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Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In this lecture we finally prove the Kadison-Singer conjecture. First, we recall the main statement of the theorem. Theorem 15.1 (Marcus, Spielman, Srivastava [MSS13b]). Given a set of vectors v1,..., vm ∈ Rd in isotropic position, if max1≤i≤m ‖vi‖2 ≤ , then there is a 2 partitioning S1, S2 of [m] such that for each j ∈ {1, 2}