To understand nonabelian Fourier analysis, we first need to introduce some notions from group representation theory. For further information on this subject, a good basic reference is the book Linear Representations of Finite Groups by Serre. A linear representation (or simply representation) of a group G over the vector space Cn is a homomorphism σ : G → GL(Cn), i.e., a map from group elements...

1 CES Utility In many economic textbooks the constant-elasticity-of-substitution (CES) utility function is defined as: U (x, y) = (αx ρ + (1 − α)y ρ) It is a tedious but straightforward application of Lagrangian calculus to demonstrate that the associated demand functions are: x(p x , p y , M) = α p x σ M α σ p 1−σ x + (1 − α) σ p 1−σ y and y(p x , p y , M) = 1 − α p y σ M α σ p 1−σ x + (1 − α)...

1. 1.1. Loops and Fundamental group. Let X be an arcwise connected topological space. Consider the homotopy classes of loops starting at a fixed point b 0 ∈ X : Ω(X, b 0). Recall that a loop is a continuous mapping σ : (I, ∂I) → (X, b 0). Composition σ * τ of two loops σ, τ is defined as (σ * τ)(t) =    σ(2t), 0 ≤ t ≤ 1/2 τ (2t − 1), 1/2 ≤ t ≤ 1 Let c be the constant loop at b 0 and σ −1 be ...

Given a compact orientable surface Σ, let S(Σ) be the set of isotopy classes of essential simple loops on Σ. We determine a complete set of relations for a function from S(Σ) to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in R S(Σ) and P (R S(Σ)) defining Thurston's space of measured laminations and Thurston's compactification of the Teichmül...

(i) b is a deformation parameter, related to the traditional ~ via b = √ ~, (ii) Σ is a two-dimensional topological surface with finitely many boundary components, (iii) HT b (Σ) is a Hilbert-space (possibly infinite-dimensional), (iv) OTb (Σ) is an algebra of bounded operators on HT b (Σ) and, (v) MTb (Σ) is a unitary projective representation of the mapping class group of Σ on HT b (Σ). The d...