Courses in Mathematical Sciences (MATH)

Note: Statistics courses (STAT) follow MATH listings. P—prerequisite; C—corequisite; R—recommended; Fall—offered fall semester; Spring—offered spring semester; Summer—offered in the summer session. For courses with no designated semester, consult the Schedule of Classes. Equiv.—course is equivalent to the indicated course taught at Indiana University Bloomington, or the indicated course taught at Purdue University, West Lafayette.

Special Developmental Courses

001 Introduction to Algebra (4 cr.) P: M010 (minimum grade of C) or eighth-grade mathematics. Fall, spring, summer. Covers the material in the first year of high school algebra. Numbers and algebra, integers, rational numbers, equations, polynomials, graphs, systems of equations, inequalities, radicals. Credit does not apply toward any degree.

002 Geometry (3 cr.) P or C: 001 or equivalent. This course is intended to provide one unit of geometry as a first encounter or as a review for those students with little or no geometry background and needing this prerequisite to pursue higher-level course work. Covers plane and solid geometry, right triangle trigonometry, and mathematical logic through a structure focused on problem-solving and critical thinking skills.

Undergraduate Level

Lower-Division Courses

111 Algebra (4 cr.) P: 001 (minimum grade of C) or one year of high school algebra. Fall, spring, summer. Real numbers, linear equations and inequalities, systems of equations, polynomials, exponents, logarithmic functions. Covers material in the second year of high school algebra.

M118 Finite Mathematics¹ (3 cr.) P: 111 or 110 (minimum grade of C) or equivalent. Fall, spring, summer. Set theory, logic, permutations, combinations, simple probability, conditional probability.

1The sequence MATH M118, 130, 132 or MATH M118, 136 fulfills the mathematics requirement for elementary education majors.

M119 Brief Survey of Calculus I (3 cr.) P: 111 or 110 (minimum grade of C) or equivalent. Fall, spring, summer. Sets, limits, derivatives, integrals, and applications.

123 Elementary Concepts of Mathematics (3 cr.) P: None. Mathematics for liberal arts students; experiments and activities that provide an introduction to inductive and deductive reasoning, number sequences, functions and curves, probability, statistics, topology, metric measurement, and computers.

130 Mathematics for Elementary Teachers I¹ (3 cr.) P: 111 or 110 (minimum grade of C) or equivalent; one year of high school geometry. Fall, spring, summer. Numeration systems, mathematical reasoning, integers, rationals, reals, properties of number systems, decimal and fractional notations, problem solving.

132 Mathematics for Elementary Teachers II1 (3 cr.) P: 130. Fall, spring, summer. Rationals, reals, geometric relationships, properties of geometric figures, one-, two-, and three-dimensional measurement and problem solving.

136 Mathematics for Elementary Teachers1 (6 cr.) P: 111 or 110 (minimum grade of C) or equivalent; one year of high school geometry. Fall, spring, summer. 136 is a one-semester version of 130 and 132. Not open to sudents with credit in 130 or 132.

151 Algebra and Trigonometry (5 cr.) P: 111 (minimum grade of B) or placement. Fall, spring, summer I. 151 is a one-semester version of 153-154. Not open to students with credit in 153 or 154. 151 covers college-level algebra and trigonometry and provides preparation for 163 and 164.

153 Algebra and Trigonometry I (3 cr.) P: 111 (minimum grade of C) or two years of high school algebra. Fall, spring, summer. 153-154 is a two-semester version of 151. Not open to students with credit in 151. 153 covers college-level algebra and provides preparation for 163 and 221.

154 Algebra and Trigonometry II (3 cr.) P: 153 (minimum grade of C) or five semesters of high school algebra. Fall, spring, summer. 153-154 is a two-semester version of 151. Not open to students with credit in 151. 154 covers college-level trigonometry and provides preparation for 163 and 221.

163 Integrated Calculus and Analytic Geometry I (5 cr.) P: 151 or 154 (minimum grade of C) or equivalent, and one year of geometry. Equiv. IU MATH M211. Fall, spring, summer I. Review of plane analytic geometry and trigonometry, functions, limits, differentiation, applications of differentiation, integration, the fundamental theorem of calculus, and applications of integration.

164 Integrated Calculus and Analytic Geometry II (5 cr.) P: 163 (minimum grade of C–). Equiv. IU MATH M212. Fall, spring, summer I. Transcendental functions, techniques of integration, indeterminant forms and improper integrals, conics, polar coordinates, sequences, infinite series, and power series.

179 Computers and Mathematics (3 cr.) P: 163. Exploration of some modern mathematical concepts, using the computer as an experimental tool. Posssible topics include iteration, fixed points, convergence, stability/instability, chaos, fractals. Function approximation: polynomials, splines, computer graphics. Calculus: numerical approximations, symbolic manipulations. Arithmetic with large integers: prime numbers, factorization, encryption, unsolved problems in number theory.

190 Topics in Applied Mathematics for Freshmen (3 cr.) Treats applied topics in mathematics at the freshman level. Prerequisites and course material vary with the applications.

221 Calculus for Technology I (3 cr.) P: 151 or 154 (minimum grade of C) or equivalent, and one year of geometry. Fall, spring, summer. Analytic geometry, the derivative and applications, the integral and applications.

222 Calculus for Technology II (3 cr.) P: 221 (minimum grade of C–). Fall, spring, summer. Differentiation of transcendental functions, methods of integration, power series, Fourier series, differential equations.

261 Multivariate Calculus (4 cr.) P: 164. Equiv. IU MATH M311. Fall, spring, summer. Spatial analytic geometry, vectors, curvilinear motion, curvature, partial differentiation, multiple integration, line integrals, Green’s theorem.

262 Linear Algebra and Differential Equations (4 cr.) P: 164. R: 261. Fall, spring, summer. First-order equations, higher-order linear equations, initial and boundary value problems, power series solutions, systems of first-order equations, Laplace transforms, applications. Requisite topics of linear algebra: vector spaces, linear independence, matrices, eigenvalues, and eigenvectors.

290 Topics in Applied Mathematics for Sophomores (3 cr.) Treats applied topics in mathematics at the sophomore level. Prerequisites and course material vary with the applications.

Upper-Division Courses

351 Elementary Linear Algebra (3 cr.) P: 261. Not open to students with credit in 511. Fall, spring. Systems of linear equations, matrices, vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, applications.

375 Theory of Interest (3 cr.) P: 261. An introduction to the theory of finance including such topics as compound interest, annuities certain, amortization schedules, sinking funds, bonds, and related securities.

390 Topics in Applied Mathematics for Juniors (3 cr.) Treats applied topics in mathematics at the junior level. Prerequisites and course material vary with the applications.

414 Numerical Methods (CSCI 414) (3 cr.) P: 262 and a course in a high-level programming language. Not open to students with credit in CSCI 512. Error analysis, solution of nonlinear equations, direct and iterative methods for solving linear systems, approximation of functions, numerical differentiation and integration, numerical solution of ordinary differential equations.

417 Discrete Modeling and Game Theory (3 cr.) P: 262 and 351 or 511 or consent of instructor. Linear programming; mathematical modeling of problems in economics, management, urban administration, and the behavioral sciences.

426 Introduction to Applied Mathematics and Modeling (3 cr.) P: 262 and PHYS152. Introduction to problems and methods in applied mathematics and modeling. Formulation of models for phenomena in science and engineering, their solution, and physical interpretation of results. Examples chosen from solid and fluid mechanics, mechanical systems, diffusion phenomena, traffic flow, and biological processes.

441 Foundations of Analysis (3 cr.) P: 261. Set theory, mathematical induction, real numbers, completeness axiom, open and closed sets in Rm, sequences, limits, continuity and uniform continuity, inverse functions, differentiation of functions of one and several variables.

442 Foundations of Analysis II (3 cr.) P: 441. Continuation of differentiation, the mean value theorem and applications, the inverse and implicit function theorems, the Riemann integral, the fundamental theorem of calculus, point-wise and uniform convergence, convergence of infinite series, series of functions.

453 Beginning Abstract Algebra (3 cr.) P: 351 or consent of the instructor. Basic properties of groups, rings, and fields, with special emphasis on polynomial rings.

456 Introduction to the Theory of Numbers (3 cr.) P: 261. Divisibility, congruences, quadratic residues, Diophantine equations, the sequence of primes.

462 Elementary Differential Geometry (3 cr.) P: 351. Calculus and linear algebra applied to the study of curves and surfaces. Curvature and torsion, Frenet-Serret apparatus and theorem, fundamental theorem of curves. Transformation of R2, first and second fundamental forms of surfaces, geodesics, parallel translation, isometries, fundamental theorem of surfaces.

463 Intermediate Euclidean Geometry for Secondary Teachers (3 cr.) P: 002 (or one year of high school geometry), and 300, or consent of instructor. History of geometry. Ruler and compass constructions, and a critique of Euclid. The axiomatic method, models, and incidence geometry. Presentation, discussion and comparison of Hilbert’s, Birkhoff’s, and SMSG’s axiomatic developments.

490 Topics in Mathematics for Undergraduates (1-5 cr.) By arrangement. Open to students only with the consent of the department. Supervised reading and reports in various fields.

1The sequence MATH M118, 130, 132 or MATH M118, 136 fulfills the mathematics requirement for elementary education majors.

S490 Senior Seminar (3 cr.)

492 Capstone Experience (1-3 cr.) By arrangement.

495 TA Instruction (0 cr.) For teaching assistants. Intended to help prepare TAs to teach by giving them the opportunity to present elementary topics in a classsroom setting under the supervision of an experienced teacher who critiques the presentations.

Undergraduate and Graduate Level

505 Intermediate Abstract Algebra (3 cr.) P: 453 or consent of the instructor. Group theory with emphasis on concrete examples and applications. Field theory: ruler and compass constructions, Galois theory, solvability of equations by radicals.

510 Vector Calculus (3 cr.) P: 261. Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green’s theorem, Stokes’ theorem, divergence theorems, applications to engineering and the physical sciences.

511 Linear Algebra with Applications (3 cr.) P: 261. Not open to students with credit in 351. Matrices, rank and inverse of a matrix, decomposition theorems, eigenvectors, unitary and similarity transformations on matrices.

519 Introduction to Probability (STAT 519) (3 cr.) P: 262. See STAT 519.

520 Boundary Value Problems of Differential Equations (3 cr.) P: 261 and 262. Sturm-Liouville theory, singular boundary conditions, orthogonal expansions, separation of variables in partial differential equations, spherical harmonics.

522 Qualitative Theory of Differential Equations (3 cr.) P: 262 and 351. Laplace transforms, systems of linear and nonlinear ordinary differential equations, brief introduction to stability theory, approximation methods, other topics.

523 Introduction to Partial Differential Equations (3 cr.) P: 262 and 510, or consent of instructor. Method of characteristics for quasilinear first-order equations; complete integral; Cauchy-Kowalewsky theory; classification of second-order equations in two variables; canonical forms; difference methods of hyperbolic and parabolic equations; Poisson integral method for elliptic equations.

525 Introduction to Complex Analysis (3 cr.) P: 261 and 262. Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; elementary conformal mapping.

526 Principles of Mathematical Modeling (3 cr.) P: 262 and 510, or consent of instructor. Ordinary and partial differential equations of physical problems, simplification, dimensional analysis, scaling, regular and singular perturbation theory, variational formulation of physical problems, continuum mechanics, and fluid flow.

527 Advanced Mathematics for Engineering And Physics I (3 cr.) P: 262. MATH527 and 528 constitute a two-semester sequence covering a broad range of topics including advanced calculus, linear algebra, complex variables, and differential equations, both ordinary and partial.

528 Advanced Mathematics for Engineering and Physics II(3 cr.) P:527. Continuation of MATH527.

530 Functions of a Complex Variable I (3 cr.) P or C: 544. Complex numbers, holomorphic functions, harmonic functions, linear transformations. Power series, elementary functions, Riemann surfaces, contour integration, Cauchy’s theorem, Taylor and Laurent series, residues. Maximum and argument principles. Special topics.

531 Functions of a Complex Variable II (3 cr.) P: 530. Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge’s theorem, Mittag-Leffler theorem, analytic continuation and Reimann surfaces, Picard theorems.

532 Elements of Stochastic Processes (STAT 532) (3 cr.) P: 519. See STAT 532.

535 Theoretical Mechanics (3 cr.) P: 262 and PHYS 152. Kinematics and dynamics of systems of particles and of rigid bodies; Lagrange and Hamilton-Jacobi equations; oscillations about equilibrium; Hamiltonian systems; integral invariants; transformation theory.

536 Perturbation and Asymptotic Analysis (3 cr.) P: 525 or 530, and 523. Matched asymptotic expansions, inner and outer expansions, strained coordinates and multiple scales, turning point analysis.

537 Applied Mathematics for Scientists and Engineers I (3 cr.) P: 261, 262, and consent of instructor. Covers theories, techniques, and applications of partial differential equations, Fourier transforms, and Laplace transforms. Overall emphasis is on applications to physical problems.

544 Real Analysis and Measure Theory (3 cr.) P: 441 or consent of instructor. Algebras of sets, real number system, Lebesgue measure, measurable functions, Lebesgue integration, differentiation, absolute continuity, Banach spaces, metric spaces, general measure and integration theory, Riesz representation theorem.

545 Principles of Analysis II (3 cr.) P: 544. Continues the study of measure theory begun in 544.

546 Introduction to Functional Analysis (3 cr.) P: 545. By arrangement. Banach spaces, Hahn-Banach theorem, uniform boundedness principle, closed graph theorem, open mapping theorem, weak topology, Hilbert spaces.

547 Analysis for Teachers I (3 cr.) P: 261. Set theory, logic, relations, functions, Cauchy’s inequality, metric spaces, neighborhoods, Cauchy sequence.

548 Analysis for Teachers II (3 cr.) P: 547. Functions on a metric space, continuity, uniform continuity, derivative, chain rule, Reimann integral, fundamental theorem of calculus, double integrals.

549 Applied Mathematics for Secondary School Teachers (3 cr.) P: 262 and 351. Summer, odd-numbered years. Applications of mathematics to problems in the physical sciences, social sciences, and the arts. Content varies. May be repeated for credit with the consent of the instructor.

550 Algebra for Teachers I (3 cr.) P: 351. Definitions and elementary properties of groups, rings, integral domains, fields. Intended for secondary school teachers.

551 Algebra for Teachers II (3 cr.) P: 550. Polynomial rings, fields, vector spaces, matrices.

552 Applied Computational Methods II (3 cr.) P: 559 and consent of instructor. The first part of the course focuses on numerical integration techniques and methods for ODEs. The second part concentrates on numerical methods for PDEs based on finite difference techniques with brief surveys of finite element and spectral methods. The final week contains a brief survey of methods for bifurcations, which combine various learned numerical techniques. Four lab assignments related to applications are assigned.

553 Introduction to Abstract Algebra (3 cr.) P: 453 or consent of instructor. Group theory: finite abelian groups, symmetric groups, Sylow theorems, solvable groups, Jordan-Hölder theorem. Ring theory: prime and maximal ideals, unique factorization rings, principal ideal domains, Euclidean rings, factorization in polynomial and Euclidean rings. Field theory: finite fields, Galois theory, solvability by radicals.

554 Linear Algebra (3 cr.) P: 351. Review of basics: vector spaces, dimension, linear maps, matrices, determinants, linear equations. Bilinear forms; inner product spaces; spectral theory; eigenvalues. Modules over principal ideal domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation.

559 Applied Computational Methods I (3 cr.) P: 262 and 351 or 511. Computer arithmetic, interpolation methods, methods for nonlinear equations, methods for solving linear systems, special methods for special matrices, linear least square methods, methods for computing eigenvalues, iterative methods for linear systems; methods for optimization and minimization.

561 Projective Geometry (3 cr.) P: 351. Projective invariants, Desargues’ theorem, cross-ratio, axiomatic foundation, duality, consistency, independence, coordinates, conics.

562 Introduction to Differential Geometry and Topology (3 cr.) P: 351 and 442. Smooth manifolds, tangent vectors, inverse and implicit function theorems, submanifolds, vector fields, integral curves, differential forms, the exterior derivative, DeRham cohomology groups, surfaces in E3, Gaussian curvature, two-dimensional Riemannian geometry, Gauss-Bonnet and Poincaré theorems on vector fields.

563 Advanced Geometry (3 cr.) P: 300 or consent of instructor. Topics in Euclidean and non-Euclidean geometry.

571 Elementary Topology (3 cr.) P: 441. Topological spaces, metric spaces, continuity, compactness, connectedness, separation axioms, nets, function spaces.

572 Introduction to Algebraic Topology (3 cr.) P: 571. Singular homology theory, Ellenberg-Steenrod axioms, simplicial and cell complexes, elementary homotopy theory, Lefschetz fixed point theorem.

578 Mathematical Modeling of Physical Systems I (3 cr.) P: 262, PHYS 152 and 251 and consent of Instructor. Linear systems modeling, mass-spring-damper systems, free and forced vibrations, applications to automobile suspension, accelerometer, seismograph, etc., RLC circuits, passive and active filters, applications to crossover networks and equalizers, nonlinear systems, stability and bifurcation, dynamics of a nonlinear pendulum, van der Pol oscillator, chemical reactor, etc., introduction to chaotic dynamics, identifying chaos, chaos suppression and control, computer simulations and laboratory experiments.

581 Introduction to Logic for Teachers (3 cr.) P: 351. Not open to students with credit in 385. Logical connectives, rules of sentential inference, quantifiers, bound and free variables, rules of inference, interpretations and validity, theorems in group theory, introduction to set theory.

583 History of Elementary Mathematics (3 cr.) P: 261. A survey and treatment of the content of major developments of mathematics through the eighteenth century, with selected topics from more recent mathematics, including non-Euclidean geometry and the axiomatic method.

585 Mathematical Logic I (CSCI 585) (3 cr.) P: 351. Formal theories for propositional and predicate calculus with study of models, completeness, compactness. Formalization of elementary number theory; Turing machines, halting problem, and the undecidability of arithmetic.

587 General Set Theory (3 cr.) P: 351. Informal axiomatization of set theory, cardinal numbers, countable sets, cardinal arithmetic, order types, well-ordered sets and ordinal numbers, axiom of choice and equivalences, paradoxes of intuitive set theory, Zermelo-Fraenkel axioms.

588 Mathematical Modeling of Physical Systems II (3 cr.) P: 578. Depending on the interests of the students, the content may vary from year to year. Emphasis will be on mathematical modeling of a variety of physical systems. Topics will be chosen from the volumes "Mathematics in Industrial Problems" by Avner Friedman. Researchers from local industries will be invited to present real-world applications. Each student will undertake a project in consultation with one of the instructors or an industrial researcher.

598 Topics in Mathematics (1-5 cr.) By arrangement. Directed study and reports for students who wish to undertake individual reading and study on approved topics.

Graduate Level

612 Methods of Applied Mathematics II (3 cr.) P: 611. Continuation of theory of linear integral equations; Sturm-Liouville and Weyl theory for second-order differential operators, distributions in n dimensions, and Fourier transforms.

626 Mathematical Formulation of Physical Problems I (3 cr.) P: Graduate standing and consent of instructor. Topics to be chosen from the following: Tensor formulation of the field equations in continuum mechanics, fluid dynamics, hydrodynamic stability, wave propagation, and theoretical mechanics.

627 Mathematical Formulation of Physical Problems II (3 cr.) P: 626. Continuation of 626.

642 Methods of Linear and Nonlinear Partial Differential Equations (3 cr.) P: 520, 523, and 611. Topics from linear and nonlinear partial differential equations, varied from time to time.

646 Functional Analysis (3 cr.) P: 546. Advanced topics in functional analysis, varying from year to year at the discretion of the instructor.

672 Algebraic Topology I (3 cr.) P: 572. Continuation of 572; cohomology, homotopy groups, fibrations, further topics.

673 Algebraic Topology II (3 cr.) P: 672. Sequel to 672 covering further advanced topics in algebraic and differential topology such as K-theory and characteristic classes.

692 Topics in Applied Mathematics (1-3 cr.)

693 Topics in Analysis (1-3 cr.)

694 Topics in Differential Equations (1-3 cr.)

697 Topics in Topology (1-3 cr.)