Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General Resultants.- Chow Varieties.- Toric Varieties.- Newton Polytopes and Chow Polytopes.- Triangulations and Secondary Polytopes.- A-Resultants and Chow Polytopes of Toric Varieties.- A-Discriminants.- Principal A-Discriminants.- Regular A-Determinants and A-Discriminants.- Classical Discriminants and Resultants.- Discriminants and Resultants for Polynomials in One Variable.- Discriminants and Resultants for Forms in Several Variables.- Hyperdeterminants.- Appendix A. Determinants.- Appendix B. A. Cayley: On the Theory of Elimination.- Bibliography.- Notes and References.- List of Notation.- Index

Discriminants, Resultants, and Multidimensional Determinants

(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares, and for tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is anticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alphabet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cognitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought processes (how the brain works). Mathematics has already played a big role towards the goal of giving a universal foundation of studies in these disciplines. We mention as examples the theory of Neural Networks going back to McCulloch and Pitts [25] and Minsky and Papert [27], the PAC learning of Valiant [40], Statistical Learning Theory as developed by Vapnik [42], and the use of reproducing kernels as in [17] among many other mathematical developments. We are heavily indebted to these developments. Recent discussions with a number of mathematicians have also been helpful. In

http://pages.cs.wisc.edu/~brecht/cs838docs/cuckersmale.pdf

On the mathematical foundations of learning

4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

4-manifolds and Kirby calculus

Introduction Classical computation Quantum computation Solutions Elementary number theory Bibliography Index.

Classical and Quantum Computation

The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with first volume the author has revised the text and added new material. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of the first volume and is suitable for beginning graduate students.

Basic Algebraic Geometry

The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties.

This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Algebraic geometry V: fano varieties

/pdf/lectures-on-minimal-models-and-birational-transformations-of-1d7ho08xms.pdf

Lectures on Minimal Models And Birational Transformations of Two Dimensional Schemes

Algebraic Geometry IV

This book aims to present a general survey of algebra, of its basic notions and main branches Now what language should we choose for this? In reply to the question ‘What does mathematics study?’, it is hardly acceptable to answer ‘structures’ or ‘sets with specified relations’; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion

/pdf/basic-notions-of-algebra-4j04vnyv4v.pdf

Igorʹ Rostislavovich Shafarevich

Papers

Basic Algebraic Geometry

Algebraic geometry V: fano varieties

Lectures on Minimal Models And Birational Transformations of Two Dimensional Schemes

Algebraic Geometry IV

Basic notions of algebra