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Spring 2002 Course DescriptionsMATH 103: Mathematical Explorations (TOPICS)
This course is for students who wish to experience how mathematical ideas naturally evolve. The homework consists of the students actively investigating mathematical ideas. The course emphasizes ideas and imagination as opposed to techniques and calculations. Topics vary depending on the instructor and are described below. Some assessment is done through writing assignments. Lecture 01: Iteration and Patterns. This section of Math 103 will use computers regularly, with easy-to-use interactive color graphics tools. Assignments will involve computer work and writing as well as straight-forward problems relating to the concepts. The class meets in a computer lab one day each week, and elsewhere for discussion on the other days. The computer work is done with partners, both in class and for homework. Writing and computation assignments are individual.The course consists of three parts, which makes a good sequence — you'll be amazed at how far we go! We assume nothing, and your skills will grow as we go. Some of the roads we will take depend on what you bring to the class, but the overall schedule is:
These all have various connections to real world natural phenomena, which we will emphasize. We spend 4 weeks on each of the three parts, and 2 weeks on a final project instead of a final exam. Lectures 02 & 03: Flatland and Flatterland. MATH 106: Calculus For The Life and Social Sciences
Course serves as an introduction to differential and integral calculus, partial derivatives, elementary differential equations. Examples from biology and the social sciences are used. MATH 111: Calculus
Course topics include: functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve. Graphing calculators are used, and their pitfalls are discussed, as applicable to the above topics. MATH 111 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 112 or 122. MATH 112: Calculus
Course focus is on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations (separation of variables, initial conditions, systems, some applications). Also covered are infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations. MATH 122: Honors Calculus
Topics covered include: differentiation and integration of elementary transcendental functions, techniques of integration, applications, polar coordinates, infinite series, and complex numbers, as well as an introduction to proving theorems. The approach is more theoretical than in MATH 112. MATH 171: Statistical Theory and Application In The Real World
This introductory statistics course discusses techniques for analyzing data occurring in the real world and the mathematical and philosophical justification for these techniques. Topics include: population and sample distributions, central limit theorem, statistical theories of point estimation, confidence intervals, testing hypotheses, the linear model, and the least squares estimator. The course concludes with a discussion of tests and estimates for regression and analysis of variance (if time permits). The computer is used to demonstrate some aspects of the theory, such as sampling distributions and the Central Limit Theorem. In the lab portion of the course, students learn and use computer-based methods for implementing the statistical methodology presented in the lectures. (No previous familiarity with computers is presumed.) MATH 191: Calculus For Engineers
Course topics include: plane analytic geometry, differential and integral calculus, and applications. MATH 191 covers essentially the same topics as 190, but is designed for students with some previous successful experience with calculus. MATH 192: Calculus For Engineers
Course topics include: polar coordinates, infinite series, and power series. Also covered are: vectors and calculus of functions of several variables through double and triple integrals. MATH 213: Calculus
Course topics include: vectors and vector-valued functions; multivariable and vector calculus including multiple and line integrals; first- and second-order differential equations with applications; systems of differential equations; and elementary partial differential equations. This course is designed for students who wish to master the basic techniques of calculus, but whose major will not require a substantial amount of mathematics. The course may emphasize different topics in the syllabus in different semesters. MATH 221: Linear Algebra and Calculus
Course covers linear algebra and differential equations. Topics include: vector algebra, linear transformations, matrices, and linear differential equations, as well as an introduction to proving theorems. This course is especially recommended for students who plan to major in mathematics or in a strongly related field. MATH 222: Calculus
Course topics include: multivariable and vector differential and integral calculus, including multiple, line, and surface integrals. This course is especially recommended for students who plan to major in mathematics or in a strongly related field. MATH 224: Honors Linear Algebra and Calculus
Course topics include: vector fields; line integrals; differential forms and exterior derivative; work, flux, and density forms; integration of forms over parametrized domains; and Green's, Stoke's, and divergence theorems. MATH 231: Linear Algebra
Course topics include: vectors, matrices, and linear transformations, affine and Euclidean spaces, transformation of matrices, and eigenvalues. MATH 293: Engineering Mathematics
Course topics include: vector fields and vector calculus; complex numbers; introduction to ordinary and partial differential equations; and Fourier series and boundary value problems. May include computer use in problem solving. MATH 294: Engineering Mathematics
Course topics include: matrix theory and linear algebra, inner product spaces; and systems of linear ordinary differential equations. May include computer use in solving problems. MATH 311: Introduction to Analysis (NEW)
Provides a transition from calculus to real analysis. Topics include: rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis will be placed upon understanding and constructing mathematical proofs. MATH 336: Applicable Algebra
An introduction to the concepts and methods of abstract algebra and number theory that are of interest in applications. Covers: basic theory of groups, rings and fields and their applications to such areas as public-key cryptography, error-correcting codes, parallel computing, and experimental designs. Also covers: elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes and Latin squares. MATH 356: Groups and Geometry
Groups were introduced in the nineteenth century as the set of symmetries of an algebraic or geometric object, and this viewpoint is a central one in modern Mathematics. This course studies the geometry of the planes and of patterns in the plane in terms of the group of symmetries ("isometries") of the plane. Prior knowledge of groups is not a prerequisite. One aim is to give students experience in modern algebra and geometry (including the geometry of complex numbers) and a sense of the unity of mathematics before they take the 400-level courses Special care is given to initiate the student into the writing of proofs and the language of mathematics. Topics include: symmetries; groups of transformations; subgroups and cosets; homomorphisms and isomorphisms; orbits and fixed points; Frieze groups, wallpaper groups ("2-dimensional crystallographic groups"); and the associated tesselations of the Euclidean plane. MATH 362: Dynamic Models in Biology (also BIOEE 362) (NEW)
Introductory survey of the development, computer implementation and applications of dynamic models in biology and ecology. Case-study format, covering a broad range of current application areas such as regulatory networks, neurobiology, cardiology, infectious disease management, and conservation of endangered species. Students will also learn how to construct and study biological systems models on the computer using a scripting and graphics environment. MATH 401: Honors Seminar: Topics In Modern Mathematics
This course is a participatory seminar primarily aimed at introducing senior and junior mathematics majors to some of the challenging problems and areas of modern mathematics. The seminar helps students develop research and expository skills in mathematics, which is important for careers in any field that makes significant use of the mathematical sciences (i.e., pure or applied mathematics, physical or biological sciences, business and industry, medicine). The content varies from year to year. MATH 403: History of Mathematics
Survey of the development of mathematics from antiquity to the present, with an emphasis on the achievements, problems, and mathematical viewpoints of each historical period and the evolution of such basic concepts as number, geometry, construction, and proof. Readings from original sources in translation. Students are required to give oral and written reports. MATH 414: Honors Introduction to Analysis II
The sequence MATH 413-414, designed for honors students, provides an introduction to the theory of functions of real variables, stressing a rigorous logical development of the subject rather than applications. Topics include: metric spaces, the real number system, continuous and differentiable functions, uniform convergence and approximation theorems, Fourier series, Riemann and Lebesgue integrals, calculus in several variables, and differential forms. MATH 418: Introduction To The Theory of Functions of One Complex Variable (REVISED)
A theoretical and rigorous introduction to complex variable theory. Topics include: complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping. Students interested in the applications of complex analysis should consider MATH 422. MATH 420: Differential Equations and Dynamical Systems
Course covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. MATH 422: Applied Complex Analysis
Course covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. MATH 424: Wavelets and Fourier Series (NEW)
Both Fourier series and wavelets provide methods to represent or approximate general functions in terms of simple building blocks. Such representations have important consequences, both for pure mathematics and for applications. Fourier series use natural sinusoidal building blocks and may be used to help solve differential equations. Wavelets use artificial building blocks that have the advantage of localization in space. A full understanding of both topics requires a background involving Lebesgue integration theory and functional analysis. This course presents as much as possible on both topics without such formidable prerequisites. The emphasis is on clear statements of results and key ideas of proofs, working out examples, and applications. Related topics that may be included in the course: Fourier transforms, Heisenberg uncertainty principle, Shannon sampling theorem, and Poisson summation formula. MATH 427: Introduction to Ordinary Differential Equations (SWITCHED SEMESTERS)
Covers the basic existence, uniqueness, and stability theory together with methods of solution and methods of approximation. Topics include: singular points, series solutions, Sturm-Liouville theory, transform methods, approximation methods, and application to physical problems. MATH 432: Introduction To Algebra II
An introduction to various topics in abstract algebra, including: groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finitely generated modules over Euclidean domains with application to canonical forms of matrices. MATH 434: Honors Introduction To Algebra II
Math 433-434 is the honors version of MATH 431-432. MATH 433-434 is more theoretical and rigorous than 431-432 and includes additional material such as multilinear and exterior algebra. MATH 452: Classical Geometries
This is an introduction to hyperbolic, spherical, and projective geometry--the classical geometries that developed as Euclidean geometry was better understood. For example, the historical problem of the independence of Euclid's fifth postulate is understood when the existence of the hyperbolic plane is realized. Straightedge (and compass) constructions and stereographic projection in Euclidean geometry can be understood within the structure of projective geometry. Topics in hyperbolic geometry include: models of the hyperbolic plane and relations to spherical geometry. Topics in projective geometry include: homogeneous coordinates and the classical theorems about conics and configurations of points and lines. Optional topics include: principles of perspective drawing, finite projective planes, orthogonal Latin squares, and the cross ratio. MATH 454: Introduction To Differential Geometry
Course covers differential geometry of curves and surfaces. Also covers curvature, geodesics, and differential forms. Serves as an introduction to n-dimensional Riemannian manifolds. This material provides some background for the study of general relativity; connections with the latter are indicated. MATH 472: Statistics
Classical and recently developed statistical procedures are discussed in a framework that emphasizes the basic principles of statistical inference and the rationale underlying the choice of these procedures in various settings. These settings include problems of estimation, hypothesis testing, and large sample theory. MATH 483: Intensional Logic (also Phil 436)
For description, see PHIL 436. MATH 486: Applied Logic (also Com S 486)
Course covers: propositional and predicate logic; compactness and completeness by tableaux, natural deduction, and resolution. Other possible topics include: equational logic; Herbrand Universes and unification; rewrite rules and equational logic, Knuth-Bendix method and the congruence-closure algorithm and lambda-calculus reduction strategies; topics in Prolog, LISP, ML, or Nuprl; and applications to expert systems and program verification. MATH 490: Supervised Reading and Research
Supervised reading and research by arrangement with individual professors. Not for material currently available in regularly scheduled courses. Graduate CoursesMany of our graduate courses are topics courses for which descriptions are not included here; however, a schedule of graduate courses to be offered in Spring 2002 will be linked from Courses before October 15, 2001. This schedule includes course descriptions that are often more detailed than those included here, as well as a means for interested students to participate in the process of selecting meeting times. MATH 505: Educational Issues In Undergraduate Mathematics
This course examines various educational issues in undergraduate mathematics and the relationship of these issues to the mathematics itself. The precise choice of topics varies, but the intent is that a balance of different views be presented and discussed. There are extensive readings in the course and occasional guest lectures. Possible topics include: nature of proof and how and when to teach it, calculus "reform," teaching mathematics to school teachers, using writing, using history, alternative assessments, alternatives to lecturing, equity issues, effective uses of technology, what is mathematical understanding and how do we recognize it, what should every mathematics major know, and research in undergraduate mathematics. MATH 507: Teaching Secondary Mathematics: Theory and Practices
This course provides direct experience of new approaches, curricula and standards in mathematics education. Discussion of articles, activities for the secondary classroom and videotape of classroom teaching is tied to in-class exploration of math problems. Experience in the computer lab, examining software environments and their use in the mathematics classroom is included. Participants are expected to write short papers, share ideas in class and present their opinions on issues. MATH 508: Mathematics For Secondary School Teachers
An examination of the principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics and current mathematics research. MATH 612: Complex Analysis
Complex analysis, Fourier analysis, and distribution theory. MATH 620: Partial Differential Equations
Course covers basic theory of partial differential equations. MATH 634: Algebra
Dedekind domains, primary decomposition, Hilbert basis theorem, and local rings. MATH 650: Lie Groups
Course topics include: topological groups, Lie groups; relation between Lie groups and Lie algebras; exponential map, homogeneous manifolds; and invariant differential operators. MATH 651: Introductory Algebraic Topology
Course covers fundamental group and covering spaces, and homology theories for complexes and spaces. MATH 662: Riemannian Geometry
Course topics include: linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form. MATH 672: Probability Theory II
Topics for MATH 671-672 include: properties and examples of probability spaces; sample space, random variables, and distribution functions; expectation and moments; independence, Borel-Cantelli lemma, zero-one law; convergence of random variables, probability measures, and characteristic functions; law of large numbers; selected limit theorems for sums of independent random variables; Markov chains, recurrent events; ergodic and renewal theorems; Martingale theory; and Brownian motion and processes with independent increments. MATH 674: Introduction To Mathematical Statistics
Topics include: an introduction to the theory of point estimation, hypothesis testing and confidence intervals, consistency, efficiency, sufficiency, and the method of maximum likelihood. Basic concepts of decision theory are discussed; asymptotic methods are introduced and developed in detail. The course is coordinated with OR&IE 670 to form the second part of a one-year course in mathematical statistics. MATH 681: Logic
Course covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems. Other topics as time permits. MATH 713: Functional Analysis
Course covers: topological vector spaces, Banach and Hilbert spaces, and Banach algebras. Additional topics selected by instructor. MATH 717: Applied Dynamical Systems (also T&AM 776)
Course topics include: review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov's method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics. MATH 732: Seminar In Algebra
MATH 735: Topics In Algebra
Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Course content varies. MATH 739: Topics In Algebra
Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Course content varies. MATH 752: Seminar In Topology
MATH 758: Topics In Topology
Selection of advanced topics from modern algebraic, differential, and geometric topology. Course content varies. MATH 762: Seminar In Geometry
MATH 767: Algebraic Geometry
MATH 772: Seminar In Probability and Statistics
MATH 778: Stochastic Processes
MATH 782: Seminar In Logic
MATH 783: Model Theory
An introduction to model theory at the level of the books by Hodges or Chang and Keisler. MATH 790: Supervised Reading and Research
Last modified: April 7, 2003 |
