Department of Mathematics
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 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 Catalog or the director of mathematical education in the Department of Mathematics. A section in the mathematics program of the Catalog suggests course selections for future high school teachers.
All majors and minors have a primary academic advisor assigned in ConnectCarolina. 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, drop in advising is available each semester in the math department. Students who have declared a math major and have completed MATH 233 are required to attend a math department advising session to discuss course selections and any other questions before a hold on registration is lifted. The department’s director of undergraduate studies, assistant director of undergraduate studies, and manager of student services (see contact tab above) 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 website.
Placement into Mathematics Courses
Standardized test scores such as the Advanced Placement (AP) or the American College Test (ACT) can be used for placement into mathematics courses. Students who do not have placement scores via the AP or ACT may take the department's ALEKS Placement Test. Please visit the placement page of the department’s website for specific information regarding placement and departmental placement tests.
Graduate School and Career Opportunities
The B.S. degree program, especially if it includes the sequences MATH 521–MATH 522 and MATH 577–MATH 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.
David Adalsteinsson, Idris Assani, Prakash Belkale, Roberto Camassa, Ivan V. Cherednik, Hans Christianson, M. Gregory Forest, Boyce Griffith, Jingfang Huang, Shrawan Kumar, Jeremy Marzuola, Richard McLaughlin, Jason Metcalfe, Sorin Mitran, Richárd Rimányi, Lev Rozansky, Michael E. Taylor, Alexander N. Varchenko, Mark Williams.
Yaiza Canzani, Jiuzu Hong, Katherine Newhall, David Rose, Justin Sawon.
Olivia Dumitrescu, Shahar Kovalsky, Casey Rodriguez, Pedro Sáenz, Andrey Smirnov.
Emily Burkhead, Linda Green, Mark McCombs, Elizabeth McLaughlin, Miranda Thomas.
Joseph A. Cima, James N. Damon, Patrick B. Eberlein, Ladnor D. Geissinger, Sue E. Goodman, Jane M. Hawkins, Robert G. Heyneman, Christopher Jones, Ancel Mewborn, Karl E. Peterson, Joseph F. Plante, Robert Proctor, Michael Schlessinger, William W. Smith, James D. Stasheff, Jonathan M. Wahl, Warren R. Wogen.
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.
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. Honors version available.
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. Honors version available.
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.
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. Honors version available.
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.
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. Honors version available.
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.
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. Honors version available.
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).
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. Honors version available.
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.
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. Honors version available.
Students will develop the conceptual framework necessary to understand waves of any kind, starting from laboratory observations. Honors version available.
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.
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.
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.
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.
What properties should a fair election have and are these properties achievable in theory and in practice? How can mathematics and statistics be used to expose election fraud and gerrymandering? Students will address these questions as they compare different election systems, evaluate their strengths, weaknesses, and abuses, and design improvements to current structures. Topics will include gerrymandering, ranked voting, approval voting, and Arrow's Impossibility Theorem.
This seminar engages students in an exploration of the interplay between mathematics, origami, and fractal symmetry. Learning objectives will include mastering basic origami folding techniques, identifying and applying fundamental symmetry operations, recognizing and analyzing fractal symmetry, and creating geometric tessellations. Students will use image editing software (Illustrator and Photoshop), mathematical imaging software (Ultra Fractal), and the laser cutter in UNC's BeAM space, to design and create modular origami and fractal tessellation artwork.
In this seminar, students will explore ideas from topology and geometry and their application to symmetry patterns. Students will learn to identify and classify two-dimensional symmetry patterns and create their own designs. Students will relate symmetry patterns to their folded-up counterparts, called orbifolds, and use tools from topology and geometry to determine which patterns are possible and which patterns can never be achieved.
Special topics course. Content will vary each semester. Honors version available.
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 does not satisfy any general education requirements. It is intended for students who need it as a prerequisite for other classes. A student cannot receive credit for this course after receiving credit for MATH 231 or higher.
This course provides just-in-time instruction and practice on basic algebra to support students in Algebra. It also provides additional practice on some of the more difficult topics from MATH 110. This course is intended for students currently enrolled in MATH 110 who need additional review of algebra.
Students will use mathematical and statistical methods to address societal problems, make personal decisions, and reason critically about the world. Authentic contexts may include voting, health and risk, digital humanities, finance, and human behavior. This course does not count as credit towards the psychology or neuroscience majors.
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.
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. A student cannot receive credit for this course after receiving credit for MATH 231 or higher.
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. A student cannot receive credit for this course after receiving credit for MATH 231 or higher .
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.
Awarded as placement credit based on test scores. Does not fulfill a graduation requirement.
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.
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.
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.
This course introduces students to the tools of linear algebra and optimization, including solving linear systems, matrices as linear transformations, eigenvalues and eigenvectors, approximations, root finding, derivatives, and optimization in multiple dimensions. This course emphasizes multidimensional thinking and applications to data science. A student cannot receive credit for this course after receiving credit for MATH 347.
Limits, derivatives, and integrals of functions of one variable. Students may not receive credit for both MATH 231 and MATH 241. Honors version available.
This course provides just-in-time instruction and review on algebra and trigonometry to support students in MATH 231. It also provides additional practice on some of the more difficult topics from Calculus 1. This course is intended to be taken by students currently enrolled in MATH 231 who need review of algebra and trigonometry.
Calculus of the elementary transcendental functions, techniques of integration, indeterminate forms, Taylor's formula, infinite series. Honors version available.
Vector algebra, solid analytic geometry, partial derivatives, multiple integrals. Honors version available.
Permission of the instructor. Elective topics in mathematics. This course has variable content and may be taken multiple times for credit.
Permission of the instructor. A seminar on a chosen topic in mathematics in which the students participate more actively than in usual courses.
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. Formerly offered as MATH 290.
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.
Algebra of matrices with applications: determinants, solution of linear systems by Gaussian elimination, Gram-Schmidt procedure, and eigenvalues. Previously offered as MATH 547.
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. Honors version available.
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). Honors version available.
Course is computational laboratory component designed to help students visualize ODE solutions in Matlab. Emphasis is on differential equations motivated by applied sciences. Some applied linear algebra will appear as needed for computation and modeling purposes.
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.
Advanced Undergraduate and Graduate-level Courses
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.
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 field work in both the high school and college environments.
Permission of the instructor. An investigation of various ways elementary concepts in mathematics can be developed. Applications of the mathematics developed will be considered.
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.
A general survey of the history of mathematics with emphasis on elementary mathematics. Some special problems will be treated in depth.
A grade of A- or better in STOR 215 may substitute for MATH 381. The real numbers, continuity and differentiability of functions of one variable, infinite series, integration. Honors version available.
Functions of several variables, the derivative as a linear transformation, inverse and implicit function theorems, multiple integration. Honors version available.
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.
Linear differential equations, power series solutions, Laplace transforms, numerical methods.
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.
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.
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.
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.
A grade of A- or better in STOR 215 may substitute for 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.
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. Students may not receive credit for both STOR 435 and STOR 535.
Counting selections, binomial identities, inclusion-exclusion, recurrences, Catalan numbers. Selected topics from algorithmic and structural combinatorics, or from applications to physics and cryptography.
Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tessellations, and fundamental group.
This course introduces analytical, computational, and statistical techniques, such as discrete models, numerical integration of ordinary differential equations, and likelihood functions, to explore various fields of biology.
This lab introduces analytical, computational, and statistical techniques, such as discrete models, numerical integration of ordinary differential equations, and likelihood functions, to explore various fields of biology.
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.
This course will provide an introduction to convex optimization, including convex sets and functions, modeling, conic problems, optimality conditions and algorithms. The second part of the course will address non-convex problems, focusing on contemporary optimization challenges in large-scale optimization and practical approaches for machine learning and deep learning.
Requires some knowledge of computer programming. Model validation and numerical simulations using ordinary, partial, stochastic, and delay differential equations. Applications to the life sciences may include muscle physiology, biological fluid dynamics, neurobiology, molecular regulatory networks, and cell biology.
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.
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 347.
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.
Permission of the instructor. Topics may focus on matrix theory, analysis, algebra, geometry, or applied and computational mathematics.
Interdisciplinary introduction to nonlinear dynamics and chaos. Fixed points, bifurcations, strange attractors, with applications to physics, biology, chemistry, finance.
Foundations of probability. Basic classical theorems. Modes of probabilistic convergence. Central limit problem. Generating functions, characteristic functions. Conditional probability and expectation.
Basic counting; partitions; recursions and generating functions; signed enumeration; counting with respect to symmetry, plane partitions, and tableaux.
Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Moebius inversion, q-analogs, combinatorial and projective geometries, codes, and designs.
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.
A rigorous treatment of complex integration, including the Cauchy theory. Elementary special functions, power series, local behavior of analytic functions.
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.
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.
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.
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.
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.
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.
Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory.
Calculus on manifolds, vector bundles, vector fields and differential equations, de Rham cohomology.
Topological spaces, product spaces, connectedness, compactness. Classification of surfaces, fundamental group, covering spaces. Simplicial homology.
Permission of the department. Directed study of an advanced topic in mathematics. Topics will vary.
Permission of the director of undergraduate studies. Readings in mathematics and the beginning of directed research on an honors thesis.
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.
Department of Mathematics
Phillips Hall, CB# 3250