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The Dynamical Mordell Lang Conjecture
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Book Synopsis The Dynamical Mordell–Lang Conjecture by : Jason P. Bell
Download or read book The Dynamical Mordell–Lang Conjecture written by Jason P. Bell and published by American Mathematical Soc.. This book was released on 2016-04-20 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.
Book Synopsis The Dynamical Mordell-Lang Conjecture for Polynomial Endomorphisms of the Affine Plane by : Junyi Xie
Download or read book The Dynamical Mordell-Lang Conjecture for Polynomial Endomorphisms of the Affine Plane written by Junyi Xie and published by . This book was released on 2017 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane over the algebraic numbers. More precisely, let f be an endomorphism of the affine plan over the algebraic numbers. Let x be a point in the affine plan and C be a curve. If the intersection of C and the orbits of x is infinite, then C is periodic.
Book Synopsis Number Theory – Diophantine Problems, Uniform Distribution and Applications by : Christian Elsholtz
Download or read book Number Theory – Diophantine Problems, Uniform Distribution and Applications written by Christian Elsholtz and published by Springer. This book was released on 2017-05-26 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy’s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.
Book Synopsis Heights in Diophantine Geometry by : Enrico Bombieri
Download or read book Heights in Diophantine Geometry written by Enrico Bombieri and published by Cambridge University Press. This book was released on 2006 with total page 676 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.
Book Synopsis Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties by : Carlo Gasbarri
Download or read book Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties written by Carlo Gasbarri and published by American Mathematical Soc.. This book was released on 2015-12-22 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.
Book Synopsis Moduli Spaces and Arithmetic Dynamics by : Joseph H. Silverman
Download or read book Moduli Spaces and Arithmetic Dynamics written by Joseph H. Silverman and published by American Mathematical Soc.. This book was released on with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Advanced Topics in the Arithmetic of Elliptic Curves by : Joseph H. Silverman
Download or read book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
Book Synopsis Nevanlinna Theory And Its Relation To Diophantine Approximation by : Min Ru
Download or read book Nevanlinna Theory And Its Relation To Diophantine Approximation written by Min Ru and published by World Scientific. This book was released on 2001-06-06 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: It was discovered recently that Nevanlinna theory and Diophantine approximation bear striking similarities and connections. This book provides an introduction to both Nevanlinna theory and Diophantine approximation, with emphasis on the analogy between these two subjects.Each chapter is divided into part A and part B. Part A deals with Nevanlinna theory and part B covers Diophantine approximation. At the end of each chapter, a table is provided to indicate the correspondence of theorems.
Book Synopsis Some Problems of Unlikely Intersections in Arithmetic and Geometry by : Umberto Zannier
Download or read book Some Problems of Unlikely Intersections in Arithmetic and Geometry written by Umberto Zannier and published by Princeton University Press. This book was released on 2012-03-25 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).
Book Synopsis Mathematical Logic in the 20th Century by : Gerald E. Sacks
Download or read book Mathematical Logic in the 20th Century written by Gerald E. Sacks and published by World Scientific. This book was released on 2003 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable book is a collection of 31 important both inideas and results papers published by mathematical logicians inthe 20th Century. The papers have been selected by Professor Gerald ESacks. Some of the authors are Gdel, Kleene, Tarski, A Robinson, Kreisel, Cohen, Morley, Shelah, Hrushovski and Woodin.
Book Synopsis The Arithmetic of Elliptic Curves by : Joseph H. Silverman
Download or read book The Arithmetic of Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
Book Synopsis Model Theory and Algebraic Geometry by : Elisabeth Bouscaren
Download or read book Model Theory and Algebraic Geometry written by Elisabeth Bouscaren and published by Springer. This book was released on 2009-03-14 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski's proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.
Book Synopsis Partial Dynamical Systems, Fell Bundles and Applications by : Ruy Exel
Download or read book Partial Dynamical Systems, Fell Bundles and Applications written by Ruy Exel and published by American Mathematical Soc.. This book was released on 2017-09-20 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.
Book Synopsis Dynamics in One Non-Archimedean Variable by : Robert L. Benedetto
Download or read book Dynamics in One Non-Archimedean Variable written by Robert L. Benedetto and published by American Mathematical Soc.. This book was released on 2019-03-05 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of complex dynamics in one variable, initiated by Fatou and Julia in the early twentieth century, concerns the iteration of a rational function acting on the Riemann sphere. Building on foundational investigations of p-adic dynamics in the late twentieth century, dynamics in one non-archimedean variable is the analogous theory over non-archimedean fields rather than over the complex numbers. It is also an essential component of the number-theoretic study of arithmetic dynamics. This textbook presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line, which is the appropriate setting for the associated Fatou and Julia sets, is developed from the ground up, as are relevant results in non-archimedean analysis. The presentation is accessible to graduate students with only first-year courses in algebra and analysis under their belts, although some previous exposure to non-archimedean fields, such as the p-adic numbers, is recommended. The book should also be a useful reference for more advanced students and researchers in arithmetic and non-archimedean dynamics.
Book Synopsis Geometry and Dynamics in Gromov Hyperbolic Metric Spaces by : Tushar Das
Download or read book Geometry and Dynamics in Gromov Hyperbolic Metric Spaces written by Tushar Das and published by American Mathematical Soc.. This book was released on 2017-04-14 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.
Book Synopsis Recurrence Sequences by : Graham Everest
Download or read book Recurrence Sequences written by Graham Everest and published by American Mathematical Soc.. This book was released on 2015-09-03 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.
Book Synopsis Fundamentals of Diophantine Geometry by : S. Lang
Download or read book Fundamentals of Diophantine Geometry written by S. Lang and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.
