# Graduate classes, Fall 2008, Mathematics

 MATH 511: Analysis I Credits: 4 − Description − Sections
Content: An introduction to fundamental analytic concepts including: The complex number system, geometry and topology of the complex plane, analytic functions, conformal mappings, complex integration, and singularities.
000MSC: E406TuTh 11:30am - 12:45pmShanshuang Yangmax 20
 MATH 515: Numerical Analysis I Credits: 4 − Description − Sections
Content: This course covers topics in numerical linear algebra, including: matrix factorizations such as LU, QR and SVD; direct and iterative methods for solving linear systems and least squares problems; numerical approaches to solving eigenvalue problems; problem sensitivity and algorithm stability. A solid theoretical background of algorithms will be balanced with practical implementation issues.
Particulars: Background in linear algebra is assumed. Some knowledge of computer architectures, applied mathematics and elementary numerical analysis would help but is not absolutely essential.
000MSC: E406MWF 10:40am - 11:30amJames Nagymax 20
 MATH 521: Algebra I Credits: 4 − Description − Sections
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.
000MSC: E408TuTh 1:00pm - 2:15pmVictoria Powersmax 20
 MATH 535: Combinatorics I Credits: 4 − Description − Sections
Content: This course will begin the development of fundamental topics in Combinatorics. Included will be: Basic enumeration theory, uses of generating functions, recurrence relations, the inclusion / exclusion principle and its applications, partition theory, general Ramsey theory, an introduction to posets, and matrices of 0's and 1's, systems of distinct representatives, introduction to block designs, codes, and general set systems.
000MSC: E408MWF 12:50pm - 1:40pmRon Gouldmax 15
 MATH 547: Differential Topology Credits: 4 − Description − Sections
Content: Smooth manifolds, tangent spaces and derivatives, Sard's theorem, inverse function theorem, transversality, intersection theory, fixed-point theory, vector fields, Euler characteristic, the Poincare-Hopf theorem, exterior algebras, differential forms, integration, Stokes' theorem, de Rham cohomology.
000MSC: W306MWF 11:45am - 12:35pmDavid Borthwick
 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). Additional topics: 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. 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. 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.
000MSC: W306TuTh 10:00am - 11:15amMichele Benzi
 MATH 578R: Seminar in Algebra Credits: 1 - 12 − Description − Sections
Content: Research topics in algebra of current interest to faculty and students.
000MSC: W303Tu 4:00pm - 4:50pmEric Brusselmax 15
 MATH 579R: Seminar in Analysis Credits: 4 − Description − Sections
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
000MSC: W306W 12:50pm - 1:40pmEldad Haber
 MATH 597R: Directed Study Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)
 MATH 599R: Master's Thesis Research Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)
 MATH 733: Probabilistic Methods Credits: 4 − Description − Sections
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535-536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the Lovasz-Local Lemma and its applications. The second-moment method. Large deviation inequalities and Derandomization.
000MSC: E408TuTh 10:00am - 11:15amVojtech Rodl
 MATH 787R: Topics in Combinatorics: Random Structures II Credits: 4 − Description − Sections
Content: Title of topic and course description to follow soon.
000MSC: W306TuTh 8:30am - 9:45amMichal Karonski
 MATH 788R: Topics in Algebra: Quadratic forms Credits: 4 − Description − Sections
Content: TOPIC TITLE: Quadratic forms We shall cover the following topics in this course: Witt group of quadratic forms, Clifford Algebras, Signatures; Pfister's theory of multiplicative forms; field invariants associated to quadratic forms--level, u-invariant, pythagoras number; central simple algebras with involution and hermitian forms.
000MSC: E406TuTh 1:00pm - 2:15pmRaman Parimala
 MATH 789R: Topics in Analysis: Numerical Methods in Imaging Credits: 4 − Description − Sections
Content: Topic title and course description to follow.
000MSC: E408MW 9:00am - 10:15amEldad Haber
 MATH 789R: Topics in Analysis: Geometric PDE, II Credits: 4 − Description − Sections
Content: In this course we study partial differential equations arising in differential geometry and applied mathematics.
001MSC: W302TuTh 2:30pm - 3:45pmVladimir Oliker
 MATH 797R: Directed Study Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)
 MATH 799R: Dissertation Research Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)