# Graduate classes, Fall 2014, Mathematics

 MATH 511: Analysis I Credits: 3 − 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: E406MW 11:30am - 12:45pmDavid Borthwickmax 20
 MATH 515: Numerical Analysis I Credits: 3 − 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: W306TuTh 10:00am - 11:15amJames Nagymax 20
 MATH 521: Algebra I Credits: 3 − Description − Sections
Content: An introduction to fundamental algebraic concepts including: Groups, homomorphisms, the action of a group on a set, Sylow theorems, group representations and characters, rings, ring homomorphisms, and basics on commutative rings.
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000MSC: E406MW 10:00am - 11:15amRaman Parimalamax 16
 MATH 535: Combinatorics I Credits: 3 − 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.
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000MSC: E406MW 1:00pm - 2:15pmRon Gouldmax 20
 MATH 557: Partial Differential Equations I Credits: 3 − 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: W304TuTh 1:00pm - 2:15pmVladimir Olikermax 20
 MATH 561: Matrix Analysis Credits: 3 − 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: R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge University Press (1985; 1991).
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000MSC: E408TuTh 11:30am - 12:45pmMichele Benzimax 20
 MATH 577R: Seminar in Combinatorics Credits: 3 − 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: W302M 4:00pm - 4:50pmDwight Duffusmax 20
 MATH 578R: Seminar in Algebra Credits: 3 − Description − Sections
Content: Research topics in algebra of current interest to faculty and students.
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000MSC: W306Tu 4:00pm - 4:50pmDavid Zureick-Brownmax 20
 MATH 579R: Seminar in Analysis Credits: 3 − Description − Sections
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
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000MSC: W301F 12:00pm - 12:50pmAlessandro Venezianimax 20
001MSC: W302Tu 4:00pm - 4:50pmVladimir Olikermax 25
 MATH 599R: Master's Thesis Research Credits: 1 - 9 − Description − Sections
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YANGMSCFaculty (TBA)
 MATH 787R: Topics in Combinatorics: Hypergraphs Credits: 3 − Description − Sections
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000MSC: E408TuTh 1:00pm - 2:15pmVojtech Rodlmax 16
000MSC: E406TuTh 10:00am - 11:15amVojtech Rodlmax 16
 MATH 788R: Topics in Algebra: Local Fields and Class Field Theory Credits: 3 − Description − Sections
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000MSC: E406TuTh 1:00pm - 2:15pmDavid Zureick-Brownmax 16
 MATH 789R: Topics in Analysis: Advanced Numerical Linear Algebra Methods with Applications Credits: 3 − Description − Sections
Content: * Theory and computation of matrix functions * Matrices, moments and quadrature * Applications to Network Science * Applications to Physics
Texts: Reading materials will be provided by the instructor.
Assessments: TBA
Prerequisites: Math 515-516.
000MSC: E408TuTh 2:30pm - 3:45pmMichele Benzimax 16