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dc.contributor.advisorSottile, Frank
dc.creatorHein, Nickolas Jason
dc.date.accessioned2013-12-16T20:02:29Z
dc.date.available2013-12-16T20:02:29Z
dc.date.created2013-08
dc.date.issued2013-06-21
dc.date.submittedAugust 2013
dc.identifier.urihttp://hdl.handle.net/1969.1/151084
dc.description.abstractThe Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. When conjectured, it sparked interest in real osculating Schubert calculus, and computations played a large role in developing the surrounding theory. Our purpose is to uncover generalizations of the Mukhin-Tarasov-Varchenko Theorem, proving them when possible. We also improve the state of the art of computationally solving Schubert problems, allowing us to more effectively study ill-understood phenomena in Schubert calculus. We use supercomputers to methodically solve real osculating instances of Schubert problems. By studying over 300 million instances of over 700 Schubert problems, we amass data significant enough to reveal generalizations of the Mukhin-Tarasov- Varchenko Theorem and compelling enough to support our conjectures. Combining algebraic geometry and combinatorics, we prove some of these conjectures. To improve the efficiency of solving Schubert problems, we reformulate an instance of a Schubert problem as the solution set to a square system of equations in a higher- dimensional space. During our investigation, we found the number of real solutions to an instance of a symmetrically defined Schubert problem is congruent modulo four to the number of complex solutions. We proved this congruence, giving a generalization of the Mukhin-Tarasov-Varchenko Theorem and a new invariant in enumerative real algebraic geometry. We also discovered a family of Schubert problems whose number of real solutions to a real osculating instance has a lower bound depending only on the number of defining flags with real osculation points. We conclude that our method of computational investigation is effective for uncovering phenomena in enumerative real algebraic geometry. Furthermore, we point out that our square formulation for instances of Schubert problems may facilitate future experimentation by allowing one to solve instances using certifiable numerical methods in lieu of more computationally complex symbolic methods. Additionally, the methods we use for proving the congruence modulo four and for producing an
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.subjectSchubert Calculus
dc.subjectSquare Systems
dc.subjectCertification
dc.subjectReal Algebraic Geometry
dc.subjectComputational Algebraic Geometry
dc.subjectEnumerative Algebraic Geometry
dc.titleReality and Computation in Schubert Calculus
dc.typeThesis
thesis.degree.departmentMathematics
thesis.degree.disciplineMathematics
thesis.degree.grantorTexas A & M University
thesis.degree.nameDoctor of Philosophy
thesis.degree.levelDoctoral
dc.contributor.committeeMemberLandsberg, Joseph
dc.contributor.committeeMemberRojas, J. Maurice
dc.contributor.committeeMemberAmato, Nancy
dc.type.materialtext
dc.date.updated2013-12-16T20:02:29Z


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