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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

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1523979975350178

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1397599845

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 v_{N} 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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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