Graduate catalog 2008  2009, 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − Description 
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.  MATH 522: Algebra II  Credits: 4  − 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 524: Theory Of Computing  Credits: 4  − Description 
 MATH 524X: Discrete Structures/Comp Sci  Credits: 4  − Description 
 MATH 531: Graph Theory I  Credits: 4  − 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: 4  − 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: 4  − Description 
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.  MATH 536: Combinatorics II  Credits: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − 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: 4  − Description 
 MATH 556: Ordinary Differen Equations II  Credits: 4  − Description 
 MATH 557: Partial Differential Equations I  Credits: 4  − 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: 4  − 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: 4  − 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 578R: Seminar in Algebra  Credits: 1  12  − Description 
Content: Research topics in algebra of current interest to faculty and students.  MATH 579R: Seminar in Analysis  Credits: 4  − Description 
Content: Topics include: numerical methods for linear algebra, inverse problems and PDE's  MATH 584: Algorithms  Credits: 4  − 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: 4  − 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: 4  − Description 
 MATH 597R: Directed Study  Credits: 1  12  − Description 
 MATH 599R: Master's Thesis Research  Credits: 1  12  − Description 
 MATH 721: Struct Of Algebraic Systems I  Credits: 4  − Description 
 MATH 730: Linear Algebra In Combinatoric  Credits: 4  − Description 
 MATH 731: Ramsey Theory  Credits: 4  − 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: 4  − 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: 4  − 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: 4  − Description 
 MATH 737: Random Graph Theory  Credits: 4  − 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: 4  − Description 
 MATH 748: Advanced Partial Differential Equations  Credits: 4  − 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: 4  − Description 
 MATH 756: Part Differential Equations II  Credits: 4  − Description 
 MATH 771: Numerical Optimization  Credits: 4  − 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: 4  − Description 
Content: Examples and classification of PDE's, initial and boundary value problems, wellposed problems, the maximum principle, finite differnce methods, varational formulations for elliptic PDE's, finite element methods, and iterative soulution methods. Prerequisites: Mathematics: 511512, 515516.  MATH 787R: Topics in Combinatorics: Random Structures  Credits: 4  − Description 
 MATH 787R: Topics in Combinatorics: Random Structures II  Credits: 4  − Description 
Content: Course description to follow soon.  MATH 787R: Topics in Combinatorics: Probabilistic & Extremal Methods  Credits: 4  − 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: Ordered Combinatorial & A  Credits: 4  − 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: 4  − Description 
 MATH 787R: Topics in Combinatorics: Random Structures  Credits: 4  − 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 788R: Topics in Algebra: Quadratic forms  Credits: 4  − Description 
 MATH 788R: Topics in Algebra: Brauer Group and Related Structures  Credits: 4  − 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: Number Theory  Credits: 4  − 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: 4  − 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: 4  − 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 789R: Topics in Analysis: Numerical Methods in Imaging  Credits: 4  − Description 
Content: Topic title and course description to follow.  MATH 789R: Topics in Analysis: Computational Fluid Dynamics  Credits: 4  − 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: Comp. Methods for Image Restoration  Credits: 4  − 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: 4  − Description 
Content: In this course we study partial differential equations arising in differential geometry and applied mathematics.  MATH 797R: Directed Study  Credits: 1  12  − Description 
 MATH 799R: Dissertation Research  Credits: 1  12  − Description 

