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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MW 11:30am  12:45pm  David Borthwick  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W306  TuTh 10:00am  11:15am  James Nagy  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MW 10:00am  11:15am  Raman Parimala  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MW 1:00pm  2:15pm  Ron Gould  max 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 Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W304  TuTh 1:00pm  2:15pm  Vladimir Oliker  max 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, 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: R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge University Press (1985; 1991). Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 11:30am  12:45pm  Michele Benzi  max 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. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W302  M 4:00pm  4:50pm  Dwight Duffus  max 20  MATH 578R: Seminar in Algebra  Credits: 3  − Description  − Sections 
Content: Research topics in algebra of current interest to faculty and students. Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W306  Tu 4:00pm  4:50pm  David ZureickBrown  max 20  MATH 579R: Seminar in Analysis  Credits: 3  − Description  − Sections 
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: W301  F 12:00pm  12:50pm  Alessandro Veneziani  max 20  001  MSC: W302  Tu 4:00pm  4:50pm  Vladimir Oliker  max 25  MATH 599R: Master's Thesis Research  Credits: 1  9  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  YANG  MSC   Faculty (TBA)   MATH 787R: Topics in Combinatorics: Hypergraphs  Credits: 3  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 1:00pm  2:15pm  Vojtech Rodl  max 16  000  MSC: E406  TuTh 10:00am  11:15am  Vojtech Rodl  max 16  MATH 788R: Topics in Algebra: Local Fields and Class Field Theory  Credits: 3  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 1:00pm  2:15pm  David ZureickBrown  max 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 515516.  000  MSC: E408  TuTh 2:30pm  3:45pm  Michele Benzi  max 16 
