We give an introduction to a topic in the “stable algebra of matrices,” as related to certain problems in symbolic dynamics. We introduce enough symbolic dynamics to explain these connections, but the algebra is of independent interest and can be followed with little attention to the symbolic dynamics. This “stable algebra of matrices” involves the study of properties and relations of square matrices over a semiring S, which are invariant under two fundamental equivalence relations: shift equivalence and strong shift equivalence. When S is a field, these relations are the same, and matrices over S are shift equivalent if and only if the nonnilpotent parts of their canonical forms are similar. We give a detailed account of these relations over other rings and semirings. When S is a ring, this involves module theory and algebraic K theory. We discuss in detail and contrast the problems of characterizing the possible spectra, and the possible nonzero spectra, of nonnegative real matrices.We also review key features of the automorphism group of a shift of finite type; the recently introduced stabilized automorphism group; and the work of Kim, Roush and Wagoner giving counterexamples to Williams’ shift equivalence conjecture.

These lecture notes provide quantum probabilistic concepts and methods for spectral analysis of graphs, in particular, for the study of asymptotic behavior of the spectral distributions of growing graphs. Quantum probability theory is an algebraic generalization of classical (Kolmogorovian) probability theory, where an element of a (not necessarily commutative) ∗-algebra is treated as a random variable. In this aspect the concepts and methods peculiar to quantum probability are applied to the spectral analysis of adjacency matrices of graphs. In particular, we focus on the method of quantum decomposition and the use of various concepts of independence. The former discloses the noncommutative nature of adjacency matrices and gives a systematic method of computing spectral distributions. The latter is related to various graph products and provides a unified aspect in obtaining the limit spectral distributions as corollaries of various central limit theorems.

This is an introduction to representation theory and harmonic analysis on finite groups. This includes, in particular, Gelfand pairs (with applications to diffusion processes à la Diaconis) and induced representations (focusing on the little group method of Mackey and Wigner). We also discuss Laplace operators and spectral theory of finite regular graphs. In the last part, we present the representation theory of GL(2, Fq), the general linear group of invertible 2 × 2 matrices with coefficients in a finite field with q elements. More precisely, we revisit the classical Gelfand–Graev representation of GL(2, Fq) in terms of the so-called multiplicity-free triples and their associated Hecke algebras. The presentation is not fully self-contained: most of the basic and elementary facts are proved in detail, some others are left as exercises, while, for more advanced results with no proof, precise references are provided.

This chapter is devoted to the study of the space of bounded harmonic functions and the Liouville property. We start with the entropic criterion for the Liouville property. We then investigate the relationship of the Liouville property with amenability, speed of the random walk, and coupling of exit measures.The central example of lamplighter groups is studied.

This chapter provides a full elementary proof of Gromov’s theorem, which states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof proceeds along the ideas laid forth by Ozawa, using the existence of a harmonic cocycle. Gromov’s theorem is then used to classify all recurrent groups. Also, consequences of harmonic cocycle to diffusivitiy of the walk are shown.

This chapter is devoted to proving the Milnor–Wolf theorem, which states that a finitely generated solvable group has polynomial growth if and only if it is actually virtually nilpotent.

In this chapter all the basic notation and concepts are introduced.The notions of nilpotent, solvable, free, linear, finitely generated, and finitely presented groups are defined and examples are provided.Spaces of bounded and Lipschitz harmonic functions are defined, as well as harmonic functions of polynomial growths. Group actions are discussed and convolutions over abstract groups are defined.

Here, we dive deeper into the realm of reversible Markov chains, via the perspective of network theory. The notions of conductance and resistance are defined, as well as voltage and current, and the corresponding mathematical theory.The Laplacian and Green function are defined and their relation to harmonic functions explained. The chapter culminates with a proof (using network theory) that recurrence and transience are essentially group properties: these properties remain invariant when changing between different reasonable random walks on the same group (specifically, symmetric and adapted with finite second moment).

This chapter dives into the theory of (discrete time) martingales.The optional stopping theorem and the martingale convergence theorem are proved.These are used to provide some initial results regarding random walks on groups and bounded harmonic functions. Specifically, the random walk on the integer line is shown to be recurrent. Also, it is shown that the space of bounded harmonic functions is either just the constant functions or has infinite dimension.