Graduate catalog 2013  2014, Mathematics
Note:
The class catalog is published every two years. Courses
listed in the catalog may not be offered in a given term.
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MATH 500: Probability  Credits: 3  − Description 
Content: This course will begin the development of fundamental topics in probability theory and its applications in combinatorics and algorithms. Included will be: events and their probabilities, random variables and their distributions, limit theorems, martingales, concentration of probability, random walks and Markov chains.  MATH 511: Analysis I  Credits: 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 II  Credits: 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 411412 sequence or the equivalent.  MATH 515: Numerical Analysis I  Credits: 3  − Description 
Content: The course will cover fundamental concepts of numerical analysis and scientific computing.
Material includes numerical methods for
1. Interpolation
2. Differentiation
3. Integration
4. Linear algebra
5. Ordinary differential equations
6. Partial differential equations Particulars: This is a "hands on" course and students will be required to demonstrate their understanding of the concepts through programming assignments (help will be given for the novice programmer). Prerequisites: Background in calculus, linear algebra and ODE's is assumed. Some knowledge of computer architectures, applied mathematics and elementary numerical analysis would help but is not absolutely essential.  MATH 516: Numerical Analysis II  Credits: 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 520: Algebra III  Credits: 3  − Description 
Content: This course will develop fundamental topics in commutative algebra and algebraic geometry, including affine algebraic varieties and their morphisms, Zariski topology, Hilbert Basis Theorem, Noether Normalizaiton, Hilbert's Nullstellensatz, equivalence between algebra and geometry, projective and quasiprojective varieties and their morphisms, Veronese, Segre, and Pl/"ucker embeddings, enumerative problems, and correspondences.  MATH 521: Algebra I  Credits: 3  − Description 
Content: Linear algebra, including canonical forms, infinitedimensional 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 II  Credits: 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 and Geometry  Credits: 3  − Description 
Content: This is an introductory course covering topics like affine and projective varieties, Hilbert's Nullstellensatz, morphism and rational maps of varieties, dimension, smoothness, algebraic curves, intersection multiplicity and Bezout's theorem. Topics from commutative algebra like integral extensions, Noether's normalisation lemma and primary decomposition will also be treated simultaneously.  MATH 524: Theory Of Computing  Credits: 3  − Description 
 MATH 524X: Discrete Structures/Comp Sci  Credits: 3  − Description 
 MATH 528: Algebraic Number Theory  Credits: 3  − Description 
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, Dirichlet’s unit theorem, valuations, completions, local fields, and the ramification theory of local fields. Particulars: Texts: Algebraic Number Theory, by Jurgen Neukirch Prerequisites: Math 521 and 522, or consent of instructor.  MATH 531: Graph Theory I  Credits: 3  − Description 
Content: I will introduce basic graphtheoretical concepts, graphs, trees, networks, cycles, independence number, chromatic number, planarity and genus, paths and cycles, etc. I will emphasize "extremal" problems and counting techniques. Particulars: Grades will be based on written assignments.  MATH 532: Graph Theory II  Credits: 3  − Description 
Content: Topics include: independence of vertices and edges (matchings), factorizations and decompositions, coloring (both vertices and edges), and classic external theory. Prerequisites: Mathematics 531.  MATH 535: Combinatorics I  Credits: 3  − Description 
Content: This is the first course of a 2course 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].  MATH 536: Combinatorics II  Credits: 3  − Description 
Content: This course is the second of the sequence of Math 535536 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: Topology I  Credits: 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: Topology II  Credits: 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 3manifolds). Chosen in accordance with the interest of students and instructor.  MATH 543: Algebraic Topology I  Credits: 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 II  Credits: 3  − Description 
Content: Singular, simplicial and cellular homology, long exact sequences in homology, MayerVietoris sequences, excision, Euler characteristic, degrees of maps, BorsukUlam theorem, Lefschetz fixed point theorem, cohomology, universal coefficient theorem, the cup product, Poincare duality  MATH 545: Introduction to Differential Geometry I  Credits: 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 3space to illustrate key concepts. Particulars: Open to undergraduates with permission of the instructor.  MATH 546: Intro. to Differential Geometry II  Credits: 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 Topology  Credits: 3  − Description 
Content: Smooth manifolds, tangent spaces and derivatives, Sard's theorem, inverse function theorem, transversality, intersection theory, fixedpoint theory, vector fields, Euler characteristic, the PoincareHopf theorem, exterior algebras, differential forms, integration, Stokes' theorem, de Rham cohomology.  MATH 550: Functional Analysis  Credits: 3  − Description 
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.  MATH 555: Intro To Applied Analysis  Credits: 3  − Description 
 MATH 556: Ordinary Differen Equations II  Credits: 3  − Description 
 MATH 557: Partial Differential Equations I  Credits: 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 557: Partial Differential Equations I  Credits: 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 558: Partial Differential Equations  Credits: 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 Analysis  Credits: 3  − Description 
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). Particulars: Textbook: C. D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000.
Additional readings:
R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge University Press (1985; 1991).
R. A. Horn and C. R. Johnson, "Topics in Matrix Analysis", Cambridge University Press (1991; 1994).
F. R. Gantmacher, "The Theory of Matrices", vols. III, Chelsea (1959; 1971).
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences", Academic Press (1979); reprinted by SIAM, 1994.
D. Serre, "Matrices. Theory and Applications", Springer, 2002.  MATH 577R: Seminar in Combinatorics  Credits: 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 Combinatorics: Network Science  Credits: 3  − Description 
 MATH 577R: Seminar in Combinatorics: Introduction to Complex Networks  Credits: 3  − Description 
Content: • Mathematical background. A brief on matrix algebra. A brief on data analysis.
• Introduction to networks. History; Basic Definitions; Walks and Paths: Eulerian and Hamiltonian networks, random walks, shortest path distance, Degree Distributions: Poisson, power law, identifying correlations; Graph Connectivity: connected components, cliques; Graph Structures: trees, bipartivity and planarity; Modern Applications.
• Adjacency relation in networks. Degree distributions; degreedegree correlations; Spectral properties of networks. Spectrum of the Adjacency Matrix, the Graph Laplacian.
• Fragments in complex networks. subgraphs, network motifs, graphlets, closed walks, combinatorics of subgraphs, clustering coefficients, combinatorics of assortativity, network hierarchy, network reciprocity
• Matrix functions on networks. exponential adjacency matrix, hyperbolic functions of the adjacency matrix, network entropy and free energy, network communicability, returnability
• Centrality measure in networks. A description of the definitions and applications of centralities like degree, Katz, eigenvector, PageRank, subgraph, closeness and betweenness.
• Global measures of networks. A description of a few global measures characterising the degree heterogeneity, fractality, and topological classes of networks. Adjacency and distancebased invariants; expansion and network classes; spectral scaling method; network bipartivity
• Community structure in networks. Network partition methods; local improvement methods; spectral partittioning; methods based on link centrality; quality criteria; methods based on modularity; the problem of resolution; clustering based on similarity; communities based on communicability; how to find bipartitions in networks.
• Random models of networks. “Classical” random networks; smallworld random networks; scalefree networks; random geometric networks.  MATH 578R: Seminar in Algebra  Credits: 3  − Description 
Content: Research topics in algebra of current interest to faculty and students.  MATH 579R: Seminar in Analysis  Credits: 3  − Description 
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's  MATH 584: Algorithms  Credits: 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 590: Teaching Seminar  Credits: 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 & Logic  Credits: 3  − Description 
 MATH 597R: Directed Study  Credits: 1  9  − Description 
 MATH 599R: Master's Thesis Research  Credits: 1  9  − Description 
 MATH 721: Struct Of Algebraic Systems I  Credits: 3  − Description 
 MATH 730: Linear Algebra In Combinatoric  Credits: 3  − Description 
 MATH 731: Ramsey Theory  Credits: 3  − Description 
Content: This course will continue the development of ramsey theory begun in Math 531532 and Math 535536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the HalesJewett 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 531532 and Math 535536 or permission of the instructor.  MATH 732: Extremal Graph Theory  Credits: 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 kconnected graphs. Matchings: fundamentals, the number of 1factors, ffactors, 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 Methods  Credits: 3  − Description 
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the LovaszLocal Lemma and its applications. The secondmoment method. Large deviation inequalities and Derandomization.  MATH 736: Random Algorithms  Credits: 3  − Description 
 MATH 737: Random Graph Theory  Credits: 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 Topology  Credits: 3  − Description 
 MATH 748: Advanced Partial Differential Equations  Credits: 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 quasilinear elliptic and parabolic equations; basic methods and results on solvability * Quasilinear 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 MongeAmpere 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 * MongeKantorovich optimal transportation theory and its connections with nonlinear PDE's Prerequisites: Mathematics 558 or permission of the instructor.  MATH 755: Ordinary Differen Equations II  Credits: 3  − Description 
 MATH 756: Part Differential Equations II  Credits: 3  − Description 
 MATH 771: Numerical Optimization  Credits: 3  − Description 
Content: This course will provide students with an overview of stateoftheart numerical methods for solving unconstrained, largescale optimization problems. Algorithm development will be emphasized, including efficient and robust implementations. In addition, students will be exposed to stateoftheart software that can be used to solve optimization problems. Prerequisites: Mathematics 511512, 515516.  MATH 772: Numerical Partial Differential Equations  Credits: 3  − Description 
Content: Partial Differential Equations are a formidable tool for describing realworld 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 problemdependent. 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: 511512, 515516.  MATH 786R: Topics in Topology  Credits: 3  − 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, EilenbergMacLane spaces and some algebraic topology. Advanced topics may include: Coxeter and Artin groups, BassSerre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasiisometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem. Particulars: TBA Prerequisites: TBA  MATH 787R: Topics in Combinatorics: Ordered Combinatorial & A  Credits: 3  − Description 
Content: The course will have two components:
(1) A series of lectures on the basics of order theory, including an introduction to finite and infinite partially ordered sets and lattices, topics from combinatorics/set systems such as Sperner theory, uses of ordering to study classes of graphs, digraphs and other relational systems.
The length of time spent on this will depend upon the backgrounds and interests of participants.
(2) Focus on topics from "Graphs and Homomorphisms" by Hell and Nesetril, particularly the lattice of graph types ordered by homomorphism. There will be time spent on recent papers motivated by the Hedetniemi Conjecture, diverse notions of chromatic number [fractional, circular], etc. A set of coherently organized papers will be made available to participants before the beginning of the seminar. The emphasis will be on several open problems [of varying degrees of accessibility]. Particulars: Participants will be expected to take an active role in presenting material and in [skeptically] attending all presentations.
There will be no text but, in addition to the Hell/Nesetril source, basic material on orderings, set systems, combinatorics and ordered sets, and lattice theory can be found in:
(1) Combinatorics of Finite Sets, I. Anderson
(2) Combinatorial Theory, M. Aigner
(3) Introduction to Lattices and Order, B. Davey and H. Priestley Prerequisites: The introductory sequence in combinatorics and basic knowledge of group theory and graph theory will be assumed.  MATH 787R: Topics in Combinatorics: Extremal Combinatorics & Optimization  Credits: 3  − Description 
 MATH 787R: Topics in Combinatorics: Random Structures  Credits: 3  − Description 
Content: The course will cover several advanced topics from the theory of random graphs, hypergraphs and other random structures, like random subsets of integers. Prerequisites: No prerequisite is required, but some knowledge of probability, graph theory and combinatorics is anticipated.  MATH 787R: Topics in Combinatorics: Random Structures  Credits: 3  − Description 
 MATH 787R: Topics in Combinatorics: Random Structures II  Credits: 3  − Description 
Content: Course description to follow soon.  MATH 787R: Topics in Combinatorics: Probabilistic & Extremal Methods  Credits: 3  − Description 
Content: This course will develop discrete probabilistic aspects of topics begun in Math 535536. We will discuss deletion method, the secondmoment method, LovaszLocal Lemma and its applications. Large deviation inequalities, Derandomization and the regularity lemma and its applications.  MATH 787R: Topics in Combinatorics: Ramsey Theory  Credits: 3  − Description 
Content: This course will continue the development of ramsey theory begun in Math 531532 and Math 535536. Included will be: Sets: Ramsey's theorem, the compactness principle. Progressions: Van der Waerden's Theorem, the HalesJewett 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.  MATH 787R: Topics in Combinatorics: Hypergraphs  Credits: 3  − Description 
 MATH 787R: Topics in Combinatorics: Extremal Combinatorics and Ramsey Theory  Credits: 3  − Description 
Content: TBA Particulars: TBA Prerequisites: TBA  MATH 787R: Topics in Combinatorics: Hypergraphs II  Credits: 3  − Description 
 MATH 788R: Topics in Algebra: Number Theory  Credits: 3  − Description 
Content: Chomology of finite and profinite groups; Galois cohomology, commutative case; nonabelian Galois chohomology and principal homogeneous spaces; cohomological dimension of fields; open questions.  MATH 788R: Topics in Algebra: Elliptic Curves  Credits: 3  − Description 
Content: This will be an introductory course on elliptic curves. The course content will include the following topics: geometry of cubic curves, Weierstrass normal form,
groups of rational points, torsion points and NagelLutz Theorem, elliptic curves over finite fields, MordellWeil theorem.  MATH 788R: Topics in Algebra: Algebraic Geometry  Credits: 3  − Description 
Content: After an introduction to affine and projective varieties, we shall cover the following topics on algebraic curves: Bezout's theorem, resolution of singularities for curves, Riemann Roch theorem. Necessary materials from commutative algebra will also be covered.  MATH 788R: Topics in Algebra: Quadratic forms  Credits: 3  − Description 
 MATH 788R: Topics in Algebra: Brauer Group and Related Structures  Credits: 3  − Description 
Content: We study the classical and cohomological Brauer group of a field, its role in algebraic geometry and number theory, and other applications. Topics include exponent and index of central simple algebras, splitting behavior, definition by descent, and Brauer Severi varieties.  MATH 788R: Topics in Algebra: Algebraic Groups  Credits: 3  − Description 
Content: This course will present linear algebraic groups from an angle oblique to the standard treatments. We will study them organically, as opposed to the more systematic treatment available in the standard texts (by Borel, Humphreys, Springer).  MATH 788R: Topics in Algebra: Modular Forms/Elliptic Curves  Credits: 3  − Description 
 MATH 788R: Topics in Algebra: Algebraic Geometry II  Credits: 3  − Description 
Content: We will cover the topics: sheaves, schemes, cohomology of sheaves, Serre duality theorems and RiemannRoch theorem for curves.  MATH 788R: Topics in Algebra: Stacks  Credits: 3  − Description 
 MATH 789R: Comp. Methods for Image Restoration  Credits: 3  − Description 
Content: In this course we review the field of image restoration and its application to medical imaging. We will discuss the mathematical background, using variational techniques and numerical optimization. The course is a "hands on" course and the students will write Matlab code and deal with actual data.  MATH 789R: Topics in Analysis: Geometric PDE, II  Credits: 3  − Description 
Content: In this course we study partial differential equations arising in differential geometry and applied mathematics.  MATH 789R: Topics in Analysis: Numerical Methods in Imaging  Credits: 3  − Description 
Content: Topic title and course description to follow.  MATH 789R: Topics in Analysis: Computational Fluid Dynamics  Credits: 3  − Description 
Content: The course provides an introduction to numerical methods for the partial differential equations in fluid dynamics. This is a dynamic research field, so attention will be given also to recent proposals. The course will cover mainly incompressible fluids, however a part will be devoted to compressible fluid dynamics. A part of the course will be given in the Computer Room.
Introduction: Derivations of basic equations of fluid mechanics, NavierStokes, Euler, potential flows. Properties of these equations.
Incompressible fluid dynamics: NavierStokes equations for incompressible fluids. Saddle point nature of the problem. Steady Stokes equations. Finite element discretization. Infsup condition. Pressure matrix method, quasicompressibility methods. Strongly consistent stabilization. Unsteady case. Projection and splitting methods. Spectral discretization, finite volume discretization.
Compressible fluid dynamics: Inviscid equations. Finite difference, Finite volumes. TVD, TVB schemes for nonlinear hyperbolic equations. Shockcapturing schemes. Particulars: Books:
• A. Quarteroni, A. Valli: Numerical Approximation of Partial Differential Equations, SpringerVerlag, Berlin 1994.
• H.C. Elman, D.J. Silvester, A.J. Wathen, Finite Elements and Fast Iterative Solvers: with applications in Incompressible Fluid Dynamics, Oxford Univ. Press, 2005.
• J. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, SpringerVerlag (1996)
• R. Le Veque, Numerical Methods for Conservation Laws, Birkhauser, 1988
• L. Quartapelle: Numerical solution of the incompressible NavierStokes equations, BirkhauserVerlag, Basel, 1993.  MATH 789R: Topics in Analysis: Iterative Methods  Credits: 3  − Description 
Content: An overview of iterative methods for largescale linear algebra
problems including stationary iterations, Krylov subspace methods,
preconditioning techniques, multilevel methods and applications. Particulars: Main textbook: Y. Saad, Iterative Methods for Sparse Linear Systems, Second Edition. SIAM, Philadelphia, 2003. Prerequisites: Math 515516 or instructor's permission  MATH 789R: Topics in Analysis: Geometric Function Theory & PDE  Credits: 3  − Description 
Content: While conformal mappings are homeomorphic solutions of CauchyRiemann systems, quasiconformal mappings can be viewed as homeomorphic solutions of Beltrami systems. In this course we will study conformal and quasiconformal mappings in Euclidean spaces from a PDE perspective. A major focus of the course will be the presentation of the rigidity theory of 1quasiconformal mappings, from the most elementary form in the complex plane to recent developments.  MATH 789R: Topics in Analysis: Computational Methods for Inverse Problems  Credits: 3  − Description 
 MATH 797R: Directed Study  Credits: 1  9  − Description 
 MATH 799R: Dissertation Research  Credits: 1  9  − Description 

