Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- Prerequisites and notation -- 1 Introduction -- 1.1 Presentation -- 1.2 Some new applications of the large sieve -- 2 The principle of the large sieve -- 2.1 Notation and terminology -- 2.2 The large sieve inequality -- 2.3 Duality and 'exponential sums' -- 2.4 The dual sieve -- 2.5 General comments on the large sieve inequality -- 3 Group and conjugacy sieves -- 3.1 Conjugacy sieves -- 3.2 Group sieves -- 3.3 Coset sieves -- 3.4 Exponential sums and equidistribution for group sieves -- 3.5 Self-contained statements -- 4 Elementary and classical examples -- 4.1 The inclusion-exclusion principle -- 4.2 The classical large sieve -- 4.3 The multiplicative large sieve inequality -- 4.4 The elliptic sieve -- 4.5 Other examples -- 5 Degrees of representations of finite groups -- 5.1 Introduction -- 5.2 Groups of Lie type with connected centres -- 5.3 Examples -- 5.4 Some groups with disconnected centres -- 6 Probabilistic sieves -- 6.1 Probabilistic sieves with integers -- 6.2 Some properties of random finitely presented groups -- 7 Sieving in discrete groups -- 7.1 Introduction -- 7.2 Random walks in discrete groups with Property -- 7.3 Applications to arithmetic groups -- 7.4 The cases of (2) and (4) -- 7.5 Arithmetic applications -- 7.6 Geometric applications -- 7.7 Explicit bounds and arithmetic transitions -- 7.8 Other groups -- 8 Sieving for Frobenius over finite fields -- 8.1 A problem about zeta functions of curves over finite fields -- 8.2 The formal setting of the sieve for Frobenius -- 8.3 Bounds for sieve exponential sums -- 8.4 Estimates for sums of Betti numbers -- 8.5 Bounds for the large sieve constants -- 8.6 Application to Chavdarov's problem -- 8.7 Remarks on monodromy groups -- 8.8 A last application -- Appendix A Small sieves.

A.1 General results -- A.2 An application -- Appendix B Local density computations over finite fields -- B.1 Density of cycle types for polynomials over finite fields -- B.2 Some matrix densities over finite fields -- B.3 Other techniques -- Appendix C Representation theory -- C.1 Definitions -- C.2 Harmonic analysis -- C.3 One-dimensional representations -- C.4 The character tables of GL(2, Fq) and SL(2, Fq) -- Appendix D Property (T) and Property (τ) -- D.1 Property (T) -- D.2 Properties and examples -- D.3 Property (τ) -- D.4 Shalom's theorem -- Appendix E Linear algebraic groups -- E.1 Basic terminology -- E.2 Galois groups of characteristic polynomials -- Appendix F Probability theory and random walks -- F.1 Terminology -- F.2 The Central Limit Theorem -- F.3 The Borel-Cantelli lemmas -- F.4 Random walks -- Appendix G Sums of multiplicative functions -- G.1 Some basic theorems -- G.2 An example -- Appendix H Topology -- H.1 The fundamental group -- H.2 Homology -- H.3 The mapping class group of surfaces -- References -- Index.