# Graduate classes, Fall 2009, 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.
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000MSC: E408MWF 10:40am - 11:30amDavid Borthwickmax 15
 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.
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000MSC: W304TuTh 10:00am - 11:15amMichele Benzimax 20
 MATH 521: Algebra I Credits: 4 − Description − Sections
Content: Linear algebra, including canonical forms, infinite-dimensional vector spaces, tensor products, and multilinear algebra. Group theory including group actions and representations.
Texts: A.W. Knapp, "Basic algebra", Birkhauser, 2006.
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000MSC: E406TuTh 1:00pm - 2:15pmSkip Garibaldimax 15
 MATH 531: Graph Theory I Credits: 4 − Description − Sections
Content: I will introduce basic graph-theoretical concepts, graphs, trees, networks, cycles, independence number, chromatic number, planarity and genus, paths and cycles, etc. I will emphasize "extremal" problems and counting techniques.
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000MSC: E406MWF 12:50pm - 1:40pmRon Gouldmax 15
 MATH 543: Algebraic Topology I Credits: 4 − Description − Sections
Content: Homotopy theory, the fundamental group, free products of groups with amalgamation, Van Kampen's Theorem, covering spaces, classification of surfaces, classifying spaces, higher homotopy groups
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000MSC: E406MWF 9:35am - 10:25amEmily Hamiltonmax 15
 MATH 557: Partial Differential Equations I Credits: 4 − Description − Sections
Content: This course will introduce some of the basic techniques for studying and solving partial differential equations (PDE's) with special emphasis on applications. Included in the course are the following topics: 1. Basic concepts, sample problems, motivation 2. Maximum principles for elliptic and parabolic equations 3. Basic concepts of the theory of distributions 4. Method of fundamental solutions; Green's functions 5. Fourier transform 6. Variational methods, eigenvalues and eigenfunctions 7. Applications; Maxwell's equations, diffusion, geometric flows, image processing
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000MSC: E406TuTh 11:30am - 12:45pmShanshuang Yangmax 15
 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).
Texts: 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.
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000MSC: E408TuTh 1:00pm - 2:15pmMichele Benzimax 16
 MATH 577R: Seminar in Combinatorics Credits: 4 − Description − Sections
Content: The seminar in combinatorics is a research seminar for students and faculty. It runs weekly, and features speakers from outside Emory who come to talk about topics of interest to the Emory faculty.
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000MSC: W306F 4:00pm - 5:00pmDwight Duffusmax 15
 MATH 578R: Seminar in Algebra Credits: 1 - 12 − Description − Sections
Content: Research topics in algebra of current interest to faculty and students.
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000MSC: W303Tu 4:00pm - 5:00pmSkip Garibaldimax 15
 MATH 579R: Seminar in Analysis Credits: 4 − Description − Sections
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
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000MSC: W306W 12:50pm - 1:40pmAlessandro Venezianimax 20
 MATH 597R: Directed Study Credits: 1 - 12 − Description − Sections
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 MATH 599R: Master's Thesis Research Credits: 1 - 12 − Description − Sections
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 MATH 786R: Topics in Topology Credits: 4 − Description − Sections
Content: In this course we will study discrete groups from a geometric perspective. (Discrete groups are groups which are defned by generators and relations, such as free groups, free abelian groups, braid groups, etc.) By treating groups as geometric objects, one can solve many algebraic problems which are much more difficult without the geometry. Introductory topics include: Free groups, group presentations, Cayley graphs and Cayley complexes, Dehn's problems, fundamental groups, hyperbolic geometry, Eilenberg-MacLane spaces and some algebraic topology. Advanced topics may include: Coxeter and Artin groups, Bass-Serre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasi-isometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem.
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000MSC: E408MWF 11:45am - 12:35pmAaron Abramsmax 15
 MATH 797R: Directed Study Credits: 1 - 12 − Description − Sections
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 MATH 799R: Dissertation Research Credits: 1 - 12 − Description − Sections
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