Table of contents

This survey concerns representations of Hecke-Shimura rings of integral positive-definite quadratic forms on spaces of polynomial harmonic vectors, and the question of simultaneous diagonalization of the corresponding Hecke operators. Explicit relations are deduced between the zeta functions of the quadratic forms in 2 and 4 variables corresponding to the harmonic eigenvectors of genera 1 and 2, and the zeta functions of Hecke and Andrianov of theta series weighted by these eigenvectors, respectively. Similar questions for single-class quadratic forms were considered earlier in the paper [1]. The general situation is discussed in the paper [2].

This is a discussion of recent progress in the theory of singular traces on ideals of compact operators, with emphasis on Dixmier traces and their applications in non-commutative geometry. The starting point is the book Non-commutative geometry by Alain Connes, which contains several open problems and motivations for their solutions. A distinctive feature of the exposition is a treatment of operator ideals in general semifinite von Neumann algebras. Although many of the results presented here have already appeared in the literature, new and improved proofs are given in some cases. The reader is referred to the table of contents below for an overview of the topics considered.

Szemerédi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem has been developed into a major new area of combinatorial number theory. This is the topic of the present survey.