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Allegretti, Dylan Gregory Lucasi.

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The Geometry of Cluster Varieties from Surfaces.

Record Type:	Electronic resources : Monograph/item
Title/Author:	The Geometry of Cluster Varieties from Surfaces./
Author:	Allegretti, Dylan Gregory Lucasi.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2016,
Description:	176 p.
Notes:	Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.
Contained By:	Dissertation Abstracts International77-12B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10150460
ISBN:	9781369053388

The Geometry of Cluster Varieties from Surfaces.
Allegretti, Dylan Gregory Lucasi.

The Geometry of Cluster Varieties from Surfaces. - Ann Arbor : ProQuest Dissertations & Theses, 2016 - 176 p.

Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.

Thesis (Ph.D.)--Yale University, 2016.

Cluster varieties are geometric objects introduced by Fock and Goncharov. They have recently found applications in several areas of mathematics and mathematical physics. The goal of this thesis is to study the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary.

ISBN: 9781369053388Subjects--Topical Terms:

515831
Mathematics.

The Geometry of Cluster Varieties from Surfaces.
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100 1 $a Allegretti, Dylan Gregory Lucasi. $3 3284407
245 1 4 $a The Geometry of Cluster Varieties from Surfaces.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2016
300 $a 176 p.
500 $a Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.
500 $a Adviser: Alexander Goncharov.
502 $a Thesis (Ph.D.)--Yale University, 2016.
520 $a Cluster varieties are geometric objects introduced by Fock and Goncharov. They have recently found applications in several areas of mathematics and mathematical physics. The goal of this thesis is to study the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary.
520 $a The main original contribution of this thesis is to develop the properties of a particular kind of cluster variety called the symplectic double . Let S be a compact oriented surface with boundary together with finitely many marked points on its boundary, and let S° denote the same surface equipped with the opposite orientation. We consider the surface SD obtained by gluing S and S° along corresponding boundary components. We show that the symplectic double is birational to a certain moduli space of local systems associated to this surface SD. We define a version of the notion of measured lamination on S D and prove that the space of all such laminations is a tropicalization of the symplectic double. We describe a canonical map from this space of laminations into the algebra of rational functions on the symplectic double. There is an explicit formula expressing this map in terms of special polynomials called F-polynomials from the theory of cluster algebras.
520 $a The second main contribution of this thesis is a proof of Fock and Goncharov's duality conjectures for quantum cluster varieties associated to a disk with finitely many marked points on its boundary. These duality conjectures identify a canonical set of elements in the quantized algebra of functions on a cluster variety satisfying a number of special properties. The results presented here grew out of joint work with Hyun Kyu Kim on quantum cluster varieties associated to punctured surfaces.
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