Part of the motivation for this book was its role in solving open problems in regular variation – in brief, the study of limiting relations of the form f(λx)/f(x) → g(x) as x → ∞ for all λ > 0 and its relatives. This was the subject of the earlier book Regular Variation by N. H. Bingham, C. M. Goldie and J. L. Teugels (BGT). So to serve as prologue to the present book, a brief summary of the many uses of regular variation is included, to remind readers of BGT and spare others needing to consult it. Topics covered include: probability (weak law of large numbers, central limit theorem, stability, domains of attraction, etc.), complex analysis (Abelian, Tauberian and Mercerian theorems, Levin–Pfluger theory), analytic number theory (prime divisor functions; results of Hardy and Ramanujan, Erdős and Kac, Rényi and Turán); the Cauchy functional equation g(λμ) = g(λ)g(μ) for all λ; μ > 0; dichotomy – the solutions are either very nice (powers) or very nasty (pathological – the ‘Hamel pathology’).

This section gives a general overview of abelian model structures and their homotopy categories. It is also meant to be a survey of the most fundamental examples of such homotopy categories. These include the chain homotopy category of a ring, the derived category of a ring, and the stable module category of a quasi-Frobenius (or Iwanaga–Gorenstein) ring.

We present basic ideas used throughout the book: sets, lists, relations, equivalence relations, and equivalence classes.

These extended notes give an introduction to the theory of finite group schemes over an algebraically closed field, with minimal prerequisites. They conclude with a brief survey of the inverse Galois problem for automorphism group schemes.

In this introductory chapter, we will formally introduce the main variants of the traveling salesman problem, symmetric and asymmetric, explain a very useful graph-theoretic view based on Euler’s theorem, and describe the classical simple approximation algorithms: Christofides’ algorithm and the cycle cover algorithm.

We also introduce basic notation, in particular from graph theory, and some fundamental combinatorial optimization problems.

We start by introducing C*-algebras associated with locally compact groups. Next, the theory of Hilbert modules, C*-correspondences, crossed-product algebras, and Morita equivalence are discussed. This is followed with applications to Mackey’s theory of induction of representations and Howe’s theory of theta correspondence. Next, K-theory and (equivariant) KK-theory are introduced. Connections between isolated points in the unitary dual and generators of the K-theory of C*-algebras of liminal groups are discussed.

This introductory chapter provides a first glimpse at the principal characters of the book: Albert algebras and octonions, but also at other members of the families of Freudenthal and composition algebras. They are presented here in the familiar surroundings of the field of real numbers and over the integers. This has the advantage of following rather closely the historical development of the subject (stretching back into the nineteenth century), and of providing a first motivation for the study of quadratic Jordan algebras. Following Zorn (1933), we define the algebra of Graves–Cayley octonions, allowing us to view the Hamiltonian quaternions as an appropriate subalgebra. We describe in detail the Z-algebras of Hurwitz quaternions (1896) and of Dickson–Coxeter octonions (1923, 1946), which in turn give rise to our first encounter with Albert algebras over the integers.