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You searched for +publisher:"The Ohio State University" +contributor:("Kennedy, Gary"). Showing records 1 – 2 of 2 total matches.

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The Ohio State University

1. Nash, Evan D., Nash. Extended Tropicalization of Spherical Varieties.

Degree: PhD, Mathematics, 2018, The Ohio State University

The first steps in defining a notion of spherical tropicalization were recently takenby Tassos Vogiannou in his thesis and by Kiumars Kaveh and Christopher Manonin a related paper. Broadly speaking, the classical notion of tropicalization concernsitself with valuations on the function field of a toric variety that are invariant underthe action of the torus. Spherical tropicalization is similar, but considers insteadspherical G-varieties and G-invariant valuations.The core idea of my dissertation is the construction of the extended tropicalizationof a spherical embedding. Vogiannou, Kaveh, and Manon only concern themselveswith subvarieties of a spherical homogeneous space G/H. My thesis describes how totropicalize a spherical embedding by tropicalizing the additional G-orbits of X andadding them to the tropicalization of G/H as limit points. This generalizes work doneby Kajiwara and Payne for toric varieties and affords a means for understanding howto tropicalize the compactification of a subvariety of G/H in X.The extended tropicalization construction can be described from three differentperspectives. The first uses the polyhedral geometry of the colored fan and the secondextends the Grobner theory definition given by Kaveh and Manon. The third methodworks by embedding the spherical variety in a specially-constructed toric variety,tropicalizing there with the standard theory, and then applying a particular piecewise-projection map. This final perspective introduces a novel means for tropicalizing a homogeneous space that allows us to prove several statements about the structure of a spherical tropicalization by transferring results from the toric world where more is known.We also suggest a definition for the tropicalization of subvarieties of a homogeneousspace whose defining equations have coefficients with non-trivial valuation. Allthe previous theory has been done in the constant coefficient case, i.e. when thecoefficients of the defining equations all have trivial valuation. Advisors/Committee Members: Kennedy, Gary (Advisor).

Subjects/Keywords: Mathematics; tropical geometry; algebraic geometry; spherical varieties; spherical homogeneous spaces

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APA (6th Edition):

Nash, Evan D., N. (2018). Extended Tropicalization of Spherical Varieties. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1523979975350178

Chicago Manual of Style (16th Edition):

Nash, Evan D., Nash. “Extended Tropicalization of Spherical Varieties.” 2018. Doctoral Dissertation, The Ohio State University. Accessed January 17, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1523979975350178.

MLA Handbook (7th Edition):

Nash, Evan D., Nash. “Extended Tropicalization of Spherical Varieties.” 2018. Web. 17 Jan 2021.

Vancouver:

Nash, Evan D. N. Extended Tropicalization of Spherical Varieties. [Internet] [Doctoral dissertation]. The Ohio State University; 2018. [cited 2021 Jan 17]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1523979975350178.

Council of Science Editors:

Nash, Evan D. N. Extended Tropicalization of Spherical Varieties. [Doctoral Dissertation]. The Ohio State University; 2018. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1523979975350178

2. Miller, Jason A. Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications.

Degree: PhD, Mathematics, 2014, The Ohio State University

This thesis draws a connection between two areas of algebraic geometry, spherical varieties and Okounkov bodies, in order to study the structure of Borel orbit closures in wonderful group compactifications. Spherical varieties are a natural generalization of many classes of varieties equipped with group actions such as flag varieties, symmetric varieties, and toric varieties. The theory of Okounkov bodies is a fascinating recent development generalizing the polytopes that appear in toric geometry to any projective algebraic variety.Let X be a projective spherical G-variety equipped with a very ample G-line bundle L. Choosing a reduced decomposition of the longest element of the Weyl group determines a valuation vN on the ring of sections, R(X,L). One can then use Okounkov theory to encode information about the G-orbits of a spherical variety in terms of the associated Newton polytope. Each G-orbit closure of X determines a face of the Newton polytope. This correspondence allows one to use the combinatorial methods of convex geometry to answer questions about the G-orbit closures of the spherical variety X. However for nontoric spherical varieties, the G-orbit structure is too coarse-grained. A great deal of information about the spherical variety, such as the intersection theory, is determined by the structure of the Borel orbits. In this thesis we consider wonderful group compactifications. We prove that one can extend the correspondence between G-orbits and faces to the Borel orbits for this class of varieties. Given any Borel orbit closure of a wonderful group compactification, we show that the Okounkov construction will yield a finite union of faces of the Newton polytope. This correspondence can be shown to enjoy many of the same nice properties as in the case of G-orbits: the dimension of the space of global sections of L is given by the number of lattice points in the union of faces, and the degree of any Borel orbit closure is the sum of the normalized volumes of the associated faces. Advisors/Committee Members: Kennedy, Gary (Advisor).

Subjects/Keywords: Mathematics; spherical varieties; Okounkov; wonderful group compactifications; Borel orbits; toric varieties; flag varieties; Newton-Okounkov; polytopes; string polytope; moment polytope; algebraic geometry; Schubert varieties; standard monomial theory; crystal basis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Miller, J. A. (2014). Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1397599845

Chicago Manual of Style (16th Edition):

Miller, Jason A. “Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications.” 2014. Doctoral Dissertation, The Ohio State University. Accessed January 17, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1397599845.

MLA Handbook (7th Edition):

Miller, Jason A. “Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications.” 2014. Web. 17 Jan 2021.

Vancouver:

Miller JA. Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications. [Internet] [Doctoral dissertation]. The Ohio State University; 2014. [cited 2021 Jan 17]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1397599845.

Council of Science Editors:

Miller JA. Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications. [Doctoral Dissertation]. The Ohio State University; 2014. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1397599845

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