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.
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.
An introduction to fundamental structures and techniques in combinatorics, including such topics as graphs, trees, colorings of graphs, extremal graphs, bipartite matchings, partially ordered sets, extremal set theory, flows in networks, and the principle of inclusion/exclusion.
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.
Elementary existence theorems, equations of first order, classification of linear second order equations, the Cauchy and Dirichlet problems, potential theory, the heat and wave equations, Green's and Riemann functions, separation of variables, systems of equations.
Topological spaces, products, quotients, subspaces, connectedness, compactness, local compactness, separation axioms, convergence.
Algebraic structures, quotient structures, substructures, product structures, groups, permutation groups, groups with operators, and the Jordan-Holder theorem.
Review of finite dimensional linear algebra, the rank of a matrix, systems of linear equations, determinants, characteristic and minimal polynomials of a matrix, canonical forms for matrices, the simplicity of the ring of linear mappings of a finite dimensional vector space, the decomposition of a vector space relative to a group of linear mappings and selected topics of a more advanced nature.
Real and complex numbers, sequences and series, continuity, differentiation, integration, and uniform convergence.
A seminar for teaching assistants on the basics of teaching mathematics at the college level as well as use of appropriate technology. Includes topics such as syllabus construction, lesson planning, grading assignments, web pages, typesetting mathematics with LaTeX. Required of all new graduate teaching assistants in mathematics.
