Reading course for graduate students in mathematics. May be repeated to a maximum of nine credits.

Reading course for graduate students in mathematics. May be repeated to a maximum of nine credits.

Reading course for graduate students in mathematics. May be repeated to a maximum of nine credits.

Reading course for graduate students in mathematics. May be repeated to a maximum of nine credits.

Reading course for graduate students in mathematics. May be repeated to a maximum of nine credits.

An introduction to the basic notions and techniques in enumerative combinatorics. The material has applications to polytopal theory, hyperplane arrangements, computational commutative algebra, representation theory and symrnetric functions. Topics include generating functions, the principle of inclusion and exclusion, bijections, recurrence relations, partially ordered sets, the Mobius function and Mobius algebra, the Lagrange inversion formula, the exponential formula and tree enumeration.

Numerical solution of matrix eigenvalue problems and applications of eigenvalues. Normal forms of Jordan and Schur. Vector and matrix norms. Perturbation theory and bounds for eigenvalues. Stable matrices and Lyapunov theorems. Nonnegative matrices. Iterative methods for solving large sparse linear systems.

Tensor products, exterior algebra, differentiable maps, manifolds, geodesics, metric properties of curves in Euclidean fundamental forms, surfaces.

Embedding and metrization, compact spaces, uniform spaces and function spaces.

Rings, fields of quotients, rings of polynomials, formal power series, modules, exact sequences, groups of homomorphisms, natural isomorphisms, algebras and tensor algebras.