Topics from classical number theory, including discussions of mathematical induction, prime numbers, division algorithms, congruences, and quadratic reciprocity.

Selected topics in geometry including Euclidean and some non-Euclidean geometries.

A beginning course, with particular emphasis on point-set topology in Euclidean spaces.

A basic course in the theory of counting and graph theory. Topics in enumerative combinatorics may include: generating functions, compositions, partitions, Fibonacci numbers, permutations, cycle structure of permutations, permutations statistics, Stirling numbers of the first and second kind, Bell numbers, or inclusion-exclusion. Topics in graph theory may include: Eulerian and Hamiltonian cycles, matrix tree theorem, planar graphs and the 4-color theorem, chromatic polynomial, Hall's marriage theorem, stable marriage theorem, Ramsey theory, or electrical networks.

Partial differentiation, Jacobians, implicit function theorem, uniform convergence of series, line and surface integrals. Green's and Stokes' theorems.

A careful and vigorous investigation of the calculus of functions of a single variable. Topics will include elementary topological properties of the real line, convergence limits, continuity, differentiation and integration. Prereq: MA 213 and MA 322; or consent of the instructor.

MA 483G is essentially an introductory course in partial differential equations designed to prepare undergraduate mathematics majors for serious work in partial differential equations and to provide Ph.D.

The course and its sequel (MA/PHY 507) are designed to develop, for first-year graduate students, familiarity with the mathematical tools useful in physics. Topics include curvilinear coordinates, infinite series, integrating and solving differential equations of physics, and methods of complex variables. Work with Green's functions, eigenvalues, matrices and the calculus of variations are included as a part of MA/PHY 506 and 507.

Mathematical and computational aspects of linear programming and combinatorial optimization. Linear optimization is introduced by presenting solution techniques (primal and dual simplex) and studying geometric properties and duality for linear systems of inequalities. Asics of combinatorial optimization, including trees, paths, flows, matchings, and matroids, and the corresponding algorithms are presented.

Review of basic linear algebra from a constructive and geometric point of view. Factorizations of Gauss, Cholesky and Gram-Schmidt. Determinants. Linear least squares problems. Rounding error analysis. Stable methods for updating matrix factorizations and for linear programming. Introduction to Hermitian eigenvalue problems and the singular value decomposition via the QR algorithm and the Lanczos process. Method of conjugate gradients.