These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, four-dimensional N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 3: Hermitian matrix model. Tensor models and Permutation centralizer al- gebras. Using permutation equivalences to count matrix/tensor invariants and compute correlators. Relations to covering spaces.)

These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, four-dimensional N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 2: Local gauge invariant operators and Hilbert space of CFTs. Young diagrams and Brane geometries. Half-BPS and quarter-BPS. Counting, construction and correlators in group theoretic combinatorics.)

These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry. (Lecture 1: Two dimensional Yang Mills theory. Exact solution. Large N expansion. Role of Schur-Weyl duality - relation between representation theory of symmetric groups and unitary groups. Hurwitz spaces and string interpretation of the large N expansion.)

**** POLYGON SEMINAR**** Permutation groups and related algebras have proved to be powerful tools for understanding the counting and correlators of gauge invariant operators in 1-Matrix and multi-matrix models. Mathematical structures such as Belyi maps underlying the mixing of trace structures have been uncovered and finite N effects have been encoded using Young diagram data. These results have found applications in studies of BPS, near-BPS and non-BPS operators in N=4 SYM and quiver gauge theories. I will review some of this work and describe some open problems.

Quivers are directed graphs which encode information about the gauge groups and matter content of a large class of gauge theories, many of which have AdS/CFT duals. The counting of local gauge invariant operators and the computation of their correlators (in the free field limit) can be done by simple diagrammatic manipulations of the quiver, with the help of permutation group theory data. This data includes Young diagrams, Littlewood-Richardson numbers and branching coefficients of permutation groups. Riemann surfaces obtained by thickening the quivers are intimately related to these computations.

Abstract : Feynman Graph counting in Quantum Field Theory (QFT) can be formulated in terms of symmetric groups. This leads to expressions for graph counting and symmetry factors in terms of topological transition amplitudes for strings with a cylinder target, related to two dimensional topological field theory. The details of the interactions in the QFT are encoded in the boundary conditions which specify how the strings wind around circles. The QFTs discussed include scalar field theories and QED, where there is no large gauge group.

Recent work on membranes of M-theory has lead to a new type of physical realization of fuzzy 2-spheres in Matrix Brane actions. In standard realizations, the fuzzy two-sphere coordinates transforming in the vector of the spherical rotation symmetry are identified with matrix variables in the adjoint representation of the unitary symmetry of the Matrix brane actions.In these new realizations, spinors of the rotational symmetry are identified with variables in bi-fundamental representations of the unitary symmetry. The vector coordinates of the fuzzy 2-sphere are recovered from bilinears in the spinors by a fuzzy version of the projection map of the Hopf fibration over the two-sphere. The outcome is the emergence of a five dimensional theory from a three dimensional one as expected from the intersection of the fundamental branes of M-Theory in ABJM quotients of eleven dimensional spacetime. The demonstration of the emergent six dimensional theory expected for the brane intersection in M-Theory without a quotient remains an open problem.

TBA