{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101142"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101142","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Generalizations of no k-equal spaces","abstract":"We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the $n^{th}$ no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of $\\N^m$. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of $\\R^d$. Next, we consider the relation between no $k$-equal spaces of $\\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the $n^{th}$ no $k$-equal space of $\\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in $\\R^d$.","abstract_html":"We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the <span class=\"etd-inline-math\">n<sup>th</sup></span> no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of <span class=\"etd-inline-math\">\\N<sup>m</sup></span>. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of <span class=\"etd-inline-math\">\\R<sup>d</sup></span>. Next, we consider the relation between no $k$-equal spaces of $\\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the <span class=\"etd-inline-math\">n<sup>th</sup></span> no $k$-equal space of $\\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in <span class=\"etd-inline-math\">\\R<sup>d</sup></span>.","abstract_has_math":true,"creators":["Kosar, Nicholas J"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Baryshnikov, Yuliy","Hirani, Anil","Schenck, Hal","Yong, Alexander"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-04T20:33:59Z","date_published":"2018-09-04T20:33:59Z","updated_at":"2026-07-22T22:24:38Z","subjects":["Polychromatic configuration spaces","k-spanning trees","discriminantal arrangements"],"languages":["en"],"rights":["Copyright 2018 Nicholas Kosar"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101142","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baryshnikov, Yuliy","Hirani, Anil","Schenck, Hal","Yong, Alexander"]},{"key":"dc:creator","label":"Author","values":["Kosar, Nicholas J"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-04T20:33:59Z","2020-09-05T09:15:16Z","2018-04-10","2018-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Polychromatic configuration spaces","k-spanning trees","discriminantal arrangements"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Nicholas Kosar"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101142"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the $n^{th}$ no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of $\\N^m$. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of $\\R^d$. Next, we consider the relation between no $k$-equal spaces of $\\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the $n^{th}$ no $k$-equal space of $\\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in $\\R^d$.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-05-01","The student, Nicholas Kosar, accepted the attached license on 2018-04-07 at 12:04.","The student, Nicholas Kosar, submitted this Dissertation for approval on 2018-04-07 at 12:05.","This Dissertation was approved for publication on 2018-04-10 at 08:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12146 on 2018-08-31 at 17:18:16","Made available in DSpace on 2018-09-04T20:33:59Z (GMT). No. of bitstreams: 2 KOSAR-DISSERTATION-2018.pdf: 535418 bytes, checksum: de4bddc35e057da1ef4ff274491c039d (MD5) LICENSE.txt: 4211 bytes, checksum: 2c07369115c366140909668669b07ab9 (MD5) Previous issue date: 2018-04-10","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:34:13Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:37:00Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:42:08Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107225 on 2020-09-05T09:15:16Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Generalizations of no k-equal spaces"]}]}],"canonical_facts":{"dc:contributor":["Baryshnikov, Yuliy","Hirani, Anil","Schenck, Hal","Yong, Alexander"],"dc:creator":["Kosar, Nicholas J"],"dc:date":["2018-09-04T20:33:59Z","2020-09-05T09:15:16Z","2018-04-10","2018-05"],"dc:description":["We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the $n^{th}$ no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of $\\N^m$. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of $\\R^d$. Next, we consider the relation between no $k$-equal spaces of $\\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the $n^{th}$ no $k$-equal space of $\\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in $\\R^d$.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-05-01","The student, Nicholas Kosar, accepted the attached license on 2018-04-07 at 12:04.","The student, Nicholas Kosar, submitted this Dissertation for approval on 2018-04-07 at 12:05.","This Dissertation was approved for publication on 2018-04-10 at 08:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12146 on 2018-08-31 at 17:18:16","Made available in DSpace on 2018-09-04T20:33:59Z (GMT). No. of bitstreams: 2 KOSAR-DISSERTATION-2018.pdf: 535418 bytes, checksum: de4bddc35e057da1ef4ff274491c039d (MD5) LICENSE.txt: 4211 bytes, checksum: 2c07369115c366140909668669b07ab9 (MD5) Previous issue date: 2018-04-10","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:34:13Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:37:00Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:42:08Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107225 on 2020-09-05T09:15:16Z."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/101142"],"dc:language":["en"],"dc:rights":["Copyright 2018 Nicholas Kosar"],"dc:subject":["Polychromatic configuration spaces","k-spanning trees","discriminantal arrangements"],"dc:title":["Generalizations of no k-equal spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:38Z"}