Five recent book reviews
Matroid theory is a relatively young field of mathematics. Its origins are in 1935 when Whitney generalized common graph theory and linear algebra principles. This book was first published in 1992 and it is in recent times the most complete introductory monograph on matroid theory. Since the topic has grown widely one cannot give a complete survey in a single book. Hence the author has had to be selective. The first few chapters were chosen in a natural way. Chapters 1-6 provide a basic overview of matroid theory and cover the materials needed for this introductory course on the topic. The later chapters contain more details on matroid connectivity, representativity and decompositions. The theory is built stepwise and proofs are presented for almost all important statements. Hence the reader gets an overview of the results and can also observe how the proof techniques work. From this point of view, the book is essential for the researcher but can also be used as a textbook for both introductory and advanced courses on matroid theory. The advantage of the book as a study text is that it contains many examples and each section is supplemented by exercises.
The main topic treated by this book is a study of the generalizations of the Poincaré-Birkhoff-Witt theorem to quadratic algebras. The book starts with a review of various definitions and facts from homological algebra (including the bar construction and a quadratic duality for quadratic algebras). In chapter 2, the authors describe various definitions and properties of Koszul algebras and Koszul modules. A notion of an infinitesimal bialgebra of a Koszul algebra is introduced here. Natural operations on quadratic algebras and modules are studied in chapter 3 (including the Segre product and Veronese powers). The PBW algebras form a special subclass of Koszul algebras and their properties are studied in chapter 4. Nonhomogeneous quadratic algebras are studied in chapter 5. In particular, there is a discussion of an analogue of quadratic duality leading to the notion of curved DG-algebras. The Kozsul deformation principle and its consequences are discussed in chapter 6. The final chapter is devoted to a surprising connection to one-dependent discrete-time stochastic processes. The book offers a nice review of the field together with some new results.
This monograph shows how homological algebra helps one to connect, understand more deeply and prove important results in commutative algebra, representation theory and combinatorics. The key tool used here is the Koszul complex and its generalizations. The book is divided into seven chapters: after a preliminary chapter 1, the second chapter deals with local ring theory and proves Serre's characterization of regular local rings and their factoriality. Chapter 3 introduces generalized Koszul complexes and presents some of their recent applications. In chapter 4 structure theorems for finite free resolutions (extending the Hilbert-Burch theorem) are proved. Chapters 5-6 and the appendix concentrate on determinantal ideals and characteristic-free representation theory. However, they also involve a good deal of combinatorics, especially when dealing with the structure of Weyl and Schur modules and their applications (via tableaux and letter place algebras). By carefully selecting the material and clearly presenting the key ideas, the authors have succeeded in writing a very nice and useful monograph both for graduate students and for experts in algebra and representation theory.
This textbook is a very accessible introduction to projective geometry based on lectures given at the University of Adelaide. The book assumes a familiarity with linear algebra, elementary group theory, partial derivatives, finite fields and elementary coordinate geometry, hence it is suitable for students in their third or fourth year at university. The beginning of the book is devoted to Desargues’ theorem, which plays a key role in projective geometry. In the following chapters, the author introduces the reader to axiomatic geometry and defines the field plane PG(2,F) and its higher dimensional generalization PG(r,F). The author also considers projective spaces of dimension 2 (known as non-Desarguesian planes). The last chapters deal with conics and quadrics in PG(3, F). The textbook uses modern concepts of projective geometry closely related to algebra and algebraic geometry with the aim of helping the reader to understand and master proof techniques. An attractive feature of the book is a number of solved examples and more than 150 exercises.
This book serves as a guide to undergraduate courses in ordinary and partial differential equations. It contains the basic theory of ordinary differential equations (existence and uniqueness theorems), the variation of parameters method and the correspondence between differential and integral equations. Concerning partial differential equations, the book introduces the reader to first and second order partial differential equations, the reduction of the more general elliptic, parabolic and hyperbolic equations to the Laplace equation, the heat equation and the wave equation, together with classical results on these three basic equations. Some additional methods are then presented (including the Neumann series expansion for integral equations, the power series method, the Fourier transform method and the phase-plane analysis). There is also an introduction to the calculus of variations. The book is very well arranged. Every section is followed by many examples and exercises for better understanding of the topic. Theorems are usually not formulated in the strongest possible form, which increases the comprehensibility of the text. The book is intended not only for students of mathematics but also for students of physics, economics and other fields where differential equations play an important role.