Graduate catalog 2006  2007, Mathematics
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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: Course material will focus on iterative methods of numerical linear algebra. Both eigenvalue problems and solving systems of equations will be covered in detail with emphasis on the algorithms currently used for large scale sparse and structured problems arising from mathematical modelling of real world applications. Links to the mathematical foundation of the methods will be made whenever possible. A solid theoretical background will be balanced with implementation and numerical stability issues. Prerequisites: Students interested in this course are strongly recommended to take MATH 515 before MATH 516.  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 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  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 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 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 590: Teaching Seminar  Credits: 4  − Description 
Content: This seminar will concentrate on effective teaching techniques in mathematics. Topics included will include:
General advise for new TA's. General advise for International TA's. Students will present several practice lectures over different levels of material. They will recieve 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 597R: Directed Study  Credits: 1  12  − Description 
 MATH 599R: Master's Thesis Research  Credits: 1  12  − 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 737: Random Graphs  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 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 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 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 797R: Directed Study  Credits: 1  12  − Description 
 MATH 799R: Dissertation Research  Credits: 1  12  − Description 

