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# Algebra

See below for a selection of the latest books from Algebra category. Presented with a red border are the Algebra books that have been lovingly read and reviewed by the experts at Lovereading. With expert reading recommendations made by people with a passion for books and some unique features Lovereading will help you find great Algebra books and those from many more genres to read that will keep you inspired and entertained. And it's all free!

## On Fusion Systems of Component Type

This memoir begins a program to classify a large subclass of the class of simple saturated 2-fusion systems of component type. Such a classification would be of great interest in its own right, but in addition it should lead to a significant simplification of the proof of the theorem classifying the finite simple groups. Why should such a simplification be possible? Part of the answer lies in the fact that there are advantages to be gained by working with fusion systems rather than groups. In particular one can hope to avoid a proof of the B-Conjecture, a important but difficult result in finite group theory, established only with great effort.

## Linear Algebra and Geometry

Linear Algebra and Geometry is organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field. The materials in Linear Algebra and Geometry have been used, field tested, and refined for over two decades. It is aimed at preservice and practicing high school mathematics teachers and advanced high school students looking for an addition to or replacement for calculus. Secondary teachers will find the emphasis on developing effective habits of mind especially helpful. The book is written in a friendly, approachable voice and contains nearly a thousand problems.

## Variations on a Theorem of Tate

Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $\mathrm{Gal}(\overline{F}/F) \to \mathrm{PGL}_n(\mathbb{C})$ lift to $\mathrm{GL}_n(\mathbb{C})$. The author takes special interest in the interaction of this result with algebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, the author studies refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois Tannakian formalisms'' monodromy (independence-of-$\ell$) questions for abstract Galois representations.

## Fundamentals of the Theory of Operator Algebras, Volume IV

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume II--Advanced Theory (Graduate Studies in Mathematics series, Volume 16). The goal of the text proper is to teach the subject and lead readers to where the vast literature--in the subject specifically and in its many applications--becomes accessible. The choice of material was made from among the fundamentals of what may be called the classical'' theory of operator algebras. This volume contains the written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume II--Advanced Theory.

## Logica as Algebra

Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed to understand the book is some basic acquaintance with algebra.

## Interpolation for Normal Bundles of General Curves

Given $n$ general points $p_1, p_2, \ldots , p_n \in \mathbb P^r$, it is natural to ask when there exists a curve $C \subset \mathbb P^r$, of degree $d$ and genus $g$, passing through $p_1, p_2, \ldots , p_n$. In this paper, the authors give a complete answer to this question for curves $C$ with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle $N_C$ of a general nonspecial curve of degree $d$ and genus $g$ in $\mathbb P^r$ (with $d \geq g + r$) has the property of interpolation (i.e. that for a general effective divisor $D$ of any degree on $C$, either $H^0(N_C(-D)) = 0$ or $H^1(N_C(-D)) = 0$), with exactly three exceptions.

## The Role of Nonassociative Algebra in Projective Geometry

There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught. On the geometric side, the book introduces coordinates in projective planes, relates coordinate properties to transitivity properties of certain automorphisms and to configuration conditions. It also classifies higher-dimensional geometries and determines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined. On the algebraic side, basic properties of alternative algebras are derived, including the classification of alternative division rings. As tools for the study of the geometries, an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included.

## Random Matrices, Frobenius Eigenvalues, and Monodromy

The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.

## Tensor Spaces and Exterior Algebra

Format: Paperback / softback Release Date: 23/05/2020

This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

## Finite Fields and Applications

Format: Paperback / softback Release Date: 23/05/2020

This book provides a brief and accessible introduction to the theory of finite fields and to some of their many fascinating and practical applications.

## Collected Papers of John Milnor, Volume V

In addition to his seminal work in topology, John Milnor is also an accomplished algebraist, producing a spectacular agenda-setting body of work related to algebraic $K$-theory and quadratic forms during the five year period 1965-1970. These papers, together with other (some of them previously unpublished) works in algebra are assembled here in this fifth volume of Milnor's Collected Papers. They constitute not only an important historical archive, but also, thanks to the clarity and elegance of Milnor's mathematical exposition, a valuable resource for work in the fields treated. In addition, Milnor's papers are complemented by detailed surveys on the current state of the field in two areas. One is on the congruence subgroup problem, by Gopal Prasad and Andrei Rapinchuk. The other is on algebraic $K$-theory and quadratic forms, by Alexander Merkurjev.|In addition to his seminal work in topology, John Milnor is also an accomplished algebraist, producing a spectacular agenda-setting body of work related to algebraic $K$-theory and quadratic forms during the five year period 1965-1970. These papers, together with other (some of them previously unpublished) works in algebra are assembled here in this fifth volume of Milnor's Collected Papers. They constitute not only an important historical archive, but also, thanks to the clarity and elegance of Milnor's mathematical exposition, a valuable resource for work in the fields treated. In addition, Milnor's papers are complemented by detailed surveys on the current state of the field in two areas. One is on the congruence subgroup problem, by Gopal Prasad and Andrei Rapinchuk. The other is on algebraic $K$-theory and quadratic forms, by Alexander Merkurjev.