Fields and vector spaces; subspaces and quotient spaces; basis and dimension; linear transformations and matrices; determinants; and the theory of a single linear transformation.

Topics covered in the sequence MA 605-606 include: elementary number theory, basic group theory, Lagrange’s theorem, isomorphism theorems, solvability, direct products, Jordan-Holder theorem, Sylow theorems, basic properties of rings, quotient rings, field of quotients of an integral domain, polynomial rings, factorization, elementary properties of fields, field extensions, and Galois theory.

The MA 615-616 sequence covers topics in numerical analysis and numerical methods including: errors and accuracy; polynomial approximation; interpolation; numerical differentiation and integration; numerical solution of differential equations; least square and minimum-maximum error approximations; nonlinear equations; simultaneous linear equations; summing series, Fourier series, filter design, the frequency approach, design of numerical tools, and statistics of error analysis; eigenvalues and eigenvectors of matrices; and the orientation throughout is toward computers.

The objective of this course is to introduce the students to the basic results of convex analysis and optimization. The properties of nonlinear non-smooth optimization models will be analyzed. Topics include: separation and representation of convex sets, properties of convex functions, subgradients, optimality conditions, saddle points, constraint qualifications, Fenchel and Lagrange duality, and sensitivity analysis.  Examples of optimization models from probability, statistics, and approximation theory will be discussed, as well as some basic models from management, finance, telecommunications, and other fields.

Strategic games and Nash equilibrium, strictly competitive (zero-sum) games and max-minimization, sStrategic games with imperfect information (Bayesian games), extensive games with perfect information (bargaining and repeated games), extensive games with imperfect information and signaling games, coalitional games (the core, stable sets, and bargaining sets), and auctions.

Metric spaces and topological spaces, bases and sub-bases, connectivity, local (path) connectivity, separation axioms, compactness and local compactness, concepts of convergence, Tychonoff’s theorem, Urysohn’s lemma, Tietze extension theorem, and selected topics as time permits.

Metric spaces and topological spaces, bases and sub-bases,   connectivity, local  (path) connectivity, separation axioms, compactness and   local compactness,  concepts of convergence, Tychonoff's theorem, Ury-sohn's   lemma, Tietze  extension theorem; homotopy type, fundamental group,   covering spaces; topology  of Euclidean space and manifold; selected topics as time   permits.            

Spring semester.

Special topics in mathematics not covered in regularly scheduled courses and suitable for both graduates and advanced undergraduates. May be taken more than once.