Arithmetic And Geometry Around Hypergeometric Functions
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Arithmetic and Geometry Around Hypergeometric Functions by Rolf-Peter Holzapfel,Muhammed Uludag,M. Yoshida Pdf
This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.
Arithmetic and Geometry Around Hypergeometric Functions by Rolf-Peter Holzapfel,Muhammed Uludag,M. Yoshida Pdf
This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.
Arithmetic and Geometry Around Galois Theory by Pierre Dèbes,Michel Emsalem,Matthieu Romagny,A. Muhammed Uludağ Pdf
This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): "Geometry and Arithmetic of Moduli Spaces of Coverings (2008)" and "Geometry and Arithmetic around Galois Theory (2009)". The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on étale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.
Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Functions by Yang Lei Pdf
Our book gives the complex counterpart of Klein's classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group 𝔊′216. It provides another beautiful example on the fundamental unity of mathematics.
On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps by E. Delaygue,T. Rivoal,J. Roques Pdf
Using Dwork's theory, the authors prove a broad generalization of his famous -adic formal congruences theorem. This enables them to prove certain -adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number and not only for almost all primes. Furthermore, using Christol's functions, the authors provide an explicit formula for the “Eisenstein constant” of any hypergeometric series with rational parameters. As an application of these results, the authors obtain an arithmetic statement “on average” of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.
This textbook on Feynman integrals starts from the basics, requiring only knowledge of special relativity and undergraduate mathematics. Feynman integrals are indispensable for precision calculations in quantum field theory. At the same time, they are also fascinating from a mathematical point of view. Topics from quantum field theory and advanced mathematics are introduced as needed. The book covers modern developments in the field of Feynman integrals. Topics included are: representations of Feynman integrals, integration-by-parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, coactions and symbols, cluster algebras, elliptic Feynman integrals, and motives associated with Feynman integrals. This volume is aimed at a) students at the master's level in physics or mathematics, b) physicists who want to learn how to calculate Feynman integrals (for whom state-of-the-art techniques and computations are provided), and c) mathematicians who are interested in the mathematical aspects underlying Feynman integrals. It is, indeed, the interwoven nature of their physical and mathematical aspects that make Feynman integrals so enthralling.
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds by Radu Laza,Matthias Schütt,Noriko Yui Pdf
In recent years, research in K3 surfaces and Calabi–Yau varieties has seen spectacular progress from both arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics—in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds, held at the Fields Institute (August 16-25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi–Yau varieties. With the big variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated. Unlike most other conferences, the 2011 Calabi–Yau workshop started with 3 days of introductory lectures. A selection of 4 of these lectures is included in this volume. These lectures can be used as a starting point for the graduate students and other junior researchers, or as a guide to the subject.
Stable Homotopy Around the Arf-Kervaire Invariant by Victor P. Snaith Pdf
Were I to take an iron gun, And ?re it o? towards the sun; I grant ‘twould reach its mark at last, But not till many years had passed. But should that bullet change its force, And to the planets take its course, ‘Twould never reach the nearest star, Because it is so very far. from FACTS by Lewis Carroll [55] Let me begin by describing the two purposes which prompted me to write this monograph. This is a book about algebraic topology and more especially about homotopy theory. Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role. As is well known, for homotopy classes of maps f : S ?? S with n? 1 the sole homotopy invariant is the degree, which characterises the homotopy class completely. The search for a continuous map between spheres of di?erent dimensions and not homotopic to the constant map had to wait for its resolution until the remarkable paper of Heinz Hopf [111]. In retrospect, ?nding 3 an example was rather easy because there is a canonical quotient map from S to 3 1 1 2 theorbitspaceofthe freecircleactionS /S =CP = S .
Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf
This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.
Basic Hypergeometric Series and Applications by Nathan Jacob Fine Pdf
The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. Today, research in $q$-hypergeometric series is very active, and there are now major interactions with Lie algebras, combinatorics, special functions, and number theory. However, the theory has been developed to such an extent and with such a profusion of powerful and general results that the subject can appear quite formidable to the uninitiated. By providing a simple approach to basic hypergeometric series, this book provides an excellent elementary introduction to the subject. The starting point is a simple function of several variables satisfying a number of $q$-difference equations.The author presents an elementary method for using these equations to obtain transformations of the original function. A bilateral series, formed from this function, is summed as an infinite product, thereby providing an elegant and fruitful result which goes back to Ramanujan. By exploiting a special case, the author is able to evaluate the coefficients of several classes of infinite products in terms of divisor sums. He also touches on general transformation theory for basic series in many variables and the basic multinomial, which is a generalization of a finite sum. These developments lead naturally to the arithmetic domains of partition theory, theorems of Liouville type, and sums of squares.Contact is also made with the mock theta-functions of Ramanujan, which are linked to the rank of partitions. The author gives a number of examples of modular functions with multiplicative coefficients, along with the beginnings of an elementary constructive approach to the field of modular equations. Requiring only an undergraduate background in mathematics, this book provides a rapid entry into the field. Students of partitions, basic series, theta-functions, and modular equations, as well as research mathematicians interested in an elementary approach to these areas, will find this book useful and enlightening. Because of the simplicity of its approach and its accessibility, this work may prove useful as a textbook.
K3 Surfaces and Their Moduli by Carel Faber,Gavril Farkas,Gerard van der Geer Pdf
This book provides an overview of the latest developments concerning the moduli of K3 surfaces. It is aimed at algebraic geometers, but is also of interest to number theorists and theoretical physicists, and continues the tradition of related volumes like “The Moduli Space of Curves” and “Moduli of Abelian Varieties,” which originated from conferences on the islands Texel and Schiermonnikoog and which have become classics. K3 surfaces and their moduli form a central topic in algebraic geometry and arithmetic geometry, and have recently attracted a lot of attention from both mathematicians and theoretical physicists. Advances in this field often result from mixing sophisticated techniques from algebraic geometry, lattice theory, number theory, and dynamical systems. The topic has received significant impetus due to recent breakthroughs on the Tate conjecture, the study of stability conditions and derived categories, and links with mirror symmetry and string theory. At the same time, the theory of irreducible holomorphic symplectic varieties, the higher dimensional analogues of K3 surfaces, has become a mainstream topic in algebraic geometry. Contributors: S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I. Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.
Compact Moduli Spaces and Vector Bundles by Valery Alexeev Pdf
This book contains the proceedings of the conference on Compact Moduli and Vector Bundles, held from October 21-24, 2010, at the University of Georgia. This book is a mix of survey papers and original research articles on two related subjects: Compact Moduli spaces of algebraic varieties, including of higher-dimensional stable varieties and pairs, and Vector Bundles on such compact moduli spaces, including the conformal block bundles. These bundles originated in the 1970s in physics; the celebrated Verlinde formula computes their ranks. Among the surveys are those that examine compact moduli spaces of surfaces of general type and others that concern the GIT constructions of log canonical models of moduli of stable curves. The original research articles include, among others, papers on a formula for the Chern classes of conformal classes of conformal block bundles on the moduli spaces of stable curves, on Looijenga's conjectures, on algebraic and tropical Brill-Noether theory, on Green's conjecture, on rigid curves on moduli of curves, and on Steiner surfaces.
