Senior And Graduate Course Offerings Senior And Graduate Course Offerings 2003 spring

MATH 4315: GRAPH THEORY WITH APPLICATIONS (Section 09798)
Time:	4:00-5:30 pm, TTH, Room 140-SR
Instructor:	S. Fajtlowicz
Prerequisites:	Discrete Mathematics.
Text(s):	The course will be based on the instructor's notes.
Description:	Planar graphs and the Four-Color Theorem. Trivalent planar graphs with applications to fullerness - new forms of carbon. Algorithms for Eulerian and Hamiltonian tours. Erdos' probabilistic method with applications to Ramsey Theory and , if time permits, network flows algorithms with applications to transportation and job assigning problems, or selected problems about trees.

MATH 4332: INTRODUCTION TO REAL ANALYSIS II (Section 11794)
Time:	1:00-2:30 pm, MW, Room 132-SR
Instructor:	V. Paulsen
Prerequisites:	Math 4331.
Text(s):	Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, Latest Edition (required); Real Analysis with Real Applications,K.R. Davidson and A.P. Donsig, Prentice Hall (ISBN 0-13-041647-9) (recommended). Note: Selected topics from the recommended text Will be introduced throughout the year.
Description:	This course begins by repeating some of the topics from MATH 3333 including properties of the real numbers and of continuous functions, but done at a deeper level. The concepts of metric spaces, countability, convergence of sequences and series of functions are all introduced. In MATH 4332, we will study functions of several variables, polynomial approximation and, time permitting, Fourier series..

MATH 4333: ABSTRACT ALGEBRA (Section 09799)
Time:	11:30-1:00 TTH, Room 345 PGH
Instructor:	J. Hausen
Prerequisites:	Math 3330 or equivalent.
Text(s):	Rings, Fidel's, and Groups, An Introduction to Abstract Algebra, By R.B.J.T. Allenby, Routledge, Chapman and Hall, Inc., New York, 1983.
Description:	Topics from Ring Theory, Field Theory and Group Theory including polynomial rings, quotient rings, field extensions and finite fields. Structure theorems from group theory as time permits.

MATH 4336: PARTIAL DIFFERENTIAL EQUATIONS II (Section 11795)
Time:	4:00-5:30 pm, MW, Room 345 PGH
Instructor:	B. Keyfitz
Prerequisites:	Ordinary Differential Equations (3331 or equivalent) and Math 4335 or consent of instructor
Text(s):	Partial Differential Equations: An Introduction, W.A. Strauss, Wiley, New York, latest edition (required), plus notes by instructor.
Description:	This is the second semester of a year-long first course in partial differential equations. The objective of the course is to explore the extensive connections between PDE and mathematical analysis, as well as the role of PDE in applied mathematics. The first semester focused on first-order equations and the method of characteristics, and on linear second-order equations in two independent variables: types of equations, d'Alembert's method and separation of variables. In the second semester, we will study energy methods and maximum principles for linear and nonlinear second-order equations; boundary-value problems; hyperbolic systems; Green's functions and Duhamel's principle; approximation methods; and applications in fluid dynamics, continuum mechanics, electromagnetics and ecology.

MATH 4351: DIFFERENTIAL GEOMETRY II (Section 09800)
Time:	12:00-1:00 MWF, Room 345 PGH
Instructor:	M. Ru
Prerequisites:	Math 4350.
Text(s):	Elementary Differential Geometry by Andrew Pressley, Springer-Verlag.
Description:	This is the second part of the year-long course, which introduces the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques. Topics in the second semester include: Gaussian curvature and the Gaussian map, geodesics, minimal surfaces, Gauss' Theorem Egrigium. Depending on the interests of the audience, some advanced topics will also be discussed.

MATH 4362: ORDINARY DIFFERENTIAL EQUATIONS (Section 11821)
Time:	1:00-2:30 pm, TTH, Room 203 AH
Instructor:	P. Walker
Prerequisites:	MATH 3331 and 3334.
Text(s):	An Introduction to Ordinary Differential Equations, by Earl A. Coddington, Dover Paperback, ISBN 0486659429.
Description:	Existence, uniqueness, and continuity of solutions of single equations and systems of equations; other topics at the discretion of the instructor.

MATH 4365: NUMERICAL ANALYSIS II (Section 09801)
Time:	5:30-7:00 pm, MW, Room. 350 PGH
Instructor:	T. Pan
Prerequisites:	Math 2331 (Linear Algebra), Math 3331 (Differential Equations). Ability to do computer assignments in either FORTRAN or C. The first semester is not a prerequisite.
Text(s):	Numerical Analysis (Seventh edition), R.L. Burden and J.D. Faires.
Description:	We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on the iterative methods for solving linear systems, approximation theory, numerical solutions of nonlinear equations, iterative methods for approximating eigenvalues, and elementary methods for ordinary differential equations with boundary conditions and partial differential equations. This is an introductory course and will be a mix of mathematics and computing.

MATH 4377: ADVANCED LINEAR ALGEBRA (Section 09802)
Time:	1:00-2:30 pm, TTH, Room 309 PGH
Instructor:	M. Friedberg
Prerequisites:	Math 2431 and minimum 3 hours of 3000 level math.
Text(s):	Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition, Prentice-Hall.
Description:	Fields, vector spaces, subspaces, linear transformations, matrices and applications to simultaneous equations.

MATH 4378: ADVANCED LINEAR ALGEBRA II (Section 09803)
Time:	2:30-4:00 pm, TTH, Room 309 PGH
Instructor:	J. Johnson
Prerequisites:	Math 4377
Text(s):	Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition, Prentice-Hall.
Description:	Topics to be covered in this course include linear equations, vector spaces, polynomials, linear transformations, and matrices.

MATH 4397: Selected Topics in Mathematics - NONLINEAR DYNAMICS II (Section 11854)
Time:	11:00-12:00 am, MWF, Room 134 SR.
Instructor:	K. Josic
Prerequisites:	MATH 4340
Text(s):	Nonlinear Dynamics and Chaos by Steven H. Strogatz, Perseus Books (2000).
Description:	Nonlinear terms enter into equations for many systems. The dynamics of such systems varies from relatively trivial, to extremely complex. Still many - often surprisingly general - principles can be used to describe nonlinear dynamics and chaos. This course will cover the theoretical foundations, numerical techniques, and some current applications of nonlinear dynamics.

MATH 6303: MODERN ALGEBRA II (Section 09858)
Time:	1:00-2:30 pm, TTH, Room 315 PGH
Instructor:	K. Kaiser
Prerequisites:	Math6302 or consent of instructor.
Text(s):	Algebra, Thomas W. Hungerford, Springer-Verlag (required). I will also circulate my own class notes.
Description:	During the first semester we will cover the basic theory of groups, rings and fields with strong emphasis on principal ideal domains. We will discuss the most important algebraic constructions from a universal algebraic as well as from a categorical point of view. The second semester will be mainly on modules over principal ideal domains, Sylow theory and field extensions.

MATH 6321: FUNCTIONS OF A REAL VARIABLE II (Section 09878)
Time:	2:30-4:00 pm, TTH, Room 202 AH
Instructor:	A. Torok
Prerequisites:	Math 6320 or consent of instructor.
Text(s):	We will use mainly Real Analysis, H.L. Royden, (3^rd Edition), Prentice Hall. Other possible text to be used are: Measure and Integration, M.E.Munroe; Lebesgue Integration on Euclidean Space, Frank Jones; Foundations of Modern Analysis, Avner Friedman; Real Analysis: Modern Techniques and their Applications, G.B. Folland.
Description:	This is the continuation of Math 6320. The topics to be discussed in Math 6320-6321 include: Lebesgue measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical L^p spaces, general measure theory, and elementary topics in functional analysis.

MATH 6325: DIFFERENTIAL EQUATIONS II (Section 11798)
Time:	10:00-11:30 am, TTh, Room 345 PGH
Instructor:	M. Golubitsky
Prerequisites:	A course in ordinary differential equations which includes some of the basic theory (such as MATH 6324) and an (undergraduate) course in linear algebra.
Text(s):	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. 42, Springer-Verlag.
Description:	This course will stress the local bifurcation theory of dynamical systems through codimension two, including Liapunov-Schmidt and center manifold reductions, normal form theory, steady-state bifurcation, Hopf bifurcation, Takens-Bogdanov bifurcations and other codimension two mode interactions. Some aspects of chaotic dynamics, including Melnikov's method and Smale horseshoes, will be presented. Emphasis will be on the mathematical ideas (rather than on formal proofs) and how to apply these ideas.

MATH 6361: APPLICABLE ANALYSIS II (Section 09881)
Time:	4:00-5:30 pm, MW, Room 309 PGH
Instructor:	G. Auchmuty
Prerequisites:	MATH 4332 [Math 6360 is not required.
Text(s):	Linear Operator Theory in Science and Engineering, Naylor and Sell, Springer-Verlag.
Description:	Topics to be covered will include: (i) An introduction to spaces. (ii) Theory of continuous linear operators on H. (iii) Solvability of linear operator equations; including linear integral equations. (iv) Spectral theorem for compact self-adjoint operators. (v) Introduction to boundary value problems for ordinary differential equations and Green's functions . The 1-d Fourier transform and its properties.

MATH 6371: NUMERICAL ANALYSIS II (Section 11818)
Time:	4:00-5:30 pm, TTH, Room 309 PGH
Instructor:	E. Dean
Prerequisites:	Graduate standing or consent of the instructor. This is the second semester of a two semester course. The first semester is not a prerequisite.
Text(s):	Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch (Springer-Verlag, (2nd or 3rd) ed.)
Description:	We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on iterative methods for numerical linear algebra, including large systems of linear equations and eigenvalue problems, numerical methods for ordinary differential equations, and a brief introduction to numerical methods for partial differential equations. This course will prepare students for other graduate level courses in numerical mathematics.

MATH 6374: NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS (Section 11796)
Time:	5:30-7:00 pm, MW, Room 301 AH
Instructor:	J. He
Prerequisites:	Numerical analysis and an undergraduate PDE course.
Text(s):	A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles, Cambridge University Press, 1996. ISBN: 0521556554
Description:	This course presents the methods and underlying theory used in the numerical solution of partial differential equations. It emphasizes finite difference methods and spectral methods for simple examples of elliptic, parabolic and hyperbolic equations; finite element and finite volume are discussed at a level that is understandable to beginning graduate students in engineering and in the sciences. In detail, topics covered include: 1) review of numerical solution of ordinary differential equations by multistep and Runge-Kutta methods; 2) finite difference and finite elements techniques for the elliptic equations: multivariate differences, Galerkin approximation, maximum principle, shape function, matrix structure; 3) algorithms to solve large sparse algebraic systems: sparse Gaussian elimination, classical and conjugate gradient iterations, multigrid iterations; 4) and methods for parabolic and hyperbolic differential equations and techniques of their analysis: explicit and implicit schemes, Lax-Richtmyer stability, von Neumann analysis, method of lines approach and relation to stiff ODEs, dissipation and dispersion error, operator splitting, convection-diffusion and reaction-diffusion equations. The point of departure is mathematical but the exposition strives to maintain a balance among theoretical, algorithmic and applied aspects of the subject.

MATH 6378: BASIC SCIENTIFIC COMPUTING (Section 11788)
Time:	5:30-7:00 pm, TTH, Room 350 PGH
Instructor:	R. Sanders
Prerequisites:	Elementary Numerical Analysis. Knowledge of C and/or Fortran. Graduate standing or consent of instructor.
Text(s):	High Performance Computing, O'Reilly, Kevin Dowd & Cbarles Severance, the 2nd edition.
Description:	Fundamental techniques in high performance scientific computation. Hardware architecture and floating point performance. Pointers and dynamic memory allocation. Data structures and storage techniques related to numerical algorithms. Parallel programming techniques. Code design. Applications to numerical algorithms for the solution of systems of equations, differential equations and optimization. Data visualization. This course also provides an introduction to computer programming issues and techniques related to large scale numerical computation.

MATH 6383: PROBABILITY MODELS AND MATHEMATICAL STATISTICS II(Section 11800)
Time:	2:30-4:00 pm, TTH, Room
Instructor:	E. Kao
Prerequisites:	Math 6382 or equivalent.
Text(s):	Statistical Inference, by George Casella and Roger Berger, Second Edition, Duxbury Press, 2002
Description:	This course is an introduction to statistical inference. In this course we will cover random samples, principles of data reduction, point estimation, hypothesis testing, interval estimation, asymptotic methods, analysis of variance and regression. Emphases will be placed on conceptual development of basic ideas. Whenever applicable, we will also address issues relating to computational statistics and topics involving financial econometrics.

MATH 6395: INTRODUCTION TO COMPLEX ANALYSIS AND GEOMETRY II (Section 11799)
Time:	11:30-1:00, TTH, Room 315 PGH
Instructor:	S. Ji
Prerequisites:	Graduate standing or consent of instructor.
Text(s):	none
Description:	We shall introduce geometric measure theory (currents, distributions, etc.), the L^2 vanishing theorem, and their applications in complex analysis and geometry.

MATH 6395: WAVELET ANALYSIS II (Section 11793)
Time:	4:00-5:30 pm, TTH, Room350 PGH
Instructor:	M. Papadakis
Prerequisites:	MATH 6320,� MATH 6395 (Wavelet analysis I ).
Text(s):	E. Hernandez and G. Weiss, A first course on wavelets, CRC Press, and P. Wojtaszyck's: A mathematical introduction to wavelets, Cambridge University press.
Description:	Orthogonality conditions, Characterizations of scaling functions, low pass filters and wavelets. The construction of Daubechies' compactly supported wavelets, transfer operators, time-domain fractal constructions of scaling functions. Translations, dilations and multiresolution in one and multidimensions (definitions). Wavelet and scaling sets. Biorthogonal multiresolution analysis, multiscaling functions and multiwavelets. Multiresolution analysis on intervals. Uncertainty principles, Weyl-Heisenberg frames, Frames with erasures in finite dimensional spaces. Non-separable multiresolution analysis. Wigner-Ville transforms, the ambiguity function and time-frequency atoms, Wavelet packets. Multiplexing and demultiplexing filter banks and applications of frame designs in cellular technology.

MATH 6395: PDES AND APPLICATIONS (Section 13313)
Time:	2:30-4:00 pm, MW, Room 106 AH
Instructor:	C. Suncica
Prerequisites:	Multivariable Calculus, Real and Complex Analysis.
Text(s):	Textbook: None required. (Texbooks that will be partially used are: Strauss's PDEs, R. LeVeques's "Conservation Laws", Renardy and Rogers' "PDEs", Research Papers)
Description:	Review of basic linear PDEs. Introduction to fundamentals of fluid mechanics (basic equations of motion: continuity, momentum, energy, vorticity). Incompressible/compressible flow examples. Analysis of quasilinear PDEs with the focus on hyperbolic conservation laws. Basic numerical methods. Special topics in modeling, analysis and numerical simulation arising in the study of blood flow through compliant blood vessels.

Math 6397: NUMBER THEORY (OnLine course) (Section 13423)
Time:	?, Rm. ?
Instructor:	M. Ru
Prerequisites:	None
Text(s):	Discovering Number Theory, by Jeffrey J. Holt and John W. Jones, W.H. Freeman and Company, New York, 2001.
Description:	Number theory is a subject that has interested people for thousand of years. This course is a one-semester long graduate course on number theory. Topics to be covered include divisibility and factorization, linear Diophantine equations, congruences, applications of congruences, solving linear congruences, primes of special forms, the Chinese Remainder Theorem, multiplicative orders, the Euler function, primitive roots, quadratic congruences, representation problems and continued fractions. There are no specific prerequisites beyond basic algebra and some ability in reading and writing mathematical proofs. The method of presentation in this course is quite different. Rather than simply presenting the material, students first work to discover many of the important concepts and theorems themselves. After reading a brief introductory material on a particular subject, students work through electronic materials that contain additional background, exercises, and Research Questions, using either mathematica, maple, or HTML with Java applets. The research questions are typically more open ended and require students to respond with a conjecture and proof. We then present the theory of the material which the students have worked on, along with the proofs. The homework problems contain both computational problems and questions requiring proofs. It is hoped that students, through this course, not only learn the material, learn how to write the proofs, but also gain valuable insight into some of the realities of mathematical research by developing the subject matter on their own.

Math 6397: STATISTICS (OnLine course) (Section 13424)
Time:	?, Rm. ?
Instructor:	C. Peters
Prerequisites:	Math 1432: Calculus II or consent of instructor.
Text(s):	Applied Statistics with Microsoft Exceel, by Gerald Keller, Duxbury 2001. ISBN: 0534382029
Description:	Fundamentals of probability and statistics. Descriptive and inferential methods of statistics.

MATH 7321: FUNCTIONAL ANALYSIS II (Section 11797)
Time:	12:00-1:00 pm, MWF, Room 128 SR
Instructor:	D. Blecher
Prerequisites:	Graduate standing.
Text(s):	There will be a xeroxed set of lecture notes available. There are several good books on the market, such as Pedersen's Analysis Now or Conway's Courses in Functional Analysis.
Description:	The first semester will be a leisurely and general presentation, starting from scratch, of the basic facts in Linear Analysis, Banach spaces and Hilbert space. The second semester will be a more technical development of the theory of linear operators on Hilbert space. We will also cover topics which the students request. We will prop ably only make it to the middle of section III in the first semester.

MATH 7394: SPLITTING METHODS IN COMPUTATIONAL MECHANICS AND PHYSICS (Section 11711)
Time:	2:30-4:00 pm, TTH, Room 314 PGH
Instructor:	R. Glowinski
Prerequisites:
Text(s):	None, since the course is self-contained. However, a preliminary reading of Neumann Control of Unstable Parabolic Systems: Numerical Approach , by R. Glowinski and J.W. He, published in the Journal of Optimization Theory and Control, 96, pp. 1-55, can help.
Description:

MATH 7394: REACTION DIFFUSION SYSTEMS II (Section 11820)
Time:	10:00-11:30 am, TTH, Room 350 PGH
Instructor:	W. Fitzgibbon
Prerequisites:	The Preceding course, Reaction Diffusion Equations or consent of instructor.
Text(s):	Notes.
Description:	This will be a seminar with a lecture format. The focus will be systems as opposed to scalar reaction diffused advection equations. Applications as well as theory will be discussed.

MATH 7396: MULTIGRID METHODS (Section 11710)
Time:	1:00-2:30 pm, MW, Room 345 PGH
Instructor:	R. Hoppe
Prerequisites:	Graduate Standing
Text(s):	1. J. K. Brainble, Multigrid Methods, Longman, Harlow, 1993. 2.Wolfgang Hackbusch; Iterative Solution of Large Sparse Systems of Equations. Appl. Math. Sciences, Vol. 95, Springer, New York, 1993 ISBN 0-387-94064-2.
Description:	Multigrid methods and related multilevel approaches are the most efficient solution techniques for the numerical solution of PDEs, integral equations, and other kind of operator equations. The course starts from an introduction to the basic principles and then proceeds to a detailed analysis of the convergence behavior of various multigrid schemes. Both linear and nonlinear problems will be addressed.

*NOTE: Teaching fellows are required to register for three regularly scheduled math courses for a total of 9 hours. Ph.D students who have passed their prelim exam are required to register for one regularly scheduled math course and 6 hours of dissertation.