Department of Mathematics
PATRICK B. EBERLEIN, Chair
Professors
Idris Assani, Roberto Camassa, Ivan V. Cherednik, Joseph A. Cima, James N. Damon, Patrick B. Eberlein, M. Gregory Forest, Sue E. Goodman, Jane M. Hawkins, Christopher Jones, Shrawan Kumar, Richard McLaughlin, Michael Minion, Karl E. Petersen, Joseph F. Plante, Robert Proctor, William W. Smith, Michael E. Taylor, Alexandre N. Varchenko, Jonathan M. Wahl, Mark Williams, Warren R. Wogen.
Associate Professors
David Adalsteinsson, Jingfang Huang, Lev Rozansky.
Assistant Professors
Dmytro Arinkin, Prakash Belkale, Jason Metcalfe, Laura Miller, Sorin Mitran, Peter Mucha, Richard Rimanyi.
Lecturers
Debra Etheridge, Mark McCombs, Elizabeth McLaughlin, Brenda Shryock.
Professors Emeriti
Robert L. Davis, Ladnor D. Geissinger, Robert G. Heyneman, Norberto Kerzman, Ancel Mewborn, John A. Pfaltzgraff, Michael Schlessinger, Johann Sonner, James D. Stasheff.
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 a variety of degrees in mathematics and the mathematical sciences, providing students a wide choice of careers in this field. Among the jobs in industry, government, and the academic world that involve mathematics as a central aspect 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. 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 the public schools 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.
Finally, the B.A. 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 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 one 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 355, 356, 435, and 555. The specific requirements for the B.A. and B.S. degrees are described below.
Programs of Study
The degrees offered are bachelor of arts and bachelor of science in mathematics. A minor in mathematics is also offered.
Majoring in Mathematics: Bachelor of Arts
All Foundations, Approaches, and Connections requirements of the
General Education curriculum apply (see the General Education section of the Bulletin). The requirements specific to the major are as
follows:
A. 1. MATH 231, 232, 233, 381, 383
2. MATH 521
3. MATH 547 or 577 (preferably before the senior year)
4. At least three more mathematics courses numbered above 500
B. The Supplemental General Education requirements of the College of Arts and Sciences, either the distributive or integrative option
C. Eighteen hours of C or better (not C-) in mathematics courses listed in Part A (1–4) numbered 233 or higher
Majoring in Mathematics: Bachelor of Science
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.
B.S. Degree with a Major in Mathematics
First and Sophomore Years
• COMP 116 or MATH 565
• MATH 231, 232, 233, 381, 383
• PHYS 104 and 105 or PHYS 116 and 117
• Foreign language through level 4 (level 4 may be taken PS/D/F)
Junior and Senior Years
• MATH 521 and one of 522, 523, 528, or 566
• MATH 547 or 577 (preferably before the senior year)
• One of MATH 533, 534, 578, or 548
• At least three more mathematics courses numbered above 520
• Four or more courses in the Division of Natural Sciences and Mathematics (beyond the General Education requirements), but not in mathematics
• Eighteen hours of C or better (not C-) in mathematics courses numbered above 520
B.S. Degree with a Major in Mathematics (Applied Option)
First and Sophomore Years
• COMP 116
• MATH 231, 232, 233, 381, 383
• PHYS 116 and 117
Junior and Senior Years
• MATH 521, 524, 528, 529, and 564
• MATH 522 or 523
• MATH 547 or 577
• MATH 566 or 661
• STOR 355 and 356 or STOR 435 and 555
• Two or more courses in the Division of Natural Sciences and Mathematics (beyond the General Education requirements), but not in mathematics
• Eighteen hours of C or better (not C-) in mathematics courses numbered above 520.
Minoring in Mathematics
• MATH 231, 232, 233, 381, 383
• Three mathematics courses numbered above 500, all with a grade of C (not C-) or better
Honors in Mathematics
Special honors (H) sections are given in some mathematics courses when student demand is sufficient. Promising students are encouraged to work toward a bachelor’s degree with honors in mathematics.
The honors program will consist of six or more courses approved by the departmental honors advisor. At some time during the semester in which he or she expects to graduate, the candidate for a degree with honors will either present an honors essay written under the direction of a faculty member or take an oral examination on courses approved by the honors advisor. Students writing an honors essay will be expected to make an oral presentation of the essay. Interested students should consult the departmental honors advisor as early as possible and in no case later than the beginning of their senior year.
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 is placed in one of the courses MATH 232 or 233 and receives a grade of C- or better in the course (on the first attempt) will be given credit (without 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. No student can receive credit for MATH 116 after receiving credit for 152.
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.
Experiential Education
Undergraduate honors research projects as well as some internships or study abroad programs might qualify.
Teaching Internships and Assistantships
Undergraduates work as research assistants in the Fluid Laboratory, as tutors in the Math Help Center, and as homework graders.
Study Abroad
Opportunities include semester or year-long 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 directed at the preparation of an honors essay.
Facilities
An extensive computer system with up-to-date software, an outstanding mathematics-physics 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 course work in areas such as statistics, computer science, economics, and operations research. Undergraduate mathematics majors are in demand in many business, industry, and government fields.
Contact Information
Karl Petersen, Director of Undergraduate Studies, CB# 3250, 300A Phillips Hall, (919) 962-2380, petersen@email.unc.edu; or Kimberley Doty-Harris, Assistant to the Director of Undergraduate Studies, CB# 3250, 356 Phillips Hall, (919) 962-0198, kdharris@email.unc.edu. Web site: www.math.unc.edu.
MATH
050 [006P] 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.
051 [006P] First-Year Seminar: “Fish Gotta Swim, Birds Gotta Fly”: The Mathematics and the 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.
052 [006P] 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.
053 [006P] 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.
054 [006P] 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.
055 [006P] 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.
056 [006P] 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.
057 [006P] 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.
058 [006P] First-Year Seminar: Math and Art: Symmetry without Fear (3). Students will mathematically classify rosette patterns, the eight frieze patterns, and the 17 wallpaper patterns. Then they will take over exhibiting patterns from various cultures and local instances.
059 [006P] 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).
060 [006P] 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.
061 [006P] 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.
062 [006P] 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.
063 [006P] First-Year Seminar: From “The Sound of Music” to “The Perfect Storm” (MASC 057) (3). In this seminar students will develop the conceptual framework necessary to understand waves, starting from laboratory observations.
064 [006P] 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 by reading the book A View of the Sea by the eminent oceanographer Hank Stommel and by examining satellite and on-site observations.
065 [006P] 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.
066 [006P] 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.
110 [010] 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 [016] 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 or 231.
117 [017] Finite Mathematics (3). Provides an introduction in as nontechnical a setting as possible to the basic concepts of finite mathematics. Basic counting problems and finite probability problems are discussed. The course is intended for the nonscience major.
118 [018] Selected Topics in Mathematics (3). Provides an introduction in as nontechnical a setting as possible to selected topics in mathematics. Topics covered will vary each semester depending on the individual instructor’s selection. The course is intended for the nonscience major.
119 [019] 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.
130 [030] 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 [022] 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.
231 [031] Calculus of Functions of One Variable I (3). Prerequisite, a grade of C- or better in MATH 130 or placement by the department. Limits, derivatives, and integrals of functions of one variable.
232 [032] Calculus of Functions of One Variable II (3). Prerequisite, a grade of C- or better in MATH 231 or placement by the department. Calculus of the elementary transcendental functions, techniques of integration, indeterminate forms, Taylor’s formula, infinite series.
233 [033] 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). 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). Prerequisite, 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. No credit will be given for MATH 283 after a student takes MATH 383.
290 Directed Exploration in Mathematics (3). By 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.
295 [098] 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.
296 [090] Undergraduate Reading and Research in Mathematics (1–3). By 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.
307 [067] Revisiting Real Numbers and Algebra (EDUC 307) (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 [081] 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 [083] 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).
401 [101] 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 [106] Mathematical Methods in Biostatistics (1). Prerequisite, MATH 232 or equivalent. 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.
411 [111] Developing Mathematical Concepts (1–21). 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 [118] 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 [107] Mathematical and Computational Models in Biology (BIOL 452) (4). Prerequisites, BIOL 201 and 202, MATH 231, and either MATH 232 or STOR 155. This course will introduce analytical, computational, and statistical techniques, such as discrete models, numerical integration of ordinary differential equations, and likelihood functions, to explore topics from various fields of biology. Laboratory is included.
515 [115] 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 [121] Advanced Calculus I (3). Prerequisites, MATH 233 and 381. The real numbers, continuity and differentiability of functions of one variable, infinite series, integration.
522 [122] 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 [123] 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 [124] Elementary Differential Equations (3). Prerequisite, MATH 383. Linear differential equations, power series solutions, Laplace transforms, numerical methods.
528 [128] Mathematical Methods for the Physical Sciences I (3). Prerequisites, MATH 383 and PHYS 104 and 105, or equivalent. 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.
529 [129] Mathematical Methods for the Physical Sciences II (3). Prerequisites, PHYS 104 and 105, and one of MATH 521, 524, or 528 or equivalents. 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.
533 [133] 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 [134] Elements of Modern Algebra (3). Prerequisite, MATH 381. Binary operations, groups, subgroups, cosets, quotient groups, rings, polynomials.
535 [126] Introduction to Probability (STOR 435) (3). Prerequisite, MATH 233. Introduction to mathematical theory of probability covering random variables; moments; binomial, Poisson, normal and related distributions; generating functions; sums and sequences of random variables; and statistical applications.
547 [147] 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 [148] Combinatorial Mathematics (3). Prerequisite, MATH 381 or equivalent, or permission of the instructor. 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 [130] Topology (3). Prerequisite, MATH 233; corequisite, MATH 383 or permission of the instructor. 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 [131] Euclidean and Non-Euclidean Geometries (3). Prerequisite, MATH 381 or permission of the instructor. Critical study of basic notions and models of Euclidean and non-Euclidean geometries: order, congruence, and distance.
555 [155] Introduction to Dynamics (3). Prerequisite, MATH 383 or permission of the instructor. 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 [145] Mathematical Modeling (3). Prerequisites, MATH 283 or 383, and some knowledge of computer programming or permission of the instructor. 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 [125] 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 [166] Introduction to Numerical Analysis (3). Prerequisites, MATH 383 and 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 [137] 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 [138] 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.
579 [157] Topics in Matrix Theory (3). Prerequisites, MATH 547 or 577 or equivalent, and some knowledge of computer programming. Quadratic and Hermitian forms, Sylvester’s theorem; applications to systems of differential equations; approximation of eigenvalues and eigenvectors; non-negative matrices. Perron-Frobenius theorem; integer matrices with applications in combinatorics.
590 [175] Topics in Analysis (3). Prerequisite, MATH 522 or permission of the instructor. Topics may include linear spaces, convexity, mathematical programming, duality, algorithms, or other subjects related to mathematical analysis.
591 [176] Topics in Algebra (3). Permission of the instructor. Topics may include number theory, algebraic number theory, field theory, or algebraic geometry.
592 [177] Topics in Geometry (3). Permission of the instructor. Topics may include non-Euclidean geometries, linear geometry, finite geometries, convexity, polytopes, topology, and algebraic geometry.
635 [195] Probability (STOR 635) (3). See STOR 635 for description.
641 [189] 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 [190] 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 [193] Introductory Analysis (3). Prerequisite, 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 [196] 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 [197] Qualitative Theory of Differential Equations (3). Prerequisites, linear algebra and MATH 653, or permission of the instructor. 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 [191] Scientific Computation I (ENVR 661) (3). Prerequisites, 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 [192] 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 [198] Methods of Applied Mathematics I (ENVR 668) (3). Prerequisite, undergraduate 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 [199] Methods of Applied Mathematics II (ENVR 669) (3). Prerequisite, MATH 668 or permission of the instructor. 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 [186] Modules, Linear Algebra, and Groups (3). Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, groups and group actions.
677 [187] Groups, Representations, and Fields (3). Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory.
680 [180] Geometry of Curves and Surfaces (3). Prerequisite, 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 [181] Introductory Topology (3). Prerequisites, MATH 653 and 676 or permission of the instructor. Topological spaces, connectedness, separation axioms, product spaces, extension theorems. Classification of surfaces, fundamental group, covering spaces.