Title:
Algebraic and semi-algebraic invariants on quadrics

dc.contributor.advisor Blekherman, Grigoriy
dc.contributor.author Jung, Jaewoo
dc.contributor.committeeMember Baker, Matthew H.
dc.contributor.committeeMember Leykin, Anton
dc.contributor.committeeMember Sinn, Rainer
dc.contributor.committeeMember Yu, Josephine
dc.contributor.department Mathematics
dc.date.accessioned 2022-08-25T13:38:33Z
dc.date.available 2022-08-25T13:38:33Z
dc.date.created 2022-08
dc.date.issued 2022-07-30
dc.date.submitted August 2022
dc.date.updated 2022-08-25T13:38:33Z
dc.description.abstract This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics. The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals. The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/67301
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Castelnuovo-Mumford regularity
dc.subject quadratic monomial ideals
dc.subject Stanley-Reisner correspondence
dc.subject Sums of squares
dc.subject Dual cone
dc.subject Hankel index
dc.title Algebraic and semi-algebraic invariants on quadrics
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Blekherman, Grigoriy
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isAdvisorOfPublication c17f90d7-b83c-46a2-807f-eba9a5e4c98f
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level Doctoral
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