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.
|000||MSC: E406||TuTh 11:30am - 12:45pm||Shanshuang Yang||max 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.
|000||MSC: E406||MWF 10:40am - 11:30am||James Nagy||max 20|
|MATH 521: Algebra I||Credits: 4||− Description||− Sections|
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.
|000||MSC: E408||TuTh 1:00pm - 2:15pm||Victoria Powers||max 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.
|000||MSC: E408||MWF 12:50pm - 1:40pm||Ron Gould||max 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.
|000||MSC: W306||MWF 11:45am - 12:35pm||David 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).
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: W306||TuTh 10:00am - 11:15am||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 - 4:50pm||Eric Brussel||max 15|
|MATH 579R: Seminar in Analysis||Credits: 4||− Description||− Sections|
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
|000||MSC: W306||W 12:50pm - 1:40pm||Eldad Haber|
|MATH 597R: Directed Study||Credits: 1 - 12||− Description||− Sections|
|MATH 599R: Master's Thesis Research||Credits: 1 - 12||− Description||− Sections|
|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.
|000||MSC: E408||TuTh 10:00am - 11:15am||Vojtech Rodl|
|MATH 787R: Topics in Combinatorics: Random Structures II||Credits: 4||− Description||− Sections|
Content: Title of topic and course description to follow soon.
|000||MSC: W306||TuTh 8:30am - 9:45am||Michal 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.
|000||MSC: E406||TuTh 1:00pm - 2:15pm||Raman Parimala|
|MATH 789R: Topics in Analysis: Numerical Methods in Imaging||Credits: 4||− Description||− Sections|
Content: Topic title and course description to follow.
|000||MSC: E408||MW 9:00am - 10:15am||Eldad 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.
|001||MSC: W302||TuTh 2:30pm - 3:45pm||Vladimir Oliker|
|MATH 797R: Directed Study||Credits: 1 - 12||− Description||− Sections|
|MATH 799R: Dissertation Research||Credits: 1 - 12||− Description||− Sections|