Introduction.-Pure state extensions in linear algebra -- Density operators and pure states -- Extensions of pure states -- State spaces and the Kadison-Singer property -- States on C*-algebras -- Pure states and characters -- Extensions of pure states -- Properties of extensions and restrictions -- Maximal abelian C*-subalgebras -- Maximal abelian C*-subalgebras -- Examples of maximal abelian C*-subalgebras -- Minimal projections in maximal abelian von Neumann algebras -- Unitary equivalence -- Minimal projections -- Subalgebras without minimal projections -- Subalgebras with minimal projections -- Classification -- Stone-Čech compactification -- Stone-Čech compactification -- Ultrafilters -- Zero-sets -- Ultra-topology.-Convergence of ultrafilters for Tychonoff spaces -- Pushforward -- Convergence of ultrafilters for compact Hausdorff spaces -- Universal property -- The continuous subalgebra and the Kadison-Singer conjecture -- Total sets of states -- Haar states -- Projections in the continuous subalgebra -- The Anderson operator -- The Kadison-Singer conjecture -- The Kadison-Singer problem -- Real stable polynomials -- Realizations of random matrices -- Orthants and absence of zeroes -- Weaver's theorem -- Paving theorems -- Proof of the Kadison-Singer conjecture -- Preliminaries -- Linear algebra -- Order theory -- Topology -- Complex analysis -- Functional Analysis and Operator Algebras -- Basic functional analysis -- Hilbert spaces -- C*-algebras -- Von Neumann algebras -- Additional material -- Transitivity theorem -- G-sets, M-sets and L-sets -- GNS-representation -- Miscellaneous -- Notes and remarks -- References.