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, 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). 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 Zureick-Brown | 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 Zureick-Brown | 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 515-516. | | 000 | MSC: E408 | TuTh 2:30pm - 3:45pm | Michele Benzi | max 16 |
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