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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  MWF 10:40am  11:30am  David Borthwick  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W304  TuTh 10:00am  11:15am  Michele Benzi  max 20  MATH 521: Algebra I  Credits: 4  − Description  − Sections 
Content: Linear algebra, including canonical forms, infinitedimensional vector spaces, tensor products, and multilinear algebra. Group theory including group actions and representations. Texts: A.W. Knapp, "Basic algebra", Birkhauser, 2006. Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 1:00pm  2:15pm  Skip Garibaldi  max 15  MATH 531: Graph Theory I  Credits: 4  − Description  − Sections 
Content: I will introduce basic graphtheoretical concepts, graphs, trees, networks, cycles, independence number, chromatic number, planarity and genus, paths and cycles, etc. I will emphasize "extremal" problems and counting techniques. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MWF 12:50pm  1:40pm  Ron Gould  max 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 Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MWF 9:35am  10:25am  Emily Hamilton  max 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 Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 11:30am  12:45pm  Shanshuang Yang  max 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, 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). 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. 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. Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 1:00pm  2:15pm  Michele Benzi  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W306  F 4:00pm  5:00pm  Dwight Duffus  max 15  MATH 578R: Seminar in Algebra  Credits: 1  12  − Description  − Sections 
Content: Research topics in algebra of current interest to faculty and students. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W303  Tu 4:00pm  5:00pm  Skip Garibaldi  max 15  MATH 579R: Seminar in Analysis  Credits: 4  − Description  − Sections 
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W306  W 12:50pm  1:40pm  Alessandro Veneziani  max 20  MATH 597R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  MSC:    Faculty (TBA)   MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  MSC:    Faculty (TBA)   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, EilenbergMacLane spaces and some algebraic topology. Advanced topics may include: Coxeter and Artin groups, BassSerre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasiisometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  MWF 11:45am  12:35pm  Aaron Abrams  max 15  MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  MSC:    Faculty (TBA)   MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  00P  MSC:    Faculty (TBA)  
