved
##### List of papers

This thesis on Topological Combinatorics contains 7 papers. All of them but paper Bare published before.In paper A we prove that!i dim ˜Hi(Ind(G);Q) ! |Ind(G[D])| for any graph G andits independence complex Ind(G), under the condition that G\D is a forest. We then use acorrespondence between the ground states with i+1 fermions of a supersymmetric latticemodel on G and ˜Hi(Ind(G);Q) to deal with some questions from theoretical physics.In paper B we generalize the topological Tverberg theorem. Call a graph on the samevertex set as a (d + 1)(q − 1)-simplex a (d, q)-Tverberg graph if for any map from thesimplex to Rd there are disjoint faces F1, F2, . . . , Fq whose images intersect and no twoadjacent vertices of the graph are in the same face. We prove that if d # 1, q # 2 is aprime power, and G is a graph on (d+1)(q −1)+1 vertices such that its maximal degreeD satisfy D(D + 1) < q, then G is a (d, q)–Tverberg graph. It was earlier known that thedisjoint unions of small complete graphs, paths, and cycles are Tverberg graphs.In paper C we study the connectivity of independence complexes. If G is a graphon n vertices with maximal degree d, then it is known that its independence complex is(cn/d + !)–connected with c = 1/2. We prove that if G is claw-free then c # 2/3.In paper D we study when complexes of directed trees are shellable and how one canglue together independence complexes for finding their homotopy type.In paper E we prove a conjecture by Björner arising in the study of simplicial polytopes.The face vector and the g–vector are related by a linear transformation. We prove thatthis matrix is totaly nonnegative. This is joint work with Michael Björklund.In paper F we introduce a generalization of Hom–complexes, called set partition complexes,and prove a connectivity theorem for them. This generalizes previous results ofBabson, Cukic, and Kozlov, and questions from Ramsey theory can be described with it.In paper G we use combinatorial topology to prove algebraic properties of edge ideals.The edge ideal of G is the Stanley-Reisner ideal of the independence complex of G. Thisis joint work with Anton Dochtermann.

1. Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory$(function(){PrimeFaces.cw("OverlayPanel","overlay216347",{id:"formSmash:j_idt503:0:j_idt507",widgetVar:"overlay216347",target:"formSmash:j_idt503:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Tverberg graphs$(function(){PrimeFaces.cw("OverlayPanel","overlay216349",{id:"formSmash:j_idt503:1:j_idt507",widgetVar:"overlay216349",target:"formSmash:j_idt503:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Independence complexes of claw-free graphs$(function(){PrimeFaces.cw("OverlayPanel","overlay216351",{id:"formSmash:j_idt503:2:j_idt507",widgetVar:"overlay216351",target:"formSmash:j_idt503:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Complexes of directed trees and independence complexes$(function(){PrimeFaces.cw("OverlayPanel","overlay216353",{id:"formSmash:j_idt503:3:j_idt507",widgetVar:"overlay216353",target:"formSmash:j_idt503:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. The *g*-theorem matrices are totally nonnegative$(function(){PrimeFaces.cw("OverlayPanel","overlay216356",{id:"formSmash:j_idt503:4:j_idt507",widgetVar:"overlay216356",target:"formSmash:j_idt503:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Set Partition Complexes$(function(){PrimeFaces.cw("OverlayPanel","overlay216358",{id:"formSmash:j_idt503:5:j_idt507",widgetVar:"overlay216358",target:"formSmash:j_idt503:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. Algebraic properties of edge ideals via combinatorial topology$(function(){PrimeFaces.cw("OverlayPanel","overlay216360",{id:"formSmash:j_idt503:6:j_idt507",widgetVar:"overlay216360",target:"formSmash:j_idt503:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1196",{id:"formSmash:lower:j_idt1196",widgetVar:"widget_formSmash_lower_j_idt1196",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1197_j_idt1199",{id:"formSmash:lower:j_idt1197:j_idt1199",widgetVar:"widget_formSmash_lower_j_idt1197_j_idt1199",target:"formSmash:lower:j_idt1197:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});