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, fixedpoint theory, vector fields, Euler characteristic, the PoincareHopf 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, 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: 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 
 00P    Faculty (TBA)   MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)   MATH 733: Probabilistic Methods  Credits: 4  − Description  − Sections 
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the LovaszLocal Lemma and its applications. The secondmoment 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 formslevel, uinvariant, 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 
 00P    Faculty (TBA)   MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)  
