Algebra, Logic and Combinatorics

The following sections are included:

• Preface
• Contents

This chapter presents a very brief introduction to enumerative combinatorics. After a section on formal power series, it discusses examples of counting subsets, partitions and permutations; techniques for solving recurrence relations; the inclusion–exclusion principle; the Möbius function of a poset; q-binomial coefficients; and orbit-counting. A section on the theory of species (introduced by André Joyal) follows. The chapter concludes with a number of exercises, some of which are worked.

The complete determination of all the finite simple groups is one of the major achievements of 20th century mathematics. Yet, even the statement of this theorem, let alone its proof, appears impenetrable to anyone except a professional group theorist. It is my aim in this chapter to demystify the statement of the theorem, by giving a practical introduction to the various types of finite simple groups.

This chapter introduces the basic representation theory of finite-dimensional algebras. We outline the basic structure theorems for finite-dimensional algebras such as Artin–Wedderburn and Krull–Schmidt. We then introduce projective and injective modules, and show how the study of representations of finite-dimensional algebras can be reduced to the study of bound quiver algebras.

Mathematicians seek to exploit all available symmetry and often encode symmetry using the language of group actions. In this chapter, we consider finite groups acting by ring automorphisms on a polynomial ring. Our goal is to understand the subring of invariant polynomials.

This chapter starts with a gentle introduction to model theory, aimed at applications in algebra. It then accelerates and turns into a survey of recent spectacular applications of model theory in diophantine geometry.