{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/110798"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/110798","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Cotangent Schubert Calculus in Grassmannians","abstract":"We find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes.","abstract_html":"We find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes.","abstract_has_math":false,"creators":["Oetjen, David Christopher"],"institution":"Virginia Tech","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Mihalcea, Constantin Leonardo"],"committee_members":["Shimozono, Mark M.","Haskell, Peter E.","Orr, Daniel D."],"year":2022,"date_issued":"2022-06-15","date_published":"2022-06-15","updated_at":"2026-07-22T22:19:07Z","subjects":["Equivariant Cohomology","Equivariant K-Theory","Bott-Samelson Variety","Segre-MacPherson Class","Motivic Segre Class","Demazure-Lusztig Operator"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:35013"],"render_values":[{"text":"vt_gsexam:35013","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/110798","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Mihalcea, Constantin Leonardo"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Shimozono, Mark M.","Haskell, Peter E.","Orr, Daniel D."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Oetjen, David Christopher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-06-16T08:00:52Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-16T08:00:52Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-06-15"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Equivariant Cohomology","Equivariant K-Theory","Bott-Samelson Variety","Segre-MacPherson Class","Motivic Segre Class","Demazure-Lusztig Operator"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:35013"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/110798"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["Schubert calculus was first introduced in the nineteenth century as a way to answer certain questions in enumerative geometry. These computations relied on the multiplication of Schubert classes in the cohomology ring of Grassmannians, which parameterize k-dimensional linear subspaces of a vector space. More recently Schubert calculus has been broadened to refer to computations in generalized cohomology theories, such as (equivariant) K-theory. In this dissertation, we study Segre-MacPherson classes and motivic Segre classes of Schubert cells in Grassmannians. Segre-MacPherson classes are related to Chern-Schwartz-MacPherson classes, which are a generalization to singular spaces of the total Chern class of the tangent bundle. Motivic Segre classes are similarly related to motivic Chern classes, which are a K-theory analogue of Chern-Schwartz-MacPherson classes. This dissertation also studies the relationship between Schubert varieties and their Bott-Samelson desingularizations, specifically their (T-equivariant) cohomology and K-theory rings. Since equivariant cohomology (or K-theory) classes can be represented by polynomials, we can represent the Segre-MacPherson (or motivic Segre) classes as rational functions. Furthermore, we use certain operators that act on such polynomials (or rational functions) to find formulas for the rational function representatives of the aforementioned classes."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Cotangent Schubert Calculus in Grassmannians"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Mihalcea, Constantin Leonardo"],"dc:contributor.committeemember":["Shimozono, Mark M.","Haskell, Peter E.","Orr, Daniel D."],"dc:contributor.department":["Mathematics"],"dc:creator":["Oetjen, David Christopher"],"dc:date.accessioned":["2022-06-16T08:00:52Z"],"dc:date.available":["2022-06-16T08:00:52Z"],"dc:date.issued":["2022-06-15"],"dc:description.abstract":["We find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes."],"dc:description.abstractgeneral":["Schubert calculus was first introduced in the nineteenth century as a way to answer certain questions in enumerative geometry. These computations relied on the multiplication of Schubert classes in the cohomology ring of Grassmannians, which parameterize k-dimensional linear subspaces of a vector space. More recently Schubert calculus has been broadened to refer to computations in generalized cohomology theories, such as (equivariant) K-theory. In this dissertation, we study Segre-MacPherson classes and motivic Segre classes of Schubert cells in Grassmannians. Segre-MacPherson classes are related to Chern-Schwartz-MacPherson classes, which are a generalization to singular spaces of the total Chern class of the tangent bundle. Motivic Segre classes are similarly related to motivic Chern classes, which are a K-theory analogue of Chern-Schwartz-MacPherson classes. This dissertation also studies the relationship between Schubert varieties and their Bott-Samelson desingularizations, specifically their (T-equivariant) cohomology and K-theory rings. Since equivariant cohomology (or K-theory) classes can be represented by polynomials, we can represent the Segre-MacPherson (or motivic Segre) classes as rational functions. Furthermore, we use certain operators that act on such polynomials (or rational functions) to find formulas for the rational function representatives of the aforementioned classes."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:35013"],"dc:identifier.uri":["http://hdl.handle.net/10919/110798"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Equivariant Cohomology","Equivariant K-Theory","Bott-Samelson Variety","Segre-MacPherson Class","Motivic Segre Class","Demazure-Lusztig Operator"],"dc:title":["Cotangent Schubert Calculus in Grassmannians"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:07Z"}