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Sequences Of Enumerative Geometry
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Book Synopsis Enumerative Geometry and String Theory by : Sheldon Katz
Download or read book Enumerative Geometry and String Theory written by Sheldon Katz and published by American Mathematical Soc.. This book was released on 2006 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics! The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry. The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology.
Book Synopsis Motivic Homotopy Theory and Refined Enumerative Geometry by : Federico Binda
Download or read book Motivic Homotopy Theory and Refined Enumerative Geometry written by Federico Binda and published by American Mathematical Soc.. This book was released on 2020-03-09 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14–18, 2018, at the Universität Duisburg-Essen, Essen, Germany. It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.
Book Synopsis Spin/pin-structures And Real Enumerative Geometry by : Xujia Chen
Download or read book Spin/pin-structures And Real Enumerative Geometry written by Xujia Chen and published by World Scientific. This book was released on 2023-12-04 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spin/Pin-structures on vector bundles have long featured prominently in differential geometry, in particular providing part of the foundation for the original proof of the renowned Atiyah-Singer Index Theory. More recently, they have underpinned the symplectic topology foundations of the so-called real sector of the mirror symmetry of string theory.This semi-expository three-part monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and presents their applications in enumerative geometry.Part I contains a systematic treatment of Spin/Pin-structures from different topological perspectives and may be suitable for an advanced undergraduate reading seminar. This leads to Part II, which systematically studies orientability problems for the determinants of real Cauchy-Riemann operators on vector bundles. Part III introduces enumerative geometry of curves in complex projective varieties and in symplectic manifolds, demonstrating some applications of the first two parts in the process. Two appendices review the Čech cohomology perspective on fiber bundles and Lie group covering spaces.
Book Synopsis 3264 and All That by : David Eisenbud
Download or read book 3264 and All That written by David Eisenbud and published by Cambridge University Press. This book was released on 2016-04-14 with total page 633 pages. Available in PDF, EPUB and Kindle. Book excerpt: 3264, the mathematical solution to a question concerning geometric figures.
Book Synopsis Hopf Algebras and Their Actions on Rings by : Susan Montgomery
Download or read book Hopf Algebras and Their Actions on Rings written by Susan Montgomery and published by American Mathematical Soc.. This book was released on 1993-10-28 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.
Download or read book Intersection Theory written by W. Fulton and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the ancient origins of algebraic geometry in the solution of polynomial equations, through the triumphs of algebraic geometry during the last two cen turies, intersection theory has played a central role. Since its role in founda tional crises has been no less prominent, the lack of a complete modern treatise on intersection theory has been something of an embarrassment. The aim of this book is to develop the foundations of intersection theory, and to indicate the range of classical and modern applications. Although a comprehensive his tory of this vast subject is not attempted, we have tried to point out some of the striking early appearances of the ideas of intersection theory. Recent improvements in our understanding not only yield a stronger and more useful theory than previously available, but also make it possible to devel op the subject from the beginning with fewer prerequisites from algebra and algebraic geometry. It is hoped that the basic text can be read by one equipped with a first course in algebraic geometry, with occasional use of the two appen dices. Some of the examples, and a few of the later sections, require more spe cialized knowledge. The text is designed so that one who understands the con structions and grants the main theorems of the first six chapters can read other chapters separately. Frequent parenthetical references to previous sections are included for such readers. The summaries which begin each chapter should fa cilitate use as a reference.
Book Synopsis Patterns in Permutations and Words by : Sergey Kitaev
Download or read book Patterns in Permutations and Words written by Sergey Kitaev and published by Springer Science & Business Media. This book was released on 2011-08-30 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the 1970s. Consideration of the patterns in question has been extremely interesting from the combinatorial point of view, and it has proved to be a useful language in a variety of seemingly unrelated problems, including the theory of Kazhdan—Lusztig polynomials, singularities of Schubert varieties, interval orders, Chebyshev polynomials, models in statistical mechanics, and various sorting algorithms, including sorting stacks and sortable permutations. The author collects the main results in the field in this up-to-date, comprehensive reference volume. He highlights significant achievements in the area, and points to research directions and open problems. The book will be of interest to researchers and graduate students in theoretical computer science and mathematics, in particular those working in algebraic combinatorics and combinatorics on words. It will also be of interest to specialists in other branches of mathematics, theoretical physics, and computational biology. The author collects the main results in the field in this up-to-date, comprehensive reference volume. He highlights significant achievements in the area, and points to research directions and open problems. The book will be of interest to researchers and graduate students in theoretical computer science and mathematics, in particular those working in algebraic combinatorics and combinatorics on words. It will also be of interest to specialists in other branches of mathematics, theoretical physics, and computational biology.
Book Synopsis Tropical and Logarithmic Methods in Enumerative Geometry by : Renzo Cavalieri
Download or read book Tropical and Logarithmic Methods in Enumerative Geometry written by Renzo Cavalieri and published by Springer Nature. This book was released on 2023-11-01 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Book Synopsis Mathematical Constants II by : Steven R. Finch
Download or read book Mathematical Constants II written by Steven R. Finch and published by Cambridge University Press. This book was released on 2018-12-06 with total page 783 pages. Available in PDF, EPUB and Kindle. Book excerpt: Famous mathematical constants include the ratio of circular circumference to diameter, π = 3.14 ..., and the natural logarithm base, e = 2.718 .... Students and professionals can often name a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here the author renews the search he began in his book Mathematical Constants, adding another 133 essays that broaden the landscape. Topics include the minimality of soap film surfaces, prime numbers, elliptic curves and modular forms, Poisson–Voronoi tessellations, random triangles, Brownian motion, uncertainty inequalities, Prandtl–Blasius flow (from fluid dynamics), Lyapunov exponents, knots and tangles, continued fractions, Galton–Watson trees, electrical capacitance (from potential theory), Zermelo's navigation problem, and the optimal control of a pendulum. Unsolved problems appear virtually everywhere as well. This volume continues an outstanding scholarly attempt to bring together all significant mathematical constants in one place.
Book Synopsis Computations in Algebraic Geometry with Macaulay 2 by : David Eisenbud
Download or read book Computations in Algebraic Geometry with Macaulay 2 written by David Eisenbud and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents algorithmic tools for algebraic geometry, with experimental applications. It also introduces Macaulay 2, a computer algebra system supporting research in algebraic geometry, commutative algebra, and their applications. The algorithmic tools presented here are designed to serve readers wishing to bring such tools to bear on their own problems. The first part of the book covers Macaulay 2 using concrete applications; the second emphasizes details of the mathematics.
Book Synopsis Recurrent Sequences by : Dorin Andrica
Download or read book Recurrent Sequences written by Dorin Andrica and published by Springer Nature. This book was released on 2020-09-23 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained text presents state-of-the-art results on recurrent sequences and their applications in algebra, number theory, geometry of the complex plane and discrete mathematics. It is designed to appeal to a wide readership, ranging from scholars and academics, to undergraduate students, or advanced high school and college students training for competitions. The content of the book is very recent, and focuses on areas where significant research is currently taking place. Among the new approaches promoted in this book, the authors highlight the visualization of some recurrences in the complex plane, the concurrent use of algebraic, arithmetic, and trigonometric perspectives on classical number sequences, and links to many applications. It contains techniques which are fundamental in other areas of math and encourages further research on the topic. The introductory chapters only require good understanding of college algebra, complex numbers, analysis and basic combinatorics. For Chapters 3, 4 and 6 the prerequisites include number theory, linear algebra and complex analysis. The first part of the book presents key theoretical elements required for a good understanding of the topic. The exposition moves on to to fundamental results and key examples of recurrences and their properties. The geometry of linear recurrences in the complex plane is presented in detail through numerous diagrams, which lead to often unexpected connections to combinatorics, number theory, integer sequences, and random number generation. The second part of the book presents a collection of 123 problems with full solutions, illustrating the wide range of topics where recurrent sequences can be found. This material is ideal for consolidating the theoretical knowledge and for preparing students for Olympiads.
Book Synopsis Enumerative Algebraic Geometry by : Steven L. Kleiman
Download or read book Enumerative Algebraic Geometry written by Steven L. Kleiman and published by American Mathematical Soc.. This book was released on 1991 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1989 marked the 150th anniversary of the birth of the great Danish mathematician Hieronymus George Zeuthen. Zeuthen's name is known to every algebraic geometer because of his discovery of a basic invariant of surfaces. However, he also did fundamental research in intersection theory, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's extraordinary devotion to his subject, his characteristic depth, thoroughness, and clarity of thought, and his precise and succinct writing style are truly inspiring. During the past ten years or so, algebraic geometers have reexamined Zeuthen's work, drawing from it inspiration and new directions for development in the field. The 1989 Zeuthen Symposium, held in the summer of 1989 at the Mathematical Institute of the University of Copenhagen, provided a historic opportunity for mathematicians to gather and examine those areas in contemporary mathematical research which have evolved from Zeuthen's fruitful ideas. This volume, containing papers presented during the symposium, as well as others inspired by it, illuminates some currently active areas of research in enumerative algebraic geometry.
Book Synopsis An Invitation to Modern Enumerative Geometry by : Andrea T. Ricolfi
Download or read book An Invitation to Modern Enumerative Geometry written by Andrea T. Ricolfi and published by Springer Nature. This book was released on 2022-12-14 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on a series of lectures given by the author at SISSA, Trieste, within the PhD courses Techniques in enumerative geometry (2019) and Localisation in enumerative geometry (2021). The goal of this book is to provide a gentle introduction, aimed mainly at graduate students, to the fast-growing subject of enumerative geometry and, more specifically, counting invariants in algebraic geometry. In addition to the more advanced techniques explained and applied in full detail to concrete calculations, the book contains the proofs of several background results, important for the foundations of the theory. In this respect, this text is conceived for PhD students or research “beginners” in the field of enumerative geometry or related areas. This book can be read as an introduction to Hilbert schemes and Quot schemes on 3-folds but also as an introduction to localisation formulae in enumerative geometry. It is meant to be accessible without a strong background in algebraic geometry; however, three appendices (one on deformation theory, one on intersection theory, one on virtual fundamental classes) are meant to help the reader dive deeper into the main material of the book and to make the text itself as self-contained as possible.
Book Synopsis The Arithmetic and Geometry of Algebraic Cycles by : B. Brent Gordon
Download or read book The Arithmetic and Geometry of Algebraic Cycles written by B. Brent Gordon and published by Springer Science & Business Media. This book was released on 2000-02-29 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This book offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods. Topics include a discussion of the arithmetic Abel-Jacobi mapping, higher Abel-Jacobi regulator maps, polylogarithms and L-series, candidate Bloch-Beilinson filtrations, applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection, motivic cohomology, Chow groups of singular varieties, and recent progress on the Hodge and Tate conjectures for Abelian varieties.
Book Synopsis Advances in Algebraic Geometry Motivated by Physics by : Emma Previato
Download or read book Advances in Algebraic Geometry Motivated by Physics written by Emma Previato and published by American Mathematical Soc.. This book was released on 2001 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Our knowledge of objects of algebraic geometry such as moduli of curves, (real) Schubert classes, fundamental groups of complements of hyperplane arrangements, toric varieties, and variation of Hodge structures, has been enhanced recently by ideas and constructions of quantum field theory, such as mirror symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants. These are some of the themes of this refereed collection of papers, which grew out of the special session, ``Enumerative Geometry in Physics,'' held at the AMS meeting in Lowell, MA, April 2000. This session brought together mathematicians and physicists who reported on the latest results and open questions; all the abstracts are included as an Appendix, and also included are papers by some who could not attend. The collection provides an overview of state-of-the-art tools, links that connect classical and modern problems, and the latest knowledge available.
Book Synopsis Handbook of Enumerative Combinatorics by : Miklos Bona
Download or read book Handbook of Enumerative Combinatorics written by Miklos Bona and published by CRC Press. This book was released on 2015-03-24 with total page 1073 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting the state of the art, the Handbook of Enumerative Combinatorics brings together the work of today's most prominent researchers. The contributors survey the methods of combinatorial enumeration along with the most frequent applications of these methods.This important new work is edited by Miklos Bona of the University of Florida where he
Book Synopsis Infinite Dimensional Groups and Manifolds by : Tilmann Wurzbacher
Download or read book Infinite Dimensional Groups and Manifolds written by Tilmann Wurzbacher and published by Walter de Gruyter. This book was released on 2008-08-22 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume is a collection of refereed research papers on infinite dimensional groups and manifolds in mathematics and quantum physics. Topics covered are: new classes of Lie groups of mappings, the Burgers equation, the Chern--Weil construction in infinite dimensions, the hamiltonian approach to quantum field theory, and different aspects of large N limits ranging from approximation methods in quantum mechanics to modular forms and string/gauge theory duality. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of important themes of research at the forefront of mathematics and theoretical physics.
