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Mirror Symmetry on Toric Surfaces vi...

Meredith, M. Brandon.

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Mirror Symmetry on Toric Surfaces via Tropical Geometry.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Mirror Symmetry on Toric Surfaces via Tropical Geometry./
Author:	Meredith, M. Brandon.
Description:	126 p.
Notes:	Source: Dissertation Abstracts International, Volume: 75-01(E), Section: B.
Contained By:	Dissertation Abstracts International75-01B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3596039
ISBN:	9781303424229

Mirror Symmetry on Toric Surfaces via Tropical Geometry.
Meredith, M. Brandon.

Mirror Symmetry on Toric Surfaces via Tropical Geometry. - 126 p.

Source: Dissertation Abstracts International, Volume: 75-01(E), Section: B.

Thesis (Ph.D.)--University of California, San Diego, 2013.

Mirror symmetry is a curious duality, first noticed by physicists and then excitedly embraced by mathematicians, between certain manifolds (the A-side) and their "mirror" spaces (the B-side). On the A-side, certain counts of objects on a manifold are carried out (the purview of enumerative geometry), while on the B-side, special functions are integrated. Interestingly, the same numbers come out on both sides.

ISBN: 9781303424229Subjects--Topical Terms:

515831
Mathematics.

Mirror Symmetry on Toric Surfaces via Tropical Geometry.
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100 1 $a Meredith, M. Brandon. $3 2049885
245 1 0 $a Mirror Symmetry on Toric Surfaces via Tropical Geometry.
300 $a 126 p.
500 $a Source: Dissertation Abstracts International, Volume: 75-01(E), Section: B.
500 $a Adviser: Mark Gross.
502 $a Thesis (Ph.D.)--University of California, San Diego, 2013.
520 $a Mirror symmetry is a curious duality, first noticed by physicists and then excitedly embraced by mathematicians, between certain manifolds (the A-side) and their "mirror" spaces (the B-side). On the A-side, certain counts of objects on a manifold are carried out (the purview of enumerative geometry), while on the B-side, special functions are integrated. Interestingly, the same numbers come out on both sides.
520 $a This dissertation will consider mirror symmetry on toric surfaces, which are two dimensional varieties with certain convenient combinatorial properties and include many well-known surfaces such as the affine and projective planes and P1xP1. These surfaces are especially suited to being exploited by tropical geometry, which is a form of algebraic geometry over the "tropical semi-ring." Dramatically, key information about both sides of mirror symmetry on toric surfaces can be gleaned from the tropics.
520 $a The mirror symmetry of a certain subclass of toric varieties, those that are Fano, has seen much progress. In the case of toric surfaces, being Fano means that tropical geometry can "see" all the curves of the surface; this makes enumerative problems more straight-forward. This dissertation attempts a generalization to non-Fano toric surfaces, where some curves are "hidden" from the viewpoint of tropical geometry. The generalization is accomplished following the Gross-Seibert Program wherein singularities are added to the tropical picture in order to "pull" these curves into view. We verify these methods by carrying out explicit calculations on the second Hirzebruch surface, which is a non-Fano toric variety.
590 $a School code: 0033.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
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690 $a 0364
710 2 $a University of California, San Diego. $b Mathematics. $3 1022364
773 0 $t Dissertation Abstracts International $g 75-01B(E).
790 $a 0033
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792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3596039