Toward a spectral theory of cellular sheaves

This paper outlines a program in what one might call spectral sheaf theory —an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to...

Full description

Saved in:
Bibliographic Details
Published in:Journal of applied and computational topology 2019-12, Vol.3 (4), p.315-358
Main Authors: Hansen, Jakob, Ghrist, Robert
Format: Article
Language:English
Subjects:
Algebraic Topology
Computational Science and Engineering
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473
cites cdi_FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473
container_end_page 358
container_issue 4
container_start_page 315
container_title Journal of applied and computational topology
container_volume 3
creator Hansen, Jakob
Ghrist, Robert
description This paper outlines a program in what one might call spectral sheaf theory —an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.
doi_str_mv 10.1007/s41468-019-00038-7
format article
fullrecord <record><control><sourceid>crossref_sprin</sourceid><recordid>TN_cdi_crossref_primary_10_1007_s41468_019_00038_7</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1007_s41468_019_00038_7</sourcerecordid><originalsourceid>FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473</originalsourceid><addsrcrecordid>eNp9j01PAjEQhhsjiQT5A576A6zOtN22ezTEDxISL3BuumUqmJUlLWj49y6u8ehp5jDPvO_D2A3CHQLY-6JRGycAawEAygl7wcZSGSvQKn35t0tzxaalbBtQKI0yBsfsdtl9hbzmgZc9xUMOLT9sqMsn3iUeqW2Pbci8bCh8UrlmoxTaQtPfOWGrp8fl7EUsXp_ns4eFiLICKzQSQVAgG-yjEijAtaxq5yRZSrKpggshYtSNs41yqtYoo6lJo0kxaqsmTA5_Y-5KyZT8Pm8_Qj55BH9W9oOy75X9j7I_Q2qASn-8e6Ps37tj3vU9_6O-AWZUWBU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Toward a spectral theory of cellular sheaves</title><source>Springer Nature</source><creator>Hansen, Jakob ; Ghrist, Robert</creator><creatorcontrib>Hansen, Jakob ; Ghrist, Robert</creatorcontrib><description>This paper outlines a program in what one might call spectral sheaf theory —an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.</description><identifier>ISSN: 2367-1726</identifier><identifier>EISSN: 2367-1734</identifier><identifier>DOI: 10.1007/s41468-019-00038-7</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Algebraic Topology ; Computational Science and Engineering ; Mathematical and Computational Biology ; Mathematics ; Mathematics and Statistics</subject><ispartof>Journal of applied and computational topology, 2019-12, Vol.3 (4), p.315-358</ispartof><rights>The Author(s) 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473</citedby><cites>FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Hansen, Jakob</creatorcontrib><creatorcontrib>Ghrist, Robert</creatorcontrib><title>Toward a spectral theory of cellular sheaves</title><title>Journal of applied and computational topology</title><addtitle>J Appl. and Comput. Topology</addtitle><description>This paper outlines a program in what one might call spectral sheaf theory —an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.</description><subject>Algebraic Topology</subject><subject>Computational Science and Engineering</subject><subject>Mathematical and Computational Biology</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>2367-1726</issn><issn>2367-1734</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9j01PAjEQhhsjiQT5A576A6zOtN22ezTEDxISL3BuumUqmJUlLWj49y6u8ehp5jDPvO_D2A3CHQLY-6JRGycAawEAygl7wcZSGSvQKn35t0tzxaalbBtQKI0yBsfsdtl9hbzmgZc9xUMOLT9sqMsn3iUeqW2Pbci8bCh8UrlmoxTaQtPfOWGrp8fl7EUsXp_ns4eFiLICKzQSQVAgG-yjEijAtaxq5yRZSrKpggshYtSNs41yqtYoo6lJo0kxaqsmTA5_Y-5KyZT8Pm8_Qj55BH9W9oOy75X9j7I_Q2qASn-8e6Ps37tj3vU9_6O-AWZUWBU</recordid><startdate>20191201</startdate><enddate>20191201</enddate><creator>Hansen, Jakob</creator><creator>Ghrist, Robert</creator><general>Springer International Publishing</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20191201</creationdate><title>Toward a spectral theory of cellular sheaves</title><author>Hansen, Jakob ; Ghrist, Robert</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algebraic Topology</topic><topic>Computational Science and Engineering</topic><topic>Mathematical and Computational Biology</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hansen, Jakob</creatorcontrib><creatorcontrib>Ghrist, Robert</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Journal of applied and computational topology</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hansen, Jakob</au><au>Ghrist, Robert</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Toward a spectral theory of cellular sheaves</atitle><jtitle>Journal of applied and computational topology</jtitle><stitle>J Appl. and Comput. Topology</stitle><date>2019-12-01</date><risdate>2019</risdate><volume>3</volume><issue>4</issue><spage>315</spage><epage>358</epage><pages>315-358</pages><issn>2367-1726</issn><eissn>2367-1734</eissn><abstract>This paper outlines a program in what one might call spectral sheaf theory —an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s41468-019-00038-7</doi><tpages>44</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 2367-1726
ispartof Journal of applied and computational topology, 2019-12, Vol.3 (4), p.315-358
issn 2367-1726
2367-1734
language eng
recordid cdi_crossref_primary_10_1007_s41468_019_00038_7
source Springer Nature
subjects Algebraic Topology
Computational Science and Engineering
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
title Toward a spectral theory of cellular sheaves
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-27T16%3A14%3A40IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-crossref_sprin&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Toward%20a%20spectral%20theory%20of%20cellular%20sheaves&rft.jtitle=Journal%20of%20applied%20and%20computational%20topology&rft.au=Hansen,%20Jakob&rft.date=2019-12-01&rft.volume=3&rft.issue=4&rft.spage=315&rft.epage=358&rft.pages=315-358&rft.issn=2367-1726&rft.eissn=2367-1734&rft_id=info:doi/10.1007/s41468-019-00038-7&rft_dat=%3Ccrossref_sprin%3E10_1007_s41468_019_00038_7%3C/crossref_sprin%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c2507-41ee0a302b1b03f0301d259882e7ef2b5a8aac1c4b87b3839412c69e416fcc473%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true