Department of Mathematics

www.math.unc.edu

RICHARD MCLAUGHLIN, Chair

Professors

Idris Assani, Prakash Belkale, Roberto Camassa, Ivan V. Cherednik, Joseph A. Cima, James N. Damon, Patrick B. Eberlein, M. Gregory Forest, Sue E. Goodman, Jane M. Hawkins, Jingfang Huang, Christopher Jones, Shrawan Kumar, Richard McLaughlin, Sorin Mitran, Peter J. Mucha, Karl E. Petersen, Joseph F. Plante, 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, Robert Lipshitz, Jeremy Marzuola, Justin Sawon.

Lecturers

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

Professors Emeriti

Ladnor D. Geissinger, Robert G. Heyneman, Norberto Kerzman, Ancel Mewborn, John A. Pfaltzgraff, Michael Schlessinger, William W. Smith, Johann Sonner, James D. Stasheff, Warren R. Wogen.

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. Today mathematics is an essential partner in fields where once it played no role. At the same time, mathematics itself continues to grow and develop through research, much of which is stimulated by interactions with other fields. Every educated person needs at least a familiarity with the language of mathematics and even some more substantial knowledge of the technical aspects than in the past. People working in many fields find that areas of mathematics only recently thought to be sophisticated and advanced have become part of the everyday tools in their spheres of activity.

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, computer analyst. Students who have an interest in working in one of these professions or who intend to pursue an advanced degree in one of the mathematical sciences should seriously consider the B.S. degree with a major in mathematics (including the applied option) or one of the related degree programs in computer science or mathematical decision sciences (actuarial science, operations research, statistics).

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.

Finally, the B.A. degree with a major in mathematics is a true liberal arts degree that opens the door to the continuing intellectual growth, enrichment, and self-fulfillment that are the goals of a liberal arts education. Students intending to enter a professional school (law, medicine, business) will find that admissions officers of such schools find an undergraduate degree in mathematics an attractive part of an applicant's history.

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.

Both the B.A. and the B.S. degrees require, beyond first-year/sophomore calculus, courses in algebra and analysis at a higher level. Students who plan a career in a technical field should also develop familiarity with computers and statistics, for example by taking COMP 116 and some of STOR 435, 455, 456, and 555. The specific requirements for the B.A. and B.S. degrees are described below.

Programs of Study

The degrees offered are the bachelor of arts and the bachelor of science with a major in mathematics. A minor in mathematics also is offered. The section "Graduate School and Career Opportunities" below provides suggested course selections for students who have interests in a particular direction, such as pure mathematics, biology, economics, or high school teaching.

Majoring in Mathematics: Bachelor of Arts

Core Requirements

Additional Requirements

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

Majoring in Mathematics:
Bachelor of Science

Core Requirements

Additional Requirements

B.S. Major in Mathematics: 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.

Minoring in Mathematics

Core Requirements

Additional Requirements

Honors in Mathematics

Special honors (H) sections are given in some mathematics courses when student demand is sufficient. Also, promising students are encouraged to work toward a bachelor's degree with honors in mathematics.

The program for graduation with honors in mathematics consists of six or more courses approved by the departmental honors advisor and satisfactory completion of an honors project. The honors project is conducted in association with a departmental faculty member on a topic approved by the departmental honors advisor. 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.

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, every major is assigned each semester a mathematics faculty member as an advisor with whom the student 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.

A Note on Advanced Placement and Sequential Credit in Mathematics Courses

A student who makes a grade of 3 or higher on the AB Advanced Placement Examination will receive credit for MATH 231. A student who makes a grade of 3 or higher on the BC Advanced Placement Examination will receive credit for both MATH 231 and 232. No credit for MATH 130 will be given on the basis of any advanced placement examination.

A student who places into MATH 232 or 233 and receives a grade of C- or better in the course (on the first attempt) will be given PL credit (0 hours with no grade) for the course in the sequence 231 and 232 that precedes the course taken.

No student can receive credit for MATH 116 or 152 or 130 after receiving credit for MATH 231 or 241. No student can receive credit for MATH 116 after receiving credit for MATH 152.

Prerequisites specified in the course list below can be replaced by equivalents, by permission of the instructor or by permission of the department.

Special Opportunities in Mathematics

Departmental Involvement

Special activities for qualified students include an undergraduate Mathematics Club, the scholastic honorary society Pi Mu Epsilon, and a Putnam Examination Team. Students interested in these activities should consult the departmental honors advisor.

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 as well as many General Education and elective requirements as they complete 10 credit hours in teaching and learning, including EDUC 403, 516 or 689, 532, 533, and 601. During their final semester, students engage in a full-time student teaching internship (EDUC 593) and participate in an education leadership seminar (EDUC 503). For more details on admission requirements, application deadlines, and submitting an online application, visit the School of Education Web site: soe.unc.edu/services/apply/ug.

Experiential Education

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

Teaching Internships and Assistantships

Qualified undergraduate students may work as research assistants in the Fluid Laboratory or as tutors in the Math Help Center.

Study Abroad

Opportunities include semester or yearlong programs in a variety of countries.

Undergraduate Awards

The Archibald Henderson Prize and the Alfred Brauer Award recognize outstanding performance and promise in mathematics.

Undergraduate Research

Students can conduct original research with the guidance of a faculty member, usually in conjunction with the preparation of an honors essay.

Facilities

An extensive computer system with up-to-date software, an outstanding science library, high-technology classrooms, and an undergraduate common room are available to students.

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 also can be excellent preparation for graduate study in many fields if the course program is complemented by electives in other areas. Professional schools of law, business, and medicine are becoming increasingly interested in broadly educated undergraduates, and a properly structured B.A. degree program in mathematics with additional courses is often taken as evidence that the student has good analytical abilities as well as a broad undergraduate background. 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 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:

Contact Information

Richárd Rimányi, Director of Undergraduate Studies, CB# 3250, 429 Chapman Hall, rimanyi@email.unc.edu; or Susan F. Stedman, Assistant to the Director of Undergraduate Studies, CB# 3250, 356 Phillips Hall, sstedman@email.unc.edu. Web site: www.math.unc.edu.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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).

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.

401 Mathematical Concepts in Art (3). Mathematical theories of proportion, perspective (projective invariants and the mathematics of visual perception), symmetry, and aesthetics will be expounded and illustrated by examples from painting, architecture, and sculpture.

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.

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.

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.

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.

452 Mathematical and Computational Models in Biology (BIOL 452) (4). See BIOL 452 for description.

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.

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

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.

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.

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

528 Mathematical Methods for the Physical Sciences I (3). Prerequisites, MATH 383; and PHYS 104 and 105, or PHYS 114 and 115, or PHYS 116 and 117, or PHYS 118 and 119. 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.

529 Mathematical Methods for the Physical Sciences II (3). Prerequisites, PHYS 104 or 114, PHYS 105 or 115, and one of 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.

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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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