Graduate catalog 2016 - 2017, Mathematics

Note: The class catalog is published every two years. Courses listed in the catalog may not be offered in a given term. For class offerings for each semester please click on the "Offerings" links.
MATH 500: ProbabilityCredits: 3− Description
MATH 511: Analysis ICredits: 3− Description
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.
MATH 512: Analysis IICredits: 3− Description
Content: Topics will include: Measure and integration theory on the real line as well as on a general measure space, Bounded linear functionals on L^p spaces. If time permits, Sobolev spaces and Fourier transforms will be introduced.
Prerequisites: Students are expected to have the background of Math 411-412 sequence or the equivalent.
MATH 515: Numerical Analysis ICredits: 3− Description
Content: Course will cover fundamental parts of numerical linear algebra including matrix factorizations, solution of linear systems and least-squares problems, the calculation of eigenvalues and eigenvectors, and basic notions on iterative methods for large-scale matrix problems. Issues pertaining to conditioning and numerical stability will be thoroughly analyzed. We will also point out and use links to other mathematical and computer science disciplines such as mathematical modelling, computer architectures and parallel computing.
Particulars: Excellent background in linear algebra is assumed. Some knowledge of computer architectures, programming and elementary numerical analysis is highly desirable.
MATH 516: Numerical Analysis IICredits: 3− Description
Content: This course covers fundamental concepts of numerical analysis and scientific computing. Material includes numerical methods for curve fitting (interpolation, splines, least squares), differentiation, integration, and differential equations. It is assumed that students have a strong background in numerical linear algebra.
Prerequisites: Math 515, undergraduate course work in multivariable calculus and ordinary differential equations. An undergraduate course in numerical analysis would help, but is not absolutely essential.
MATH 517: Iterative Methods for Linear SystemsCredits: 3− Description
Prerequisites: Prerequisite MATH 516
MATH 521: Algebra ICredits: 3− Description
Content: Linear algebra, including canonical forms, infinite-dimensional vector spaces, tensor products, and multilinear algebra. Group theory including group actions and representations.
Particulars: Text: A.W. Knapp, "Basic algebra", Birkhauser, 2006.
MATH 522: Algebra IICredits: 3− Description
Content: Continuation of 521. Topics: Modules, especially modules over a principal ideal domain, fields, Galois theory, representation of finite groups, Commutative algebra.
Prerequisites: Math 521.
MATH 523: Commutative Algebra & Algebraic GeometryCredits: 3− Description
MATH 524: Theory of ComputingCredits: 3− Description
MATH 528: Algebraic Number TheoryCredits: 3− Description
MATH 531: Graph Theory ICredits: 3− Description
MATH 532: Graph Theory IICredits: 3− Description
MATH 535: Combinatorics ICredits: 3− Description
MATH 536: Combinatorics IICredits: 3− Description
Content: This course is the second of the sequence of Math 535-536 and as such will continue to develop the topics from the first semester. Specific topics will include finite geometries, Hadermard matrices, Latin Squares, an introduction to design theory, extremal set theory and an introduction to combinatorial coding theory.
MATH 541: General Topology ICredits: 3− Description
Content: An introduction to some of the fundamental concepts of topology required for basic courses in analysis. Some of these comcepts are: Baire category, topological spaces, completeness, compact and locally compact sets, connected and locally connected sets, characterizations of arcs, Jordan curves, and Peano continua, completeness, metric spaces, separability, countable bases, open and closed mappings, and homeomorphisms. Other selected topics if time permits from algebraic topology or continua theory.
MATH 542: General Topology IICredits: 3− Description
Content: The content of 542 may vary. Standard topics include Algebraic Topology (the fundamental group and covering spaces, homology and cohomology); Differential Topology (manifolds, transversality, intersection theory, integration on manifolds); and Geometric Topology (hyperbolic geometry knots and 3-manifolds). Chosen in accordance with the interest of students and instructor.
MATH 543: Algebraic Topology ICredits: 3− Description
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
MATH 544: Algebraic Topology IICredits: 3− Description
Content: Singular, simplicial and cellular homology, long exact sequences in homology, Mayer-Vietoris sequences, excision, Euler characteristic, degrees of maps, Borsuk-Ulam theorem, Lefschetz fixed point theorem, cohomology, universal coefficient theorem, the cup product, Poincare duality
MATH 545: Introduction to Differential Geometry ICredits: 3− Description
Content: An introduction to Riemannian geometry. The main goal is an understanding of the nature and uses of curvature, which is the local geometric invariant that measures the departure from Euclidean geometry. No previous experience in differential geometry is assumed, and we will rely heavily on pictures of surfaces in 3-space to illustrate key concepts.
Particulars: Open to undergraduates with permission of the instructor.
MATH 546: Intro. to Differential Geometry IICredits: 3− Description
Content: An introduction to Riemannian geometry and global analysis. Topics to be covered: Manifolds, Riemannian metrics, Connections, Curvature; Geodesics, Convexity, Topics in Global Analysis.
MATH 547: Differential TopologyCredits: 3− Description
Content: Smooth manifolds, tangent spaces and derivatives, Sard's theorem, inverse function theorem, transversality, intersection theory, fixed-point theory, vector fields, Euler characteristic, the Poincare-Hopf theorem, exterior algebras, differential forms, integration, Stokes' theorem, de Rham cohomology.
MATH 550: Functional AnalysisCredits: 3− Description
Content: An introduction to concepts and applications including: metric and normed spaces, Hilbert and Banach spaces, linear operators and functionals, compactness in metric and normed spaces, Fredholm's solvability theory, spectral theory, calculus in metric and normed spaces, selected applications.
Prerequisites: Math 511, Math 512.
MATH 555: Intro To Applied AnalysisCredits: 3− Description
MATH 556: Ordinary Differen Equations IICredits: 3− Description
MATH 557: Partial Differential Equations ICredits: 3− Description
MATH 558: Partial Differential EquationsCredits: 3− Description
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
MATH 561: Matrix AnalysisCredits: 3− Description
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).
MATH 576R: Seminar in TopologyCredits: 1 - 3− Description
MATH 577R: Seminar in CombinatoricsCredits: 3− Description
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.
MATH 577R: Seminar in CombinatoricsCredits: 3− Description
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.
MATH 578R: Seminar in AlgebraCredits: 3− Description
Content: Research topics in algebra of current interest to faculty and students.
MATH 578R: Seminar in AlgebraCredits: 3− Description
Content: Research topics in algebra of current interest to faculty and students.
MATH 579R: Seminar in AnalysisCredits: 1 - 12− Description
MATH 579R: Seminar in AnalysisCredits: 3− Description
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's
MATH 584: AlgorithmsCredits: 3− Description
Content: This course is a graduate level introduction to the design and analysis of algorithms. Although we will review some undergraduate level material, we will instead emphasize reading and experimentation at a level appropriate for the initiation of research. This course will have both theoretical and practical content. As course highlights, students will be expected to implement and analyze the performance of a fundamental data structure, starting with a close reading of the original research paper.
MATH 587G: Number TheoryCredits: 3− Description
MATH 590: Teaching SeminarCredits: 3− Description
Content: This seminar will concentrate on effective teaching techniques in mathematics. Topics included will include: General advice for new TA's. General advice for International TA's. Students will present several practice lectures over different levels of material. They will receive practice on quiz and test preparation. Syllabus information on courses most likely to be taught by new TA's will be supplied. General professional development information will also be included.
MATH 593: Computability & LogicCredits: 3− Description
MATH 597R: Directed StudyCredits: 1 - 9− Description
MATH 597R: Directed StudyCredits: 1 - 9− Description
MATH 599R: ResearchCredits: 1 - 9− Description
MATH 599R: Master's Thesis ResearchCredits: 1 - 9− Description
MATH 721: Struct Of Algebraic Systems ICredits: 3− Description
MATH 730: Linear Algebra In CombinatoricCredits: 3− Description
MATH 731: Ramsey TheoryCredits: 3− Description
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. The symmetric hypergraph Theorem, Schur and Rado numbers, higher ramsey numbers. Bipartite ramsey theory, induced ramsey theory, restricted results, Euclidean and Graph ramsey theory. Topological Dynamics, Ultra filters, the infinite.
Prerequisites: Math 531-532 and Math 535-536 or permission of the instructor.
MATH 732: Extremal Graph TheoryCredits: 3− Description
Content: Continue the development of extremal in Graph Theory begun in Math 532. Included will be: Connectivity: structure of 2 and 3 connected graphs, minimally k-connected graphs. Matchings: fundamentals, the number of 1-factors, f-factors, coverings. Cycles: Graphs with large girth and large min. degree, vertex disjoint cycles, edge disjoint cycles, cycles of specific lengths, circumference. Diameter: Graphs with large subgraphs of small diameter, factors of small diameter, ties to connectivity. Colorings: General colorings, sparse graphs of large chromatic no., perfect graphs. Turan type Extremal Theory.
Prerequisites: Math 532 Graph Theory II or permission of the instructor.
MATH 733: Probabilistic MethodsCredits: 3− Description
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535-536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the Lovasz-Local Lemma and its applications. The second-moment method. Large deviation inequalities and Derandomization.
MATH 736: Random AlgorithmsCredits: 3− Description
MATH 737: Random Graph TheoryCredits: 3− Description
Content: This course will introduce the fundamental topics in Random Graph Theory. Included will be: Basic Models: binomial, uniform. The method of moments: subgraph counts Thresholds for large subgraphs: connectedness, perfect matchings, hamiltonicity. Phase transitions. Advanced tools: martingales, Janson's inequality. Chromatic Number and other partition and extremal properties.
MATH 741: Geometric TopologyCredits: 3− Description
MATH 748: Advanced Partial Differential EquationsCredits: 3− Description
Content: This course will discuss advanced topics in the modern theory of nonlinear partial differential equations and their applications. Included in the course are many of the following topics: * Basic concepts, sample problems in physics, biology, and geometry * Linear and quasi-linear elliptic and parabolic equations; basic methods and results on solvability * Quasi-linear geometric problems: mean curvature problem, Christoffel's problem, evolution by mean curvature, prescribing scalar curvature, Yamabe's problem * Convexity and elliptic and parabolic equations of Monge-Ampere type, Aleksandrov's geometric methods, Calabi's problem and Chern's classes, Fully nonlinear problems * N. Krylov's and C. Evans's results on nonlinear problems * The reflector mapping problem, the Gauss curvature problem, the Weyl problem, the Minkowski problem * Variational problems associated with some nonlinear PDE's * Monge-Kantorovich optimal transportation theory and its connections with nonlinear PDE's
Prerequisites: Mathematics 558 or permission of the instructor.
MATH 755: Ordinary Differen Equations IICredits: 3− Description
MATH 756: Part Differential Equations IICredits: 3− Description
MATH 771: Numerical OptimizationCredits: 3− Description
Content: This course will provide students with an overview of state-of-the-art numerical methods for solving unconstrained, large-scale optimization problems. Algorithm development will be emphasized, including efficient and robust implementations. In addition, students will be exposed to state-of-the-art software that can be used to solve optimization problems.
Prerequisites: Mathematics 511-512, 515-516.
MATH 772: Numerical Partial Differential EquationsCredits: 3− Description
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.
Prerequisites: Mathematics: 511-512, 515-516.
MATH 786R: Topics in TopologyCredits: 1 - 4− Description
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, Eilenberg-MacLane spaces and some algebraic topology. Advanced topics may include: Coxeter and Artin groups, Bass-Serre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasi-isometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem.
Particulars: TBA
Prerequisites: TBA
MATH 787R: Topics in Combinatorics: HypergraphsCredits: 3− Description
MATH 787R: Topics in CombinatoricsCredits: 3− Description
MATH 788R: Topics in Algebra: Algebraic GroupsCredits: 3− Description
Content: The course will consist of the following distinct components : 1. Structure theory of reductive groups over algebraically closed fields : Overview of objects and notions such as tori, solvable groups, Lie algebras, Jordan decomposition. Conjugacy of Borel subgroups and maximal tori, root systems, Bruhat decomposition, representation and classification of semi simple groups in terms of root systems. 2. Galois cohomology of linear algebraic groups : Classical groups and algebras with involutions, Steinberg's theorem, dimension two fields and Serre's conjecture, cohomological invariants and a discussion on some open questions in this area.
MATH 788R: Topics in Algebra: Representation TheoryCredits: 3− Description
MATH 789R: Topics in Analysis: Bayesian Inverse Problems and Uncertainty QualificationCredits: 3− Description
Content: This special topics course introduces basic concepts as well as more recent advances in Bayesian methods for solving inverse problems. Motivated by real-world applications, we will contrast the frequentists and the Bayesian approach to inverse problems and emphasize the role of regularization/priors. Also, we will explore sampling techniques used for uncertainty quantification.
Particulars: Literature: Somersalo and Calvetti, An Introduction to Bayesian Scientific Computing, Springer, 2007 Kaipio and Somersalo, Statistical and Computational Inverse Problems, Springer 2004
MATH 797R: Directed StudyCredits: 1 - 9− Description
MATH 799R: Dissertation ResearchCredits: 1 - 9− Description