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 411412 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 quasiprojective 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 535536 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 3manifolds). 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, HamiltonCayley Theorem, localization of eigenvalues, Gerschgorin's Theorem. Unitary, Hermitian and skewHermitian matrices. Normal matrices and the Spectral Theorem. Orthogonalization. Householder matrices and the QR factorization.
MoorePenrose pseudoinverse. Applications to the solution of under and overdetermined systems of linear equations. Other generalized inverses. Applications to data fitting (leastsquares 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. CourantFischer Theorem. Positive definite and semidefinite matrices. Applications to statistics (covariance matrices) and numerical analysis (PDE's).
Additional topics:
Nonnegative matrices. The spectral radius. Positive matrices. Directed graphs. Nonnegative irreducible matrices. PerronFrobenius Theorem. Mmatrices. Applications to probability theory (Markov chains), economics (Leontiev's inputoutput 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.
Additional readings:
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. III, 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 
 00P    Faculty (TBA)   MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)   MATH 731: Ramsey Theory  Credits: 4  − Description  − Sections 
Content: This course will continue the development of ramsey theory begun in Math 531532 and Math 535536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the HalesJewett 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 531532 and Math 535536 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 kconnected graphs. Matchings: fundamentals, the number of 1factors, ffactors, 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 quasilinear elliptic and parabolic equations; basic methods and results on solvability * Quasilinear 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 MongeAmpere 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 * MongeKantorovich 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 
 00P    Faculty (TBA)   MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)  
