These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The former studies the ideal of polynomial identities satisfied by a ring. The latter studies the properties of rings which satisfy a polynomial identity.

The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.

The intended audience is graduate students in algebra, and researchers in algebra, combinatorics and invariant theory.

Covers a gap in the literature which was caused by the fast development in the field over the last 15-20 years Addressed to researchers in both pure and applied mathematics

Inhalt
1 Distributions and Fourier Transform.- 1.1 Introduction.- 1.2 Distributions and generalized functions.- 1.3 Tempered distributions.- 1.4 Homogeneous distributions.- 1.5 Manifolds and differential forms.- 1.6 Push down and pull back.- 1.7 More on the Fourier transform.- 1.8 Bandlimited functions and interpolation.- 2 Radon Transform.- 2.1 Properties.- 2.2 Inversion formulae.- 2.3 Alternative formulae.- 2.4 Range conditions.- 2.5 Frequency analysis.- 2.6 Radon transform of differential forms.- 3 The Funk Transform.- 3.1 Factorable mappings.- 3.2 Spaces of constant curvature.- 3.3 Inversion of the Funk transform.- 3.4 Radon's inversion via Funk's inversion.- 3.5 Unified form.- 3.6 Funk-Radon transform and wave fronts.- 3.7 Integral transform of boundary discontinuities.- 3.8 Nonlinear artifacts.- 3.9 Pizetti formula for arbitrary signature.- 4 Reconstruction from Line Integrals.- 4.1 Pencils of lines and John's equation.- 4.2 Sources at infinity.- 4.3 Reduction to the Radon transform.- 4.4 Rays tangent to a surface.- 4.5 Sources on a proper curve.- 4.6 Reconstruction from plane integrals of sources.- 4.7 Line integrals of differential forms.- 4.8 Exponential ray transform.- 4.9 Attenuated ray transform.- 4.10 Inversion formulae.- 4.11 Range conditions.- 5 Flat Integral Transform.- 5.1 Reconstruction problem.- 5.2 Odd-dimensional subspaces.- 5.3 Even dimension.- 5.4 Range of the flat transform.- 5.5 Duality for the Funk transform.- 5.6 Duality in Euclidean space.- 6 Incomplete Data Problems.- 6.1 Completeness condition.- 6.2 Radon transform of Gabor functions.- 6.3 Reconstruction from limited angle data.- 6.4 Exterior problem.- 6.5 The parametrix method.- 7 Spherical Transform and Inversion.- 7.1 Problems.- 7.2 Arc integrals in the plane.- 7.3 Hemispherical integralsin space.- 7.4 Incomplete data.- 7.5 Spheres centred on a sphere.- 7.6 Spheres tangent to a manifold.- 7.7 Characteristic Cauchy problem.- 7.8 Fundamental solution for the adjoint operator.- 8 Algebraic Integral Transform.- 8.1 Problems.- 8.2 Special cases.- 8.3 Multiplicative differential equations.- 8.4 Funk transform of Leray forms.- 8.5 Differential equations for hypersurface integrals.- 8.6 Howard's equations.- 8.7 Range of differential operators.- 8.8 Decreasing solutions of Maxwell's system.- 8.9 Symmetric differential forms.- 9 Notes.- Notes to Chapter 1.- Notes to Chapter 2.- Notes to Chapter 3.- Notes to Chapter 4.- Notes to Chapter 5.- Notes to Chapter 6.- Notes to Chapter 7.- Notes to Chapter 8.