Department of Mathematics

www.math.unc.edu

Phillips Hall, CB# 3250; (919) 962-1294

RICHARD MCLAUGHLIN, Chair

Introduction

Mathematics has always been a fundamental component of human thought and culture, and the growth of technology in recent times has further increased its importance. UNC–Chapel Hill offers several degrees in mathematics and the mathematical sciences, providing students a choice of careers in this field. Among the jobs in industry, government, and the academic world that involve mathematics are actuary, analyst, modeler, optimizer, statistician, and computer analyst.

Students majoring in mathematics may enter either the B.A. or the B.S. program. The B.A. program is more flexible than the B.S. program; it allows students to specialize in mathematics and at the same time either to follow a broad liberal arts program or to specialize in a second area (possibly even taking a second major). The B.S. program is more comprehensive; it provides solid preparation for work or for further study in mathematics and related fields. Within the B.S. program there is an applied option, which is designed for students who are primarily interested in using mathematics for the study of other sciences. The specific requirements for the B.A. and B.S. degrees are described below.

Students intending to teach mathematics in elementary and middle school and students enrolled in the School of Education who intend to major in mathematics should consult the School of Education section of the Bulletin or the director of mathematical education in the Department of Mathematics. A section below suggests course selections for future high school teachers.

Mathematics Major, B.A.

Core Requirements

Additional Requirements

All Foundations, Approaches, Connections, and Supplemental General Education requirements apply (see the "General Education Requirements" in this bulletin).

Mathematics Major, B.S.

Core Requirements

Additional Requirements

Mathematics Major, B.S.–Applied Option

Core Requirements

Additional Requirements

Students must complete either the B.S. or B.S.–Applied Option for a B.S. degree with a major in mathematics. All Foundations, Approaches, and Connections requirements of the General Education curriculum apply to students in both options.

Mathematics Minor

Core Requirements

Additional Requirements

Advising

All majors and minors have a primary academic advisor in Steele Building. Students should regularly meet with their advisors and review their Tar Heel Trackers to be sure that they are satisfying distribution and degree requirements. In addition, junior and senior majors are assigned each semester a mathematics faculty member as an advisor with whom they must meet to discuss course selections and any other questions before a hold on registration is lifted. The department's director of undergraduate studies and assistant (see contact information below) are also available by appointment. Further information on courses, undergraduate research opportunities, the honors program, careers, and graduate schools may be obtained from the department's Web site: www.math.unc.edu.

Placement into Mathematics Courses

Standardized test scores such as the Advanced Placement (AP), Scholastic Achievement Test Subject Levels I or II (SAT), or the American College Test (ACT) can be used for placement into mathematics courses. Please visit the placement page of the department's Web site for specific information regarding placement: math.unc.edu/for-undergrads/placement-info.

Special Opportunities in Mathematics

Special activities for qualified students include an undergraduate Mathematics Club, problem solving seminars, the Putnam Mathematical Competition, and the Virginia Tech Mathematics Contest. Students interested in these activities should consult the departmental honors advisor.

Qualified undergraduate students may work as research assistants in the Fluid Laboratory or as tutors in the Math Help Center. Students can conduct original research with the guidance of a faculty member, usually in conjunction with the preparation of an honors project. Study Abroad opportunities include semester or yearlong programs in a variety of countries. The Archibald Henderson Medal and the Alfred Brauer Prize recognize outstanding performance and promise in mathematics.

Undergraduate honors research projects as well as some internships or study abroad programs might qualify for experiential education credit. MATH 290, 396, and 410 satisfy this requirement.

Honors in Mathematics

Special honors (H) sections are given in some mathematics courses when student demand is sufficient (for example, 62H, 233H, 383H).

Promising students are encouraged to work toward a bachelor's degree with honors in mathematics. This program consists of six or more courses approved by the departmental honors advisor and satisfactory completion of an honors project completed over the two semesters of the senior year. The honors project is conducted in association with a departmental faculty member on a topic approved by the departmental honors advisor, and spans two semesters of independent research, during which time the honors candidate must be enrolled in 691H and 692H. The final report on the project includes both a written description and an oral presentation before a committee of three faculty (including the project advisor) approved by the departmental honors advisor. The committee will then report to the departmental honors advisor, who, in conjunction with a subcommittee of the undergraduate committee, will make the final recommendation on awarding a degree with honors or highest honors. The candidate must have a 3.5 grade point average in mathematics courses to begin an honors project and must maintain the 3.5 average through the completion of the senior year.

UNC–BEST

The UNC Baccalaureate Education in Science and Teaching (UNC–BEST) Program is a collaboration between the School of Education and the College of Arts and Sciences and is designed to allow undergraduate mathematics (and science) majors interested in teaching high school mathematics the opportunity to earn their degree and obtain licensure as a North Carolina high school mathematics teacher in four years. UNC–BEST students meet all the degree requirements for their mathematics major and complete a teaching methods class (MATH 410). Students also fulfill teaching licensure coursework requirements (see list below) as well as many General Education and elective requirements. For more details on admission requirements and application processes, visit the School of Education Web site: soe.unc.edu/services/apply/ug.

Graduate School and Career Opportunities

The B.S. degree program, especially if it includes the sequences MATH 521–522 and 577–578, is excellent preparation for graduate study in the mathematical sciences. The B.A. degree can be excellent preparation for graduate study in many fields, including admission into professional schools of law, business, and medicine. Both degrees are viewed by many employers as attractive, especially when accompanied by electives in areas such as statistics, computer science, economics, and operations research. Undergraduate mathematics majors with critical thinking skills and good analytical abilities are in demand in many business, industry, and government fields.

Following are suggested course selections (within the degree requirements) for students who have an interest in a particular direction.

B.A. or B.S. degree with a major in mathematics, suggestions for pure mathematics:

These courses provide a solid theoretical understanding of central mathematics and excellent preparation for graduate study in mathematics or the mathematical sciences.

Those planning graduate study in mathematics or the mathematical sciences may consider taking some of MATH 653, 676, 680, or subsequent courses.

B.S. degree with a major in mathematics, suggestions for mathematical biology:

For students interested in careers or further study in mathematical life sciences.

B.A. degree with a major in mathematics, suggestions for mathematical economics:

Suitable for students planning to go on to graduate school in economics or a related area, or pursue a career in economics, business, or finance. Note: With three more ECON courses numbered above 400, the requirements for the B.A. in economics could also be satisfied.

B.A. degree with a major in mathematics, suggestions for future high school teachers:

Faculty

Professors

Idris Assani, Prakash Belkale, Roberto Camassa, Ivan V. Cherednik, James N. Damon, M. Gregory Forest, Jane M. Hawkins, Jingfang Huang, Christopher Jones, Shrawan Kumar, Richard McLaughlin, Sorin Mitran, Peter J. Mucha, Karl E. Petersen, Robert Proctor, Richárd Rimányi, Lev Rozansky, Michael E. Taylor, Alexander N. Varchenko, Jonathan M. Wahl, Mark Williams.

Associate Professors

David Adalsteinsson, Jason Metcalfe, Laura Miller.

Assistant Professors

Hans Christianson, Ellen Eischen, Boyce Griffith, Jeremy Marzuola, Katherine Newhall, Nancy Rodruguez, Justin Sawon.

Lecturers

Debra Etheridge, Linda Green, Mark McCombs, Elizabeth McLaughlin, Brenda Shryock, Miranda Thomas.

Professors Emeriti

Joseph A. Cima, Patrick B. Eberlein, Ladnor D. Geissinger, Sue E. Goodman, Robert G. Heyneman, Norberto Kerzman, Ancel Mewborn, Joseph F. Plante, Michael Schlessinger, William W. Smith, Johann Sonner, James D. Stasheff, Warren R. Wogen.

Contact Information

Richárd Rimányi, Director of Undergraduate Studies, CB# 3250, 429 Chapman Hall, rimanyi@email.unc.edu.

Susan Stedman, Assistant to the Director of Undergraduate Studies, CB# 3250, 356 Phillips Hall, sstedman@email.unc.edu.

Courses

MATH–Mathematics

MATH 50 First-Year Seminar: The Predictability of Chance and Its Applications in Applied Mathematics (3). This seminar will examine the ways in which some types of behavior of random systems cannot only be predicted, but also applied to practical problems.

MATH 51 First-Year Seminar: "Fish Gotta Swim, Birds Gotta Fly": Mathematics and Mechanics of Moving Things (3). This seminar allows students to have hands-on exposure to a class of physical and computer experiments designed to challenge intuition on how motion is achieved in nature.

MATH 52 First-Year Seminar: Fractals: The Geometry of Nature (3). Many natural objects have complex, infinitely detailed shapes in which smaller versions of the whole shape are seen appearing throughout. Such a shape is a fractal, the topic of study.

MATH 53 First-Year Seminar: Symmetry and Tilings (3). Through projects using software programs, Web sites, and readings, students will discover the geometric structure of tilings, learn to design their own patterns, and explore the many interdisciplinary connections.

MATH 54 First-Year Seminar: The Science of Conjecture: Its Math, Philosophy, and History (3). Seminar will cover the history and philosophy of probability, evidence, and conjecture, consider the development of the field of probability, and look at current and future uses of probability.

MATH 55 First-Year Seminar: Geometry and Symmetry in Nature (3). The nature of space imposes striking constraints on organic and inorganic objects. This seminar examines such constraints on both biological organisms and regular solids in geometry.

MATH 56 First-Year Seminar: Information and Coding (3). With the growth of available information on almost anything, can it be reliably compressed, protected, and transmitted over a noisy channel? Students will take a mathematical view of cryptography throughout history and information handling in modern life.

MATH 57 First-Year Seminar: The Fourth Dimension (3). The idea of a fourth dimension has a rich and varied history. This seminar explores the concept of fourth (and higher) dimensions both mathematically and more widely in human thought.

MATH 58 First-Year Seminar: Math, Art, and the Human Experience (3). Students will explore the relevance of mathematical ideas to fields typically perceived as "nonmathematical" (e.g., art, music, film, literature) and how these "nonmathematical" fields influence mathematical thought.

MATH 59 First-Year Seminar: The Mystery and Majesty of Ordinary Numbers (3). Problems arising from the arithmetic of ordinary counting numbers have for centuries fascinated both mathematicians and nonmathematicians. This seminar will consider some of these problems (both solved and unsolved).

MATH 60 First-Year Seminar: Simulated Life (3). This seminar introduces students to the thought process that goes into developing computational models of biological systems. It will also expose students to techniques for simulating and analyzing these models.

MATH 61 First-Year Seminar: The Language of Mathematics: Making the Invisible Visible (3). This course will consider mathematics to be the science of patterns and will discuss some of the different kinds of patterns that give rise to different branches of mathematics.

MATH 62 First-Year Seminar: Combinatorics (3). Students will discuss combinatorics' deep roots in history, its connections with the theory of numbers, and its fundamental role for natural science, as well as various applications, including cryptography and the stock market.

MATH 63 First-Year Seminar: From "The Sound of Music" to "The Perfect Storm" (MASC 57) (3). See MASC 57 for description.

MATH 64 First-Year Seminar: A View of the Sea: The Circulation of the Ocean and Its Impact on Coastal Water (3). Why is the Gulf Stream so strong, why does it flow clockwise, and why does it separate from the United States coast at Cape Hatteras? Students will study the circulation of the ocean and its influence on coastal environments through readings and by examining satellite and on-site observations.

MATH 65 First-Year Seminar: Colliding Balls and Springs: The Microstructure of How Materials Behave (3). Students will follow the intellectual journey of the atomic hypothesis from Leucippus and Democritus to the modern era, combining the history, the applications to science, and the mathematics developed to study particles and their interactions.

MATH 66 First-Year Seminar: Non-Euclidean Geometry in Nature and History (3). The seminar will investigate non-Euclidean geometry (hyperbolic and spherical) from historical, mathematical, and practical perspectives. The approach will be largely algebraic, in contrast to the traditional axiomatic method.

MATH 67 First-Year Seminar: The Mathematics of Climate Change: Can We Predict the Future of Our Planet? (3). Is the earth warming? Predictions are based largely on mathematical models. We shall consider the limitations of models in relation to making predictions. Examples of chaotic behavior will be presented.

MATH 89 First-Year Seminar: Special Topics (3). Special topics course. Content will vary each semester.

MATH 110 Algebra (3). Placement by achievement test. Provides a one-semester review of the basics of algebra. Basic algebraic expressions, functions, exponents, and logarithms are included, with an emphasis on problem solving. This course should not be taken by those with a suitable score on the achievement test.

MATH 116 Intuitive Calculus (3). Provides an introduction in as nontechnical a setting as possible to the basic concepts of calculus. The course is intended for the nonscience major. A student may not receive credit for this course after receiving credit for MATH 152, 231, or 241.

MATH 117 Aspects of Finite Mathematics (3). Introduction to basic concepts of finite mathematics, including topics such as counting methods, finite probability problems, and networks. The course is intended for the nonscience major.

MATH 118 Aspects of Modern Mathematics (3). Introduction to mathematical topics of current interest in society and science, such as the mathematics of choice, growth, finance, and shape. The course is intended for the nonscience major.

MATH 119 Introduction to Mathematical Modeling (3). Provides an introduction to the use of mathematics for modeling real-world phenomena in a nontechnical setting. Models use algebraic, graphical, and numerical properties of elementary functions to interpret data. This course is intended for the nonscience major.

MATH 129P Precalculus Mathematics (0). Awarded as placement credit based on test scores. Does not fulfill a graduation requirement.

MATH 130 Precalculus Mathematics (3). Prerequisite, MATH 110. Covers the basic mathematical skills needed for learning calculus. Topics are calculating and working with functions and data, introduction to trigonometry, parametric equations, and the conic sections. A student may not receive credit for this course after receiving credit for MATH 231.

MATH 152 Calculus for Business and Social Sciences (3). Prerequisite, MATH 110. An introductory survey of differential and integral calculus with emphasis on techniques and applications of interest for business and the social sciences. This is a terminal course and not adequate preparation for MATH 232. A student cannot receive credit for this course after receiving credit for MATH 231 or 241.

MATH 190 Special Topics in Mathematics (3). An undergraduate seminar course that is designed to be a participatory intellectual adventure on an advanced, emergent, and stimulating topic within a selected discipline in mathematics. This course does not count as credit towards the mathematics major.

MATH 231 Calculus of Functions of One Variable I (3). Prerequisites, MATH 110 and 130. Requires a grade of C- or better in MATH 130 or placement by the department. Limits, derivatives, and integrals of functions of one variable. Students may not receive credit for both MATH 231 and MATH 241.

MATH 232 Calculus of Functions of One Variable II (3). Requires a grade of C- or better in MATH 231 or 241 or placement by the department. Calculus of the elementary transcendental functions, techniques of integration, indeterminate forms, Taylor's formula, infinite series.

MATH 233 Calculus of Functions of Several Variables (3). Prerequisite, MATH 232 or 283. Vector algebra, solid analytic geometry, partial derivatives, multiple integrals.

MATH 241 BioCalculus I (3). Requires a grade of C- or better in MATH 130 or placement by the department. Limits, derivatives, and integrals of functions of one variable, motivated by and applied to discrete-time dynamical systems used to model various biological processes.

MATH 283 BioCalculus II (3). Requires a grade of C- or better in either MATH 231 or 241, or placement by the department. Techniques of integration, indeterminate forms, Taylor's series; introduction to linear algebra, motivated by and applied to ordinary differential equations; systems of ordinary differential equations used to model various biological processes. A student cannot receive credit for this course after receiving credit for MATH 383.

MATH 290 Directed Exploration in Mathematics (0–3). Permission of the director of undergraduate studies. Experimentation or deeper investigation under the supervision of a faculty member of topics in mathematics that may be, but need not be, connected with an existing course. No one may receive more than seven semester hours of credit for this course.

MATH 294 Undergraduate Seminar in Mathematics (0–3). Permission of the instructor. A seminar on a chosen topic in mathematics in which the students participate more actively than in usual courses.

MATH 307 Revisiting Real Numbers and Algebra (3). Central to teaching precollege mathematics is the need for an in-depth understanding of real numbers and algebra. This course explores this content, emphasizing problem solving and mathematical reasoning.

MATH 381 Discrete Mathematics (3). Prerequisite, MATH 232 or 283. This course serves as a transition from computational to more theoretical mathematics. Topics are from the foundations of mathematics: logic, set theory, relations and functions, induction, permutations and combinations, recurrence.

MATH 383 First Course in Differential Equations (3). Prerequisite, MATH 233. Introductory ordinary differential equations, first- and second-order differential equations with applications, higher-order linear equations, systems of first-order linear equations (introducing linear algebra as needed).

MATH 396 Undergraduate Reading and Research in Mathematics (1–3). Permission of the director of undergraduate studies. This course is intended mainly for students working on honors projects. No one may receive more than three semester hours credit for this course.

MATH 406 Mathematical Methods in Biostatistics (1). Prerequisite, MATH 232. Special mathematical techniques in the theory and methods of biostatistics as related to the life sciences and public health. Includes brief review of calculus, selected topics from intermediate calculus, and introductory matrix theory for applications in biostatistics.

MATH 410 Teaching and Learning Mathematics (4). Study of how people learn and understand mathematics, based on research in mathematics, mathematics education, psychology, and cognitive science. This course is designed to prepare undergraduate mathematics majors to become excellent high school mathematics teachers. It involves fieldwork in both the high school and college environments.

MATH 411 Developing Mathematical Concepts (3). Permission of the instructor. An investigation of various ways elementary concepts in mathematics can be developed. Applications of the mathematics developed will be considered.

MATH 418 Basic Concepts of Analysis for High School Teachers (3). Prerequisites, MATH 233 and 381. An examination of high school mathematics from an advanced perspective, including number systems and the behavior of functions and equations. Designed primarily for prospective or practicing high school teachers.

MATH 452 Mathematical and Computational Models in Biology (BIOL 452) (3). See BIOL 452 for description.

MATH 452L Mathematical and Computational Models in Biology Laboratory (BIOL 452L) (1). See BIOL 452L for description.

MATH 515 History of Mathematics (3). Prerequisite, MATH 381. A general survey of the history of mathematics with emphasis on elementary mathematics. Some special problems will be treated in depth.

MATH 521 Advanced Calculus I (3). Prerequisites, MATH 233 and 381. The real numbers, continuity and differentiability of functions of one variable, infinite series, integration.

MATH 522 Advanced Calculus II (3). Prerequisites, MATH 383 and 521. Functions of several variables, the derivative as a linear transformation, inverse and implicit function theorems, multiple integration.

MATH 523 Functions of a Complex Variable with Applications (3). Prerequisite, MATH 383. The algebra of complex numbers, elementary functions and their mapping properties, complex limits, power series, analytic functions, contour integrals, Cauchy's theorem and formulae, Laurent series and residue calculus, elementary conformal mapping and boundary value problems, Poisson integral formula for the disk and the half plane.

MATH 524 Elementary Differential Equations (3). Prerequisite, MATH 383. Linear differential equations, power series solutions, Laplace transforms, numerical methods.

MATH 528 Mathematical Methods for the Physical Sciences I (3). Prerequisite, MATH 383. Theory and applications of Laplace transform, Fourier series and transform, Sturm-Liouville problems. Students will be expected to do some numerical calculations on either a programmable calculator or a computer. This course has an optional computer laboratory component: MATH 528L.

MATH 528L Laboratory for Mathematical Methods for the Physical Sciences I (1). Prerequisite, MATH 383; pre- or corequisite, MATH 528. Training in the use of symbolic and numerical computing packages and their application to the MATH 528 lecture topics. Students will need a CCI-compatible computing device.

MATH 529 Mathematical Methods for the Physical Sciences II (3). Prerequisite, MATH 521, 524, or 528. Introduction to boundary value problems for the diffusion, Laplace and wave partial differential equations. Bessel functions and Legendre functions. Introduction to complex variables including the calculus of residues. This course has an optional computer laboratory component: MATH 529L.

MATH 529L Laboratory for Mathematical Methods for the Physical Sciences II (1). Prerequisite, MATH 383; pre- or corequisite, MATH 529. Training in the use of symbolic and numerical computing packages and their application to the MATH 529 lecture topics. Students will need a CCI-compatible computing device.

MATH 533 Elementary Theory of Numbers (3). Prerequisite, MATH 381. Divisibility, Euclidean algorithm, congruences, residue classes, Euler's function, primitive roots, Chinese remainder theorem, quadratic residues, number-theoretic functions, Farey and continued fractions, Gaussian integers.

MATH 534 Elements of Modern Algebra (3). Prerequisite, MATH 381. Binary operations, groups, subgroups, cosets, quotient groups, rings, polynomials.

MATH 535 Introduction to Probability (STOR 435) (3). See STOR 435 for description.

MATH 547 Linear Algebra for Applications (3). Prerequisite, MATH 233 or 283. Algebra of matrices with applications: determinants, solution of linear systems by Gaussian elimination, Gram-Schmidt procedure, eigenvalues. MATH 416 may not be taken for credit after credit has been granted for MATH 547.

MATH 548 Combinatorial Mathematics (3). Prerequisite, MATH 381. Topics chosen from generating functions, Polya's theory of counting, partial orderings and incidence algebras, principle of inclusion-exclusion, Moebius inversion, combinatorial problems in physics and other branches of science.

MATH 550 Topology (3). Prerequisites, MATH 233 and 381; corequisite, MATH 383. Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tessellations, and fundamental group.

MATH 551 Euclidean and Non-Euclidean Geometries (3). Prerequisite, MATH 381. Critical study of basic notions and models of Euclidean and non-Euclidean geometries: order, congruence, and distance.

MATH 555 Introduction to Dynamics (3). Prerequisite, MATH 383. Topics will vary and may include iteration of maps, orbits, periodic points, attractors, symbolic dynamics, bifurcations, fractal sets, chaotic systems, systems arising from differential equations, iterated function systems, and applications.

MATH 564 Mathematical Modeling (3). Prerequisite, MATH 283 or 383. Requires some knowledge of computer programming. Model validation and numerical simulations using differential equations, probability, and iterated maps. Applications may include conservation laws, dynamics, mixing, geophysical flows and climate change, fluid motion, epidemics, ecological models, population biology, cell biology, and neuron dynamics.

MATH 565 Computer-Assisted Mathematical Problem Solving (3). Prerequisite, MATH 383. Personal computer as tool in solving a variety of mathematical problems, e.g., finding roots of equations and approximate solutions to differential equations. Introduction to appropriate programming language; emphasis on graphics.

MATH 566 Introduction to Numerical Analysis (3). Prerequisite, MATH 383. Requires some knowledge of computer programming. Iterative methods, interpolation, polynomial and spline approximations, numerical differentiation and integration, numerical solution of ordinary and partial differential equations.

MATH 577 Linear Algebra (3). Prerequisites, MATH 381 and 383. Vector spaces, linear transformations, duality, diagonalization, primary and cyclic decomposition, Jordan canonical form, inner product spaces, orthogonal reduction of symmetric matrices, spectral theorem, bilinear forms, multilinear functions. A much more abstract course than MATH 416 or 547.

MATH 578 Algebraic Structures (3). Prerequisite, MATH 547 or 577. Permutation groups, matrix groups, groups of linear transformations, symmetry groups; finite abelian groups. Residue class rings, algebra of matrices, linear maps, and polynomials. Real and complex numbers, rational functions, quadratic fields, finite fields.

MATH 590 Topics in Mathematics (3). Permission of the instructor. Topics may focus on matrix theory, analysis, algebra, geometry, or applied and computational mathematics.

MATH 594 Nonlinear Dynamics (PHYS 594) (3). See PHYS 594 for description.

MATH 635 Probability (STOR 635) (3). See STOR 635 for description.

MATH 641 Enumerative Combinatorics (3). Prerequisite, MATH 578. Basic counting; partitions; recursions and generating functions; signed enumeration; counting with respect to symmetry, plane partitions, and tableaux.

MATH 643 Combinatorial Structures (3). Prerequisite, MATH 578. Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Moebius inversion, q-analogs, combinatorial and projective geometries, codes, and designs.

MATH 653 Introductory Analysis (3). Requires knowledge of advanced calculus. Elementary metric space topology, continuous functions, differentiation of vector-valued functions, implicit and inverse function theorems. Topics from Weierstrass theorem, existence and uniqueness theorems for differential equations, series of functions.

MATH 656 Complex Analysis (3). Prerequisite, MATH 653. A rigorous treatment of complex integration, including the Cauchy theory. Elementary special functions, power series, local behavior of analytic functions.

MATH 657 Qualitative Theory of Differential Equations (3). Prerequisite, MATH 653. Requires knowledge of linear algebra. Existence and uniqueness theorems, linear and nonlinear systems, differential equations in the plane and on surfaces, Poincare-Bendixson theory, Lyapunov stability and structural stability, critical point analysis.

MATH 661 Scientific Computation I (ENVR 661) (3). Requires some programming experience and basic numerical analysis. Error in computation, solutions of nonlinear equations, interpolation, approximation of functions, Fourier methods, numerical integration and differentiation, introduction to numerical solution of ODEs, Gaussian elimination.

MATH 662 Scientific Computation II (COMP 662, ENVR 662) (3). Prerequisite, MATH 661. Theory and practical issues arising in linear algebra problems derived from physical applications, e.g., discretization of ODEs and PDEs. Linear systems, linear least squares, eigenvalue problems, singular value decomposition.

MATH 668 Methods of Applied Mathematics I (ENVR 668) (3). Requires an undergraduate course in differential equations. Contour integration, asymptotic expansions, steepest descent/stationary phase methods, special functions arising in physical applications, elliptic and theta functions, elementary bifurcation theory.

MATH 669 Methods of Applied Mathematics II (ENVR 669) (3). Prerequisite, MATH 668. Perturbation methods for ODEs and PDEs, WKBJ method, averaging and modulation theory for linear and nonlinear wave equations, long-time asymptotics of Fourier integral representations of PDEs, Green's functions, dynamical systems tools.

MATH 676 Modules, Linear Algebra, and Groups (3). Requires knowledge of linear algebra and algebraic structures. Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, groups and group actions.

MATH 677 Groups, Representations, and Fields (3). Prerequisite, MATH 676. Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory.

MATH 680 Geometry of Curves and Surfaces (3). Requires advanced calculus. Topics include (curves) Frenet formulas, isoperimetric inequality, theorems of Crofton, Fenchel, Fary-Milnor; (surfaces) fundamental forms, Gaussian and mean curvature, special surfaces, geodesics, Gauss-Bonnet theorem.

MATH 681 Introductory Topology (3). Prerequisites, MATH 653 and 680. Topological spaces, connectedness, separation axioms, product spaces, extension theorems. Classification of surfaces, fundamental group, covering spaces.

MATH 690 Topics in Mathematics (3). Permission of the department. Directed study of an advanced topic in mathematics. Topics will vary.

MATH 691H Honors Research in Mathematics (3). Permission of the director of undergraduate studies. Readings in mathematics and the beginning of directed research on an honors thesis.

MATH 692H Honors Thesis in Mathematics (3). Permission of the director of undergraduate studies. Completion of an honors thesis under the direction of a member of the faculty. Required of all candidates for graduation with honors in mathematics.