Symmetric groups and their group algebras, along with concepts in representation theory including Schur-Weyl duality, have been applied to the study of half-BPS states in N=4 SYM theory, in particular to clarify the holographic map of the AdS/CFT correspondence. These algebraic techniques have also been used in characterizing less supersymmetric states (e.g. quarter and eighth BPS states), computing correlators in quiver gauge theories, counting feynman graphs. A very recent direction is to use these techniques to study the complete spectrum of conformal primary fields in higher dimensional conformal field theories: this problem reveals unexpected links to the algebraic geometry of Calabi-Yau spaces. The proposed project will involve developing and applying these algebraic tecniques to questions in CFT and AdS/CFT.