Tropical and Non-Archimedean Curves - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Tropical and Non-Archimedean Curves write by Ralph Elliott Morrison. This book was released on 2015. Tropical and Non-Archimedean Curves available in PDF, EPUB and Kindle. Tropical geometry is young field of mathematics that connects algebraic geometry and combinatorics. It considers "combinatorial shadows" of classical algebraic objects, which preserve information while being more susceptible to discrete methods. Tropical geometry has proven useful in such subjects as polynomial implicitization, scheduling problems, and phylogenetics. Of particular interesest in this work is the application of tropical geometry to study curves (and other varieties) over non-Archimedean fields, which can be tropicalized to tropical curves (and other tropical varieties). Chapter 1 presents background material on tropical geometry, and presents two perspectives on tropical curves: the embedded perspective, which treats them as balanced polyhedral complexes in Euclidean space, and the abstract perspective, which treats them as metric graphs. This chapter also presents the background on curves over non-Archimedean fields necessary for the rest of this work, including the moduli space of curves of a given genus and the Berkovich analytic space associated to a curve. Chapters 2 and 3 study tropical curves embedded in the plane. Chapter 2 deals with tropical plane curves that intersect non-transversely, and opens with a result on which configurations of points in such an intersection can be lifted to intersection points of classical curves. It then moves on to present a joint work with Matthew Baker, Yoav Len, Nathan Pflueger, and Qingchun Ren that builds up a theory of bitangents of smooth tropical plane quartic curves in parallel to the classical theory. Chapter 3 presents joint work with Sarah Brodsky, Michael Joswig, and Bernd Sturmfels, and is a study of which metric graphs arise as skeletons of smooth tropical plane curves. We begin by defining the moduli space of tropical plane curves, which is the tropical analog of Castryck and Voight's space of nondegenerate curves in [CV09]. The first main theorem is that our space is full-dimensional inside of the tropicalization of the corresponding classical space, a result proved using honeycomb curves. The chapter proceeds to a computational study of the moduli space of tropical plane curves, and explicitly computes the spaces for genus up to 5. The chapter closes with both theoretical and computational results on tropical hyperelliptic curves that can be embedded in the plane. Chapter 4 presents joint work with Qingchun Ren and is an algorithmic treatment of a special family of curves over a non-Archimedean field called Mumford curves. These are of particular interest in tropical geometry, as they are the curves whose tropicalizations can have genus-many cycles. We build up a family of algorithms, implemented in sage [S+13], for computing many objects associated to such a curve over the field of p-adic numbers, including its Jacobian, its Berkovich skeleton, and points in its canonical embedding. Chapter 5 is joint work with Ngoc Tran, and is a departure from studying tropical curves. It considers what it means for matrix multiplication to commute tropically, both in the context of tropical linear algebra and by considering the tropicalization of the classical commuting variety, whose points are pairs of commuting matrices. We give necessary and sufficient conditions for small matrices to commute, and illustrate three different tropical spaces, each of which has some claim to being "the" space of tropical commuting matrices.
Tropical and Non-Archimedean Geometry
Tropical and Non-Archimedean Geometry - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Tropical and Non-Archimedean Geometry write by Omid Amini. This book was released on 2014-12-26. Tropical and Non-Archimedean Geometry available in PDF, EPUB and Kindle. Over the past decade, it has become apparent that tropical geometry and non-Archimedean geometry should be studied in tandem; each subject has a great deal to say about the other. This volume is a collection of articles dedicated to one or both of these disciplines. Some of the articles are based, at least in part, on the authors' lectures at the 2011 Bellairs Workshop in Number Theory, held from May 6-13, 2011, at the Bellairs Research Institute, Holetown, Barbados. Lecture topics covered in this volume include polyhedral structures on tropical varieties, the structure theory of non-Archimedean curves (algebraic, analytic, tropical, and formal), uniformisation theory for non-Archimedean curves and abelian varieties, and applications to Diophantine geometry. Additional articles selected for inclusion in this volume represent other facets of current research and illuminate connections between tropical geometry, non-Archimedean geometry, toric geometry, algebraic graph theory, and algorithmic aspects of systems of polynomial equations.
Nonarchimedean and Tropical Geometry
Nonarchimedean and Tropical Geometry - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Nonarchimedean and Tropical Geometry write by Matthew Baker. This book was released on 2016-08-18. Nonarchimedean and Tropical Geometry available in PDF, EPUB and Kindle. This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, which include high-level surveys as well as original research, mirror the main themes of the two Symposia. Topics covered in this volume include: Differential forms and currents, and solutions of Monge-Ampere type differential equations on Berkovich spaces and their skeletons; The homotopy types of nonarchimedean analytifications; The existence of "faithful tropicalizations" which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.
Tropical Algebraic Geometry
Tropical Algebraic Geometry - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Tropical Algebraic Geometry write by Ilia Itenberg. This book was released on 2009-05-30. Tropical Algebraic Geometry available in PDF, EPUB and Kindle. These notes present a polished introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The notes are based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.
Tropical Curves and Metric Graphs
Tropical Curves and Metric Graphs - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Tropical Curves and Metric Graphs write by Melody Tung Chan. This book was released on 2012. Tropical Curves and Metric Graphs available in PDF, EPUB and Kindle. In just ten years, tropical geometry has established itself as an important new field bridging algebraic geometry and combinatorics whose techniques have been used to attack problems in both fields. It also has important connections to areas as diverse as geometric group theory, mirror symmetry, and phylogenetics. Our particular interest here is the tropical geometry associated to algebraic curves over a field with nonarchimedean valuation. This dissertation examines tropical curves from several angles. An abstract tropical curve is a vertex-weighted metric graph satisfying certain conditions (see Definition 2.2.1), while an embedded tropical curve takes the form of a 1-dimensional balanced polyhedral complex in Rn̂. Both combinatorial objects inform the study of algebraic curves over nonarchimedean fields. The connection between the two perspectives is also very rich and is developed e.g. in [Pay09] and [BPR11]; we give a brief overview in Chapter 1 as well as a contribution in Chapter 4. Chapters 2 and 3 are contributions to the study of abstract tropical curves. We begin in Chapter 2 by studying the moduli space of abstract tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map, as initiated in [BMV11]. We provide a detailed combinatorial and computational study of these objects and give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. In Chapter 3, we study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. Our work ties together two strands in the tropical geometry literature, namely the study of the tropical moduli space of curves and tropical Brill-Noether theory. Our methods are graph-theoretic and extend much of the work of Baker and Norine [BN09] on harmonic morphisms of graphs to the case of metric graphs. We also provide new computations of tropical hyperelliptic loci in the form of theorems describing their specific combinatorial structure. Chapter 4 presents joint work with Bernd Sturmfels and is a contribution to the study of tropical curves as balanced embedded 1-dimensional polyhedral complexes. We say that a plane cubic curve, defined over a field with valuation, is in honeycomb form if its tropicalization exhibits the standard hexagonal cycle shown in Figure 4.1. We explicitly compute such representations from a given j-invariant with negative valuation, we give an analytic characterization of elliptic curves in honeycomb form, and we offer a detailed analysis of the tropical group law on such a curve. Chapter 5 is joint work with Anders Jensen and Elena Rubei and is a departure from the subject of tropical curves. In this chapter, we study tropical determinantal varieties and prevarieties. After recalling the definitions of tropical prevarieties, varieties, and bases, we present a short proof that the 4×4 minors of a 5×n matrix of indeterminates form a tropical basis. The methods are combinatorial and involve a study of arrangements of tropical hyperplanes. Our result together with the results in [DSS05], [Shi10], [Shi11] answer completely the fundamental question of when the r × r minors of a d × n matrix form a tropical basis; see Table 5.1.
