Note: 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 Master Schedule. Equiv. - course is equivalent to the indicated course taught at Indiana University Bloomington, or the indicated course taught at Purdue University, West Lafayette. For Statistics Courses, please click here.

Undergraduate and Graduate Level

504 Real Analysis (3 cr.) P: 444 or consent of the instructor. Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, the Arzela-Ascoli theorem.

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. Spring, Summer. 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. Fall, Spring, Summer. Not open to students with credit in 351. Matrices, rank and inverse of a matrix, decomposition theorems, eigenvectors, unitary and similarity transformations on matrices.

517 Discrete Mathematics 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.

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. Spring. 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. Fall. Nonlinear ODEs, critical points, stability and bifurcations, perturbations, averaging, nonlinear oscillations and chaos, Hamiltonian systems.

523 Introduction to Partial Differential Equations (3 cr.) P: 262 and 510, or consent of instructor. Fall. 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. Fall. 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 R: 511. Linear algebra, systems of ordinary differential equations, Laplace transforms, Fourier series and transforms, and partial differential equations.

528 Advanced Mathematics for Engineering and Physics II(3 cr.) P:527. Divergence theorem, Stokes' Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.

530 Functions of a Complex Variable I (3 cr.) P or C: 544. Spring. 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: 444 or consent of instructor. Fall. 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. Spring. 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. 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.

553 Introduction to Abstract Algebra (3 cr.) P: 453 or consent of instructor. Fall. 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. Spring. 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 systems of nonlinear equations.

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 445. 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.

567Dynamical Systems (3 cr.) P: 545, 571 Fundamental concepts and examples, one-dimensional systems, symbolic dynamics, topological entropy, hyperbolicity, structural stability, bifurcations, invariant measures, ergodicity.

571 Elementary Topology (3 cr.) P: 444. Fall. Fundamentals of point set topology with a brief introduction to the fundamental group and related topics; topological and metric spaces; compactness and connectedness; separation properties; local compactness; introduction to function spaces; basic notions involving deformations of continuous paths.

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.

574 Mathematical Physics (3 cr.) P: 545 Topics in special functions, representation theory, spectral theory, modern differential geometry and topology, rigorous results in statistical physics.

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

611 Methods of Applied Mathematics I (3 cr.) P: consent of instructor. Introduction to Banach and Hilbert spaces, linear integral equations with Hilbert-Schmidt kernels, eigenfunction expansions, and Fourier transforms.

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.

667 Dynamical Systems II (3 cr.) P: 567 Topics in dynamics. Continuation of Math 567.

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.

674 Mathematical Physics II (3 cr.) P: 574 Topics in mathematical physics. Continuation of Math 574.

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.)