Graduate classes, Fall 2010, Mathematics

MATH 511: Analysis ICredits: 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: Functions of One Complex Variables I, by John B. Conway. Springer Publishing, ISBN: 3-540-90328-3.
Assessments: TBA
Prerequisites: TBA
000MSC: E408TuTh 10:00am - 11:15amShanshuang Yangmax 16
MATH 515: Numerical Analysis ICredits: 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: Matrix Computations by Gene H. Golub & Charles F. Van Loan. John Hopkins University Press. ISBN# 0-8018-5414-8.
Assessments: TBA
Prerequisites: TBA
000MSC: W306TuTh 2:30pm - 3:45pmJames Nagymax 16
MATH 521: Algebra ICredits: 4− Description− Sections
Content: Linear algebra, including canonical forms, infinite-dimensional vector spaces, tensor products, and multilinear algebra. Group theory including group actions and representations.
Texts: Abstract Algebra, by Dummit adn Foote. Wiley Publishing. ISBN # 978-0-471143334-7
Assessments: TBA
Prerequisites: TBA
000MSC: E408TuTh 11:30am - 12:45pmRaman Parimalamax 16
MATH 528: Algebraic Number TheoryCredits: 4− Description− Sections
Content: Topics will include: Algebraic numbers and integers, Dedekind domains, discriminants, norm and trace, cyclotomic fields, factorization of ideals in Dedekind domains, decomposition of prime ideals in integral extensions, Hilbert ramification theory, lattice methods, class group, finiteness of class number, Dirichletís unit theorem, valuations, completions, local fields, and the ramification theory of local fields.
Texts: (1) Number Fields, by Daniel Marcus. Springer Publishing. ISBN # 10-03787902-791. (2) Introductory Algebraic Number Theory, by Saban Alaca and Kenneth Williams. Cabridge University Press. ISBN 10-05211540119.
Assessments: TBA
Prerequisites: Math 521 and 522, or consent of instructor.
000MSC: E406MWF 12:50pm - 1:40pmKen Onomax 16
MATH 535: Combinatorics ICredits: 4− Description− Sections
Content: This is the first course of a 2-course sequence on combinatorial mathematics. We shall focus on mostly introductory material on enumeration, relational structures [graphs, directed graphs, partially ordered sets, hypergraphs, matroids], and configurations [designs, codes, permutation groups].
Texts: The main reference will be P. J. Cameron, "Combinatorics: Topics, Techniques, Algorithms" [ISBN 0 521 45761 0]
Assessments: The course grade will be determined by written assignments, collected throughout the term, and a combination take-home and in-class final exam. We may replace some assignments and exams with student presentations.
Prerequisites: The course does not assume background in combinatorics. It is assumed that all students have good undergraduate background in linear algebra, abstract algebra, and analysis.
000MSC: E406MWF 10:40am - 11:30amDwight Duffusmax 16
MATH 543: Algebraic Topology ICredits: 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: Introduction to Manifolds, by John Lee. Springer Publishing. ISBN# 0-387-95026-5.
Assessments: TBA
Prerequisites: TBA
000MSC: E408TuTh 1:00pm - 2:15pmAaron Abramsmax 16
MATH 545: Introduction to Differential Geometry ICredits: 4− Description− Sections
Content: TBA
Texts: TBA
Assessments: TBA
Prerequisites: TBA
000MSC: E406MWF 11:45am - 12:35pmDavid Borthwickmax 16
MATH 561: Matrix AnalysisCredits: 4− 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: Matrix Analysis, by Horn & Johnson. Cambridge University Press. ISBN #0-521-38632-2.
Assessments: TBA
Prerequisites: TBA
000MSC: W306TuTh 1:00pm - 2:15pmMichele Benzimax 16
MATH 577R: Seminar in CombinatoricsCredits: 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
000MSC: W306F 4:00pm - 4:50pmDwight Duffusmax 30
MATH 578R: Seminar in AlgebraCredits: 1 - 12− Description− Sections
Content: Research topics in algebra of current interest to faculty and students.
Texts: TBA
Assessments: TBA
Prerequisites: TBA
000MSC: W303Tu 4:00pm - 4:50pmRaman Parimalamax 30
MATH 579R: Seminar in AnalysisCredits: 4− Description− Sections
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
Texts: TBA
Assessments: TBA
Prerequisites: TBA
000MSC: W301Tu 4:00pm - 4:50pmVladimir Olikermax 30
001MSC: W306W 12:50pm - 1:40pmJames Nagymax 30
MATH 597R: Directed StudyCredits: 1 - 12− Description− Sections
Content: TBA
Texts: TBA
Assessments: TBA
Prerequisites: TBA
00PTBAFaculty (TBA)max 999
MATH 599R: Master's Thesis ResearchCredits: 1 - 12− Description− Sections
Content: TBA
Texts: TBA
Assessments: TBA
Prerequisites: TBA
00PTBAFaculty (TBA)max 999
MATH 772: Numerical Partial Differential EquationsCredits: 4− Description− Sections
Content: Partial Differential Equations are a formidable tool for describing real-world problems ranging from fluid dynamics to economical studies. Unfortunately, in many cases an explicit solution of these equations cannot be found. Numerical techniques are mandatory for finding an approximate solution. The choice of the most appropriate method is usually problem-dependent. In this course we will consider different classes of differential problems and discuss possible methods of solution, their features in terms of accuracy, computational cost, implementation. Particular emphasis will be given to the Galerkin class of methods (Finite Elements, Spectral Methods), even if Finite Volume and Finite Differences methods will be considered as well. The course will be partially carried out in the Computer Lab, where theoretical properties of the different methods will be verified with Matlab and FreeFem codes.
Texts: Numerical Models for Differential Problems, by Alfio Quarteroni. Springer Publishing. ISBN 10: 8847010705.
Assessments: TBA
Prerequisites: TBA
000MSC: W304TuTh 10:00am - 11:15amAlessandro Venezianimax 16
MATH 787R: Topics in Combinatorics: Ramsey TheoryCredits: 4− Description− Sections
Content: This course will continue the development of ramsey theory begun in Math 531-532 and Math 535-536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the Hales-Jewett Theorem, spaces - affine and vector, Roth's Theorem and Szemeredi's Theorem. Equations: Schur's Theorem, regular homogeneous equations and systems, Rado's Theorem, finite sums and unions, Folkman's Theorem. Numbers: exact ramsey numbers, asymptotics, Van der Waerden numbers. Schur and Rado numbers, higher ramsey numbers. Bipartite ramsey theory, induced ramsey theory, restricted results, Euclidean and Graph ramsey theory. Topological Dynamics, Ultrafilters, the infinite.
Texts: TBA
Assessments: TBA
Prerequisites: TBA
000MSC: E406TuTh 8:30am - 9:45amVojtech Rodlmax 16
MATH 797R: Directed StudyCredits: 1 - 12− Description− Sections
Content: TBA
Texts: TBA
Assessments: TBA
Prerequisites: TBA
00PTBAFaculty (TBA)max 999
MATH 799R: Dissertation ResearchCredits: 1 - 12− Description− Sections
Content: TBA
Texts: TBA
Assessments: TBA
Prerequisites: TBA
00PTBAFaculty (TBA)max 999