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.

A continuation of MA 533. Topics may include hypoelliptic operators and interior regularity of solutions; P(D)-convexity and existence theorems; regularity up to the boundary; applications of the maximum principle; semi-group theory for evolution equations; perturbation methods; well-posed and improperly posed problems; equations with analytic coefficients; a symptotic behavior of solutions; nonlinear problems.

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.

Sequences and series of real and complex numbers, sequences of functions. Riemann-Stieltjes integration, Lebesque measure and integration.

Review of recent research in discrete mathematics. May be repeated to a maximum of nine credits.

May be repeated to a maximum of six semesters.