Graduate classes, Spring 2007, Mathematics
|MATH 512: Analysis II||Credits: 4||− Description||− Sections|
Content: Topics will include: Measure and integration theory on the real line as well as on a general measure space, Bounded linear functionals on L^p spaces. If time permits, Sobolev spaces and Fourier transforms will be introduced.
Prerequisites: Students are expected to have the background of Math 411-412 sequence or the equivalent.
|000||MSC: E406||TuTh 10:00am - 11:15am||Shanshuang Yang|
|MATH 516: Numerical Analysis II||Credits: 4||− Description||− Sections|
Content: Course material will focus on iterative methods of numerical linear algebra. Both eigenvalue problems and solving systems of equations will be covered in detail with emphasis on the algorithms currently used for large scale sparse and structured problems arising from mathematical modelling of real world applications. Links to the mathematical foundation of the methods will be made whenever possible. A solid theoretical background will be balanced with implementation and numerical stability issues.
Prerequisites: Students interested in this course are strongly recommended to take MATH 515 before MATH 516.
|000||MSC: W306||MWF 10:40am - 11:30am||James Nagy|
|MATH 520: Algebra III||Credits: 4||− Description||− Sections|
Content: This course will develop fundamental topics in commutative algebra and algebraic geometry, including affine algebraic varieties and their morphisms, Zariski topology, Hilbert Basis Theorem, Noether Normalizaiton, Hilbert's Nullstellensatz, equivalence between algebra and geometry, projective and quasi-projective varieties and their morphisms, Veronese, Segre, and Pl/"ucker embeddings, enumerative problems, and correspondences.
|000||MSC: E408||TuTh 10:00am - 11:15am||Eric Brussel|
|MATH 522: Algebra II||Credits: 4||− Description||− Sections|
Content: Continuation of 521. Topics: Modules, especially modules over a principal ideal domain, fields, Galois theory, representation of finite groups, Commutative algebra.
Prerequisites: Math 521.
|000||MSC: E408||MWF 2:00pm - 2:50pm||Faculty (TBA)|
|MATH 536: Combinatorics||Credits: 4||− Description||− Sections|
Content: This course is the second of the sequence of Math 535-536 and as such will continue to develop the topics from the first semester. Specific topics will include finite geometries, Hadermard matrices, Latin Squares, an introduction to design theory, extremal set theory and an introduction to combinatorial coding theory.
|000||MSC: W302||MWF 12:50pm - 1:40pm||Ron Gould|
|MATH 542: Topology II||Credits: 4||− Description||− Sections|
Content: The content of 542 may vary. Standard topics include Algebraic Topology (the fundamental group and covering spaces, homology and cohomology); Differential Topology (manifolds, transversality, intersection theory, integration on manifolds); and Geometric Topology (hyperbolic geometry knots and 3-manifolds). Chosen in accordance with the interest of students and instructor.
|000||MSC: E406||MWF 9:35am - 10:25am||William Mahavier|
|MATH 546: Intro. to Differential Geometry II||Credits: 4||− Description||− Sections|
Content: An introduction to Riemannian geometry and global analysis. Topics to be covered: Manifolds, Riemannian metrics, Connections, Curvature; Geodesics, Convexity, Topics in Global Analysis.
|000||MSC: E406||TuTh 11:30am - 12:45pm||Gideon Maschler|
|MATH 561: Matrix Analysis||Credits: 4||− Description||− Sections|
Content: Main topics:
Eigenvalues and eigenvectors of matrices, invariant subspaces, Schur triangular form, diagonalizable matrices, minimal polynomial, characteristic polynomial, Hamilton-Cayley Theorem, localization of eigenvalues, Gerschgorin's Theorem. Unitary, Hermitian and skew-Hermitian matrices. Normal matrices and the Spectral Theorem. Orthogonalization. Householder matrices and the QR factorization.
Moore-Penrose pseudoinverse. Applications to the solution of under- and over-determined systems of linear equations. Other generalized inverses. Applications to data fitting (least-squares approximation).
The Singular Value Decomposition. Matrix norms: spectral norm and Frobenius norm. Solution to matrix nearness problems. Applications to signal processing and information retrieval. Jordan canonical form. An algorithmic proof. Powers of matrices. Matrix functions. Applications to systems of differential equations. Bilinear and quadratic forms. Hermitian forms. Congruence. Sylvester's Law of Inertia. Rayleigh's principle. Courant-Fischer Theorem. Positive definite and semidefinite matrices. Applications to statistics (covariance matrices) and numerical analysis (PDE's).
Nonnegative matrices. The spectral radius. Positive matrices. Directed graphs. Nonnegative irreducible matrices. Perron-Frobenius Theorem. M-matrices. Applications to probability theory (Markov chains), economics (Leontiev's input-output model), and numerical analysis (iterative methods for linear systems). Structured matrices: circulant, Toeplitz, Hankel, Cauchy, Vandermonde, others. Block generalizations. Applications in signal processing, image processing, and numerical analysis (PDE's, interpolation).
Particulars: Textbook: C. D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000.
R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge University Press (1985; 1991).
R. A. Horn and C. R. Johnson, "Topics in Matrix Analysis", Cambridge University Press (1991; 1994).
F. R. Gantmacher, "The Theory of Matrices", vols. I-II, Chelsea (1959; 1971).
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences", Academic Press (1979); reprinted by SIAM, 1994.
D. Serre, "Matrices. Theory and Applications", Springer, 2002.
|000||MSC: W302||TuTh 1:00pm - 2:15pm||Michele Benzi|
|MATH 578R: Seminar in Algebra||Credits: 1 - 12||− Description||− Sections|
Content: Research topics in algebra of current interest to faculty and students.
|000||MSC: W303||Tu 4:00pm - 5:00pm||Skip Garibaldi|
|MATH 590: Teaching Seminar||Credits: 4||− Description||− Sections|
Content: This seminar will concentrate on effective teaching techniques in mathematics. Topics included will include:
General advise for new TA's. General advise for International TA's. Students will present several practice lectures over different levels of material. They will recieve practice on quiz and test preparation. Syllabus information on courses most likely to be taught by new TA's will be supplied. General professional development information will also be included.
|000||MSC: W304||M 3:00pm - 3:50pm||James Nagy|
|MATH 597R: Directed Study||Credits: 1 - 12||− Description||− Sections|
|MATH 599R: Master's Thesis Research||Credits: 1 - 12||− Description||− Sections|
|MATH 731: Ramsey Theory||Credits: 4||− Description||− Sections|
Content: This course will continue the development of ramsey theory begun in Math 531-532 and Math 535-536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the Hales-Jewett Theorem, spaces - affine and vector, Roth's Theorem and Szemeredi's Theorem. Equations: Schur's Theorem, regular homogeneous equations and systems, Rado's Theorem, finite sums and unions, Folkman's Theorem. Numbers: exact ramsey numbers, asymptotics, Van der Waerden numbers. The symmetric hypergraph Theorem, Schur and Rado numbers, higher ramsey numbers. Bipartite ramsey theory, induced ramsey theory, restricted results, Euclidean and Graph ramsey theory. Topological Dynamics, Ultra filters, the infinite.
Prerequisites: Math 531-532 and Math 535-536 or permission of the instructor.
|000||MSC: W302||TuTh 8:30am - 9:45am||Vojtech Rodl|
|MATH 732: Extremal Graph Theory||Credits: 4||− Description||− Sections|
Content: Continue the development of extremal in Graph Theory begun in Math 532. Included will be: Connectivity: structure of 2 and 3 connected graphs, minimally k-connected graphs. Matchings: fundamentals, the number of 1-factors, f-factors, coverings. Cycles: Graphs with large girth and large min. degree, vertex disjoint cycles, edge disjoint cycles, cycles of specific lengths, circumference. Diameter: Graphs with large subgraphs of small diameter, factors of small diameter, ties to connectivity. Colorings: General colorings, sparse graphs of large chromatic no., perfect graphs. Turan type Extremal Theory.
Prerequisites: Math 532 Graph Theory II or permission of the instructor.
|000||MSC: W302||WF 3:15pm - 4:30pm||Ron Gould|
|MATH 748: Advanced Partial Differential Equations||Credits: 4||− Description||− Sections|
Content: This course will discuss advanced topics in the modern theory of nonlinear partial differential equations and their applications. Included in the course are many of the following topics: * Basic concepts, sample problems in physics, biology, and geometry * Linear and quasi-linear elliptic and parabolic equations; basic methods and results on solvability * Quasi-linear geometric problems: mean curvature problem, Christoffel's problem, evolution by mean curvature, prescribing scalar curvature, Yamabe's problem * Convexity and elliptic and parabolic equations of Monge-Ampere type, Aleksandrov's geometric methods, Calabi's problem and Chern's classes, Fully nonlinear problems * N. Krylov's and C. Evans's results on nonlinear problems * The reflector mapping problem, the Gauss curvature problem, the Weyl problem, the Minkowski problem * Variational problems associated with some nonlinear PDE's * Monge-Kantorovich optimal transportation theory and its connections with nonlinear PDE's
Prerequisites: Mathematics 558 or permission of the instructor.
|000||MSC: W301||TuTh 2:30pm - 3:45pm||Vladimir Oliker|
|MATH 788R: Topics in Algebra: Number Theory||Credits: 4||− Description||− Sections|
Content: Chomology of finite and profinite groups; Galois cohomology, commutative case; nonabelian Galois chohomology and principal homogeneous spaces; cohomological dimension of fields; open questions.
|000||MSC: E408||MWF 10:00am - 11:15am||Faculty (TBA)|
|MATH 797R: Directed Study||Credits: 1 - 12||− Description||− Sections|
|MATH 799R: Dissertation Research||Credits: 1 - 12||− Description||− Sections|