Siu, Yum-Tong2011-08-222009Siu, Yum-Tong. 2009. Techniques for the analytic proof of the finite generation of the canonical ring. In Current Developments in Mathematics 2007, ed. David Jerison, Barry Mazur, Wilfried Schmid, Shing-tung Yau, Richard Stanley, and Tomasz Mrowka. Somerville: International Press of Boston.97815714613461089-6384http://nrs.harvard.edu/urn-3:HUL.InstRepos:5112792This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the main techniques and explains how they are put together in the proof. Of the various main techniques some special attention is given to (i) the technique of discrepancy subspaces and (ii) the technique of subspaces of minimum additional vanishing.en-USMonograph or Book2011-08-2210.4310/CDM.2007.v2007.n1.a5