Graduate classes, Fall 2012, 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: John B. Conway, "Functions of one complex variables I", Springer Publishing. ISBN: 3540903283, 2nd edition. Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 10:00am  11:15am  Shanshuang Yang  max 16  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: Gene H. Golub, "Matrix Computations", Johns Hopkins Studies in Mathematical Sciences, 3rd Edition. Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 1:00pm  2:15pm  James Nagy  max 16  MATH 521: Algebra I  Credits: 4  − 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  MWF 9:35am  10:25am  Suresh Venapally  max 16  MATH 535: Combinatorics I  Credits: 4  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E406  MW 11:45am  1:00pm  Ron Gould  max 16  MATH 550: Functional Analysis  Credits: 4  − Description  − Sections 
Content: An introduction to concepts and applications including: metric and normed spaces. Sobolev spaces, linear operators, and functionals, compactness in metric and normed spaces. Fredholm's solvability theory, spectral theory, calculus in metric and normed spaces, selected application. Texts: MaaCluer. "Elementary Functional Analysis", Springer Publishing (2009). Assessments: TBA Prerequisites: TBA  000  MSC: W306  MWF 10:40am  11:30am  David Borthwick  max 16  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: R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge University Press (1985; 1991). Assessments: TBA Prerequisites: TBA  000  MSC: E406  TuTh 11:30am  12:45pm  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: W303  F 4:00pm  5:00pm  Dwight Duffus  max 30  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: W306  W 3:00pm  4:00pm  Raman Parimala  max 16  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 30  001  MSC: W301  Tu 4:00pm  5:00pm  Vladimir Oliker  max 30  MATH 597R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  0ONO    Ken Ono  max 999  DUFF  MSC:    Dwight Duffus  max 999  GOUL  MSC: Other   Ron Gould  max 999  NAGY  MSC:    James Nagy  max 999  VENE  MSC:    Alessandro Veneziani  max 999  YANG  MSC:    Shanshuang Yang  max 999  MATH 772: Numerical Partial Differential Equations  Credits: 4  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 2:30pm  3:45pm  Alessandro Veneziani  max 16  MATH 787R: Topics in Combinatorics: Hypergraphs  Credits: 4  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 8:30am  9:45am  Vojtech Rodl  max 16  MATH 788R: Topics in Algebra: Stacks  Credits: 4  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  000  MSC: E408  TuTh 1:00pm  2:15pm  David ZureickBrown  max 16  MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  BENZ  MSC:    Michele Benzi  max 999  BORT  MSC:    David Borthwick  max 999  GARI  MSC:    Skip Garibaldi  max 999  NAGY  MSC:    James Nagy  max 999  RAMA  MSC: Other   Raman Parimala  max 999  RODL  MSC:    Vojtech Rodl  max 999  VENE  MSC:    Alessandro Veneziani  max 999  MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
Content: TBA Texts: TBA Assessments: TBA Prerequisites: TBA  BENZ  MSC:    Michele Benzi  max 999  BORT  MSC:    David Borthwick  max 999  GOUL  MSC:    Ron Gould  max 999  NAGY  MSC:    James Nagy  max 999  OLIK  MSC:    Vladimir Oliker  max 999  ONO  MSC:    Ken Ono  max 999  RAMA  MSC:    Raman Parimala  max 999  RODL  MSC:    Vojtech Rodl  max 999  VENE  MSC:    Alessandro Veneziani  max 999  YANG  MSC:    Shanshuang Yang  max 999 
