IUPUI Campus Bulletin 2023-2024

Undergraduate

Lower-Division

• MATH 00100 Introduction to Algebra (4 cr.) Covers the material taught 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. This course is no longer offered at IUPUI.

• MATH 11000 Fundamentals of Algebra (4 cr.) Intended primarily for liberal arts and business majors. Integers, rational and real numbers, exponents, decimals, polynomials, equations, word problems, factoring, roots and radicals, logarithms, quadratic equations, graphing, linear equations in more than one variable, and inequalities. This course satisfies the prerequisites needed for MATH-M 118, MATH-M 119, MATH 13000, MATH 13600, and STAT 30100.

• MATH 11100 Algebra (4 cr.) Real numbers, linear equations and inequalities, systems of equations, polynomials, exponents, and logarithmic functions. Covers material in the second year of high school algebra. This course satisfies the prerequisites needed for MATH-M 118, MATH-M 119, MATH 13000, MATH 13600, MATH 15300, MATH 15400, and STAT 30100. MATH 00100 (with a minimum grade of C) or placement.

• MATH-M 118 Finite Mathematics (3 cr.) P: MATH 11100 or MATH 11000 (with a minimum grade of C-) or placement. Set theory, logic, permutations, combinations, simple probability, conditional probability, Markov chains.

• MATH-M 119 Brief Survey of Calculus I (3 cr.) P: MATH 11100 or MATH 11000 (with a minimum grade of C-) or placement. Sets, limits, derivatives, integrals, and applications.

• MATH 12300 Elementary Concepts of Mathematics (3 cr.) 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. This course is not offered through IUPUI.

• MATH 13000 Mathematics for Elementary Teachers I (3 cr.) P: MATH 11000 or MATH 11100 taken within the last 3 terms with a grade of C- or better or an appropriate ALEKS placement score taken within last 12 months. Numeration systems, mathematical reasoning, integers, rationals, reals, properties of number systems, decimal and fractional notations, and problem solving.

• MATH 13100 Mathematics for Elementary Teachers II (3 cr.) P: MATH 13000 or MATH 13600 taken within the last 3 terms with a grade of C- or better or an appropriate ALEKS placement score taken within last 12 months. Number systems: numbers of arithmetic, integers, rationals, reals, mathematical systems, decimal and fractional notations; probability, simple and compound events, algebra review.

• MATH 13200 Mathematics for Elementary Teachers III (3 cr.) P: MATH 13000 taken within the last 3 terms with a minimum grade of C- or better or an appropriate ALEKS placement score taken within last 12 months. Rationals, reals, geometric relationships, properties of geometric figures, one-, two-, and three-dimensional measurement, and problem solving.

• MATH 13600 Mathematics for Elementary Teachers (6 cr.) P: MATH 11000 or MATH 11100 taken within the last 3 terms with a grade of C- or better or an appropriate ALEKS placement score taken within last 12 months. MATH 13600 is a one-semester version of MATH 13000 and MATH 13200. Not open to students with credit in MATH 13000 or MATH 13200.

• MATH 15300 College Algebra (3 cr.) P: MATH 11100 (not MATH 11000) taken within last 3 terms with a grade of C or better or an appropriate ALEKS placement score taken within last 12 months. MATH 15300-15400 is a two-semester version of MATH 15900. Not open to students with credit in MATH 15900. This course covers college-level algebra and, together with MATH 15400, provides preparation for MATH 16500, MATH 22100, and MATH 23100.

• MATH 15400 Trigonometry (3 cr.) P: MATH 15300 with a grade of C or better taken within the last 3 terms. MATH 15300-15400 is a two-semester version of MATH 15900. Not open to students with credit in MATH 15900. This course covers college-level trigonometry and, together with MATH 15300, provides preparation for MATH 16500, MATH 22100, and MATH 23100.

• MATH 15900 Precalculus (5 cr.) P: MATH 11100 (not MATH 11000) taken within the last 3 terms with a grade of B or better or an appropriate ALEKS placement score taken within the last 12 months.

  MATH 15900 is a one-semester version of MATH 15300-15400. Not open to students with credit in MATH 15300 or MATH 15400. This course covers college-level algebra and trigonometry and provides preparation for MATH 16500, MATH 22100, and MATH 23100.

• MATH 16500 Analytic Geometry and Calculus I (4 cr.) P: MATH 15900 (or MATH 15300 and MATH 15400) taken within the last 3 terms with a grade of C or better or an appropriate ALEKS placement score taken within last 12 months. Introduction to differential and integral calculus of one variable, with applications.

• MATH-S 165 Honors Analytic Geometry and Calculus I (4 cr.) P: Precalculus or trigonometry and consent of instructor. This course covers the same topics as MATH 16500. However, it is intended for students having a strong background in mathematics who wish to study the concepts of calculus in more depth and who are seeking mathematical challenge.

• MATH 16600 Analytic Geometry and Calculus II (4 cr.) P: MATH 16500 taken within the last 3 terms with a grade of C- or better. Continuation of MATH 16500. Inverse functions, exponential, logarithmic, and inverse trigonometric functions. Techniques of integration, applications of integration, differential equations, and infinite series.

• MATH-S 166 Honors Analytic Geometry and Calculus II (4 cr.) P: MATH-S 165 (with a minimum grade of B-) or MATH 16500 (with a minimum grade of A-), and consent of instructor. This course covers the same topics as MATH 16600. However, it is intended for students having a strong interest in mathematics who wish to study the concepts of calculus in more depth and who are seeking mathematical challenge.

• MATH 17100 Multidimensional Mathematics (3 cr.) P: MATH 15900 or (MATH 15300 and MATH 15400) taken within the last 3 terms with a grade of C or better or an appropriate ALEKS score taken within last 12 months. An introduction to mathematics in more than two dimensions. Graphing of curves, surfaces and functions in three dimensions. Two and three dimensional vector spaces with vector operations. Solving systems of linear equations using matrices. Basic matrix operations and determinants.

• MATH 19000 Topics in Mathematics for First Year Students (1 - 3 cr.) P: Prerequisites and course material vary with the topics. Treats topics in mathematics at the freshman level.

• MATH 22100 Calculus for Technology I (3 cr.) P: MATH 15900 (or MATH 15300 and MATH 15400) taken within last 3 terms with a grade of C or better or an appropriate ALEKS score taken within last 12 months. Analytic geometry, the derivative and applications, and the integral and applications.

• MATH 22200 Calculus for Technology II (3 cr.) P: MATH 22100 or equivalent taken within the last 3 terms with a grade of C- or better. Differentiation of transcendental functions, methods of integration, power series, Fourier series, and differential equations.

• MATH 23100 Calculus for Life Sciences I (3 cr.) P: MATH 15900 (or MATH 15300 and MATH 15400) taken within last 3 terms with a grade of C or better or an appropriate ALEKS score taken within last 12 months. Limits, derivatives and applications. Exponential and logarithmic functions. Integrals, antiderivatives, and the Fundamental Theorem of Calculus. Examples and applications are drawn from the life sciences.

• MATH 23200 Calculus for Life Sciences II (3 cr.) P: MATH 23100 or equivalent taken within the last 3 terms with a grade of C- or better. Matrices, functions of several variables, differential equations and solutions with applications. Examples and applications are drawn from the life sciences.

• MATH 24100 Calculus for Data Science I (3 cr.) P: College algebra and trigonometry, or pre-calculus, with a grade of C or better, taken within the past 12 months; or a recent proficiency/placement test indicating placement into a trigonometry-based calculus course This is the first course in a three course sequence for data science majors.  Topics include: functions, limits, epsilon-delta argument, differentiation and applications to data science, anti-derivatives, Fundamental Theorem of Calculus, introduction to integration, and inverse functions.

• MATH 24200 Calculus for Data Science II (3 cr.) P: MATH 24100 or equivalent Calculus I course with a grade of C- or better. This is the second course in a three course sequence for data science majors. Topics include: transcendental functions, techniques of integration, improper integrals, applications to data science, probability and expected value, introduction to differential equations, infinite series and power series, and partial derivatives.

• MATH 24300 Linear Algebra for Data Science (3 cr.) P: MATH 24200 or equivalent Calculus II course with a grade of C- or better. This is the third course in a three course sequence for data science majors. Topics include: vectors, multiple integrals, systems of linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, and applications to data science.

• MATH 26100 Multivariate Calculus (4 cr.) P: MATH 16500, MATH 16600 and MATH 17100 taken within the last 3 terms with grades of C- or better. Spatial analytic geometry, vectors, space curves, partial differentiation, applications, multiple integration, vector fields, line integrals, Green's theorem, Stokes' theorem, and the Divergence Theorem. An honors option may be available in this course.

• MATH-S 261 Honors Multivariate Calculus (4 cr.) P: MATH 16600 or MATH-S 166 with a minimum grade of B and MATH 17100 and permission of the instructor. This is an honors level version of third semester calculus (MATH 26100). It is intended for students who have strong motivation and a desire for additional challenge. The theory of multivariate calculus is developed as rigorously as possible and studied in greater depth than in MATH 26100.

• MATH 26600 Ordinary Differential Equations (3 cr.) P: MATH 16500, MATH 16600 and MATH 17100 taken within the last 3 terms with grades of C- or better. First order equations, second and n-th order linear equations, series solutions, solution by Laplace transform, systems of linear equations.

• MATH 27600 Discrete Math (3 cr.) P: or C: MATH 16500. Logic, sets, functions, integer algorithms, applications of number theory, mathematical induction, recurrence relations, permutations, combinations, finite probability, relations and partial ordering, and graph algorithms.

• MATH 29000 Topics in Mathematics for Sophomores (1 - 3 cr.) P: Prerequisites and course material vary with the topics. Treats topics in mathematics at the sophomore level.

Upper-Division

• MATH 30000 Logic and the Foundations of Algebra (3 cr.) P: or C: MATH 16600 and MATH 17100. MATH 27600 is recommended. Logic and the rules of reasoning, theorem proving. Applications to the study of the integers; rational, real, and complex numbers; and polynomials. Bridges the gap between elementary and advanced courses. This is a prerequisite for 300-level and 400-level pure mathematics courses.

• MATH 32101 Elementary Topology (3 cr.) P: MATH 26100. Introduction to topology, including metric spaces, abstract topological spaces, continuous functions, connectedness, compactness, curves, Cantor sets, continua, and the Baire Category Theorem. Also, an introduction to surfaces, including spheres, tori, the Mobius band, the Klein bottle and a description of their classification.

• MATH 33300 Chaotic Dynamical Systems (3 cr.) P: MATH 16600 or MATH 22200 or MATH 23200. The goal of the course is to introduce some of the spectacular new discoveries that have been made in the past twenty years in the field of mathematics known as dynamical systems. It is intended for undergraduate students in mathematics, science, or engineering. It will include a variety of computer experiments using software that is posted on the Web.

• MATH 35100 Elementary Linear Algebra (3 cr.) P: MATH 16600 and MATH 17100. Not open to students with credit in MATH 51100. Systems of linear equations, matrices, vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and applications.

• MATH 35300 Linear Algebra II with Applications (3 cr.) P: MATH 35100 or MATH 51100. This course involves the development of mathematics with theorems and their proofs. This course also includes several important applications, which will be used to create a mathematical model, prove theorems that lead to the solution of problems in the model, and interpret the results in terms of the original problem.

• MATH 35400 Linear Algebra II for Data Science (3 cr.) P: MATH 35100 or MATH 51100 or consent of instructor. In this course, we will explore a number of contemporary applications of linear algebra (all of which have arisen since the dawn of the Internet Age and most are still under development) in information retrieval, website ranking, text processing, community detection, pattern recognition, and recommender systems for e-commerce, all largely based on matrix factorizations, that should be of interest to students in pure and applied mathematics, actuarial science, computer & information science, and engineering.

• MATH 37300 Financial Mathematics (3 cr.) P: MATH 26100. Fundamental concepts of financial mathematics and economics, and their application to business situations and risk management. Valuing investments, capital budgeting, valuing contingent cash flows, modified duration, convexity, immunization, financial derivatives. Provides preparation for the SOA/CAS Exam FM/2.

• MATH 39000 Topics in Mathematics for Juniors (1 - 3 cr.) P: Prerequisites and course material vary with the topics. Treats topics in mathematics at the junior level.

• MATH 39800 Internship in Professional Practice (0-3 cr.) P: Approval of Department of Mathematical Sciences. Professional work experience involving significant use of mathematics or statistics. Evaluation of performance by employer and Department of Mathematical Sciences. May count toward major requirements with approval of the Department of Mathematical Sciences for a total of 6 credits.

• MATH 41400 Numerical Methods (3 cr.) P: MATH 26600 and a course in a high-level programming language. Not open to students with credit in CSCI 51200. Error analysis, solution of nonlinear equations, direct and iterative methods for solving linear systems, approximation of functions, numerical differentiation and integration, and numerical solution of ordinary differential equations.

• MATH 42100 Linear Programming and Optimization Techniques (3 cr.) P: MATH 26100 and MATH 35100. This course covers a variety of topics in operations research, including solution of linear programming problems by the simplex method, duality theory, transportation problems, assignment problems, network analysis, dynamic programming.

• MATH 42300 Discrete Modeling (3 cr.) P: MATH 26600 and MATH 35100 or MATH 51100 or consent of instructor. Linear programming, mathematical modeling of problems in economics, management, urban administration, and the behavioral sciences.

• MATH 42500 Elements of Complex Analysis (3 cr.) P: MATH 26100. Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; elementary conformal mapping.

• MATH 42600 Introduction to Applied Mathematics and Modeling (3 cr.) P: MATH 26600 and PHYS 15200. Introduction to problems and methods in applied mathematics and modeling. Formulation of models for phenomena in science and engineering, their solutions, and physical interpretation of results. Examples chosen from solid and fluid mechanics, mechanical systems, diffusion phenomena, traffic flow, and biological processes.

• MATH 44400 Foundations of Analysis (3 cr.) P: MATH 26100, MATH 30000. 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.

• MATH 44500 Foundations of Analysis II (3 cr.) P: MATH 44400. 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, and series of functions.

• MATH 45300 Beginning Abstract Algebra (3 cr.) P: MATH 35100, MATH 30000. Basic properties of groups, rings,and fields, with special emphasis on polynomial rings.

• MATH 45400 Galois Theory (3 cr.) P: MATH 45300. An introduction to Galois Theory, covering both its origins in the theory of roots of polynomial equation and its modern formulation in terms of abstract algebra. Topics include field extensions and their symmetries, ruler and compass constructions, solvable groups, and the solvability of polynomial equations by radical operations.

• MATH 45600 Introduction to the Theory of Numbers (3 cr.) P: MATH 26100. Divisibility, congruences, quadratic residues, Diophantine equations, and the sequence of primes.

• EDUC-M 457 Methods of Teaching Senior High/Junior High/Middle School Mathematics (3 cr.) P: 30 credit hours of mathematics. Study of methodology, heuristics of problem solving, curriculum design, instructional computing, professional affiliations, and teaching of daily lessons in the domain of secondary and/or junior high/ middle school mathematics.

• MATH 46200 Elementary Differential Geometry (3 cr.) P: MATH 35100. Calculus and linear algebra applied to the study of curves and surfaces. Curvature and torsion, Frenet-Serret apparatus and theorem, and fundamental theorem of curves. Transformation of R2, first and second fundamental forms of surfaces, geodesics, parallel translation, isometries, and fundamental theorem of surfaces.

• MATH 46300 Intermediate Euclidean Geometry for Secondary Teachers (3 cr.) P: MATH 30000. 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. Discussion of the teaching of Euclidean geometry.

• MATH 49000 Topics in Mathematics for Undergraduates (1-5 cr.) P: By arrangement. Open to students only with the consent of the department. Supervised reading and reports in various fields.

• MATH 49100 Seminar in Competitive Math Problem-Solving (1-3 cr.) P: Approval of the director of undergraduate programs is required. This seminar is designed to prepare students for various national and regional mathematics contests and examinations such as the Putnam Mathematical Competition, the Indiana College Mathematical Competition and the Mathematical Contest in Modeling (MCM), among others. May be repeated twice for credit.

• MATH 49200 Capstone Experience (1-3 cr.) By arrangement. Must submit Course Request Form.

• MATH 49500 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 classroom setting under the supervision of an experienced teacher who critiques the presentations.

• MATH 36601 Ordinary Differential Equations (3 cr.) P: Prerequisites: MATH 16500 and MATH 16600 and MATH 17100 with a grade of C or better in each course. C: Corequisite: MATH 35100. Introduction to differential equations for students majoring in Mathematics.  Ordinary differential equations, first and second order equations, linear systems, series solutions, existence and uniqueness, numerical methods, applications to physical problems.  Will be required of majors in Pure and Applied Mathematics Concentrations starting Fall 2024.

• MATH-M 303 Linear Algebra for Undergraduates (3 cr.) P: MATH-M 230 or MATH-M 216 or equivalent. Introduction to the theory of real vector spaces. Coordinate systems, linear dependence, bases. Linear transformations and matrix calculus. Determinants and rank. Eigenvalues and eigenvectors.

• MATH-M 463 Introduction to Probability I (3 cr.) P: (MATH-M301 or MATH-M303 or MATH 35100 or MATH 51100) and (MATH-M311 or MATH 26100 or MATH-S261). Counting techniques, the meaning of probability. Random experiments, conditional probability, independence. Random variables, expected values and standard deviations, moment generating functions, important discrete and continuous distributions. Poisson processes. Multivariate distributions, basic limit laws such as the central limit theorem.

• MATH-M 466 Introduction to Mathematical Statistics (3 cr.) P: MATH-M463 or consent of instructor. Rigorous mathematical treatment of problems in sampling and statistical inference. Method of maximum likelihood, efficiency, sufficient statistics, exponential family distributions, likelihood ratio tests, most powerful tests, minimum variance unbiased estimators, shortest confidence intervals, linear models and analysis of variance, nonparametric methods.

Advanced Undergraduate and Graduate

• MATH 50400 Real Analysis (3 cr.) P: MATH 44400 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, and the Arzela-Ascoli theorem.

• MATH 50500 Intermediate Abstract Algebra (3 cr.) P: 45300. Group theory with emphasis on concrete examples and applications. Field theory: ruler and compass constructions, Galois theory, and solvability of equations by radicals.

• MATH 51000 Vector Calculus (3 cr.) P: 26100. Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green's theorem, Stokes's theorem, divergence theorems, and applications to engineering and the physical sciences.

• MATH 51100 Linear Algebra with Applications (3 cr.) P: 26100. Not open to students with credit in MATH 35100. Matrices, rank and inverse of a matrix, decomposition theorems, eigenvectors, unitary and similarity transformations on matrices.

• MATH 51400 Numerical Analysis (3 cr.) P: MATH 26600 and MATH 35100 or MATH 51100, or consent of instructor and familiarity with one of the high-level programming languages: Fortran 77/90/95, C, C++, Matlab. Numerical Analysis is concerned with finding numerical solutions to problems, especially those for which analytical solutions do not exist or are not readily obtainable.  This course provides an introduction to the subject and treats the topics of approximating functions by polynomials, solving linear systems of equations, and of solving nonlinear equations.  These topics are of great practical importance in science, engineering and finance, and also have intrinsic mathematical interest.  The course concentrates on theoretical analysis and on the development of practical algorithms.

• MATH 51800 Advanced Discrete Mathematics (3 cr.) P: 26600. This course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods.

• MATH 52000 Boundary Value Problems of Differential Equations (3 cr.) P: 26100 and 26600. Sturm-Liouville theory, singular boundary conditions, orthogonal expansions, separation of variables in partial differential equations, and spherical harmonics.

• MATH 52200 Qualitative Theory of Differential Equations (3 cr.) P: 26600 and 35100. Nonlinear ODEs, critical points, stability and bifurcations, perturbations, averaging, nonlinear oscillations and chaos, and Hamiltonian systems.

• MATH 52300 Introduction to Partial Differential Equations (3 cr.) P: 26600 and 26100 or 51000. 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, and Poisson integral method for elliptic equations.

• MATH 52500 Introduction to Complex Analysis (3 cr.) P: MATH 26100 and MATH 26600. Instructor consent required for any undergraduate student. Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; and elementary conformal mapping.

• MATH 52600 Principles of Mathematical Modeling (3 cr.) P: MATH 26600 and MATH 42600 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.

• MATH 52800 Advanced Mathematics for Engineering and Physics II (3 cr.) P: MATH 53700 Divergence theorem, Stokes' Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.

• MATH 53000 Functions of a Complex Variable I (3 cr.) P or C: MATH 54400. Complex numbers, holomorphic functions, harmonic functions, and linear transformations. Power series, elementary functions, Riemann surfaces, contour integration, Cauchy's theorem, Taylor and Laurent series, and residues. Maximum and argument principles. Special topics.

• MATH 53100 Functions of a Complex Variable II (3 cr.) P: MATH 53000 Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge's theorem, Mittag-Leffler theorem, analytic continuation and Riemann surfaces, and Picard theorems.

• MATH 53500 Theoretical Mechanics (3 cr.) P: MATH 26600 and MATH 35100 or MATH 51100 or consent of the instructor. Continuum mechanics deals with the analysis of the motion of materials modeled as a continuous mass rather than as discrete particles. Applications of continuum mechanics are ubiquitous in science and engineering, and are getting more and more popular in medicine too. The goal of this course is to study the basic principles of continuum mechanics for deformable bodies, including conservation laws and constitutive equations, while discussing the mathematical challenges in solving these equations analytically and/or numerically.

• MATH 53700 Applied Mathematics for Scientists and Engineers I (3 cr.) P: MATH 26100 and MATH 26600. Covers theories, techniques, and applications of partial differential equations, Fourier transforms, and Laplace transforms. Overall emphasis is on applications to physical problems.

• MATH 54400 Real Analysis and Measure Theory (3 cr.) P: MATH 44400 Algebras of sets, real number system, Lebesgue measure, measurable functions, Lebesgue integration, differentiation, absolute continuity, Banach spaces, metric spaces, general measure and integration theory, and Riesz representation theorem.

• MATH 54500 Principles of Analysis II (3 cr.) P: MATH 54400 Continues the study of measure theory begun in 54400.

• MATH 54600 Introduction to Functional Analysis (3 cr.) P: MATH 54500 Banach spaces, Hahn-Banach theorem, uniform boundedness principle, closed graph theorem, open mapping theorem, weak topology, and Hilbert spaces.

• MATH 54700 Analysis for Teachers I (3 cr.) P: MATH 26100 Set theory, logic, relations, functions, Cauchy's inequality, metric spaces, neighborhoods, and Cauchy sequence.

• MATH 54900 Applied Mathematics for Secondary School Teachers (3 cr.) P: MATH 26600 and MATH 35100. 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. Course is offered on an as needed basis.

• MATH 55200 Applied Computational Methods II (3 cr.) P: 55900 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.

• MATH 55300 Introduction to Abstract Algebra (3 cr.) P: 45300 Group theory: finite abelian groups, symmetric groups, Sylow theorems, solvable groups, Jordan-Holder theorem. Ring theory: prime and maximal ideals, unique factorization rings, principal ideal domains, Euclidean rings, and factorization in polynomial and Euclidean rings. Field theory: finite fields, Galois theory, and solvability by radicals.

• MATH 55400 Linear Algebra (3 cr.) P: 35100. Review of basics: vector spaces, dimension, linear maps, matrices, determinants, and linear equations. Bilinear forms, inner product spaces, spectral theory, and eigenvalues. Modules over principal ideal domain, finitely generated abelian groups, and Jordan and rational canonical forms for a linear transformation.

• MATH 55555 Introduction to Biomathematics (3 cr.) P: MATH 26600, MATH 35100 (or MATH 51100), MATH 42600, or consent of instructor. The class will explore how mathematical methods can be applied to study problems in life-sciences. No prior knowledge of life-sciences is required. Wide areas of mathematical biology will be covered at an introductory level. Several selected topics, such as dynamical systems and partial differential equations in neuroscience and physiology, and mathematical modeling of biological flows and tissues, will be explored in depth.

• MATH 55900 Applied Computational Methods I (3 cr.) P: 26600 and 35100 or 51100. 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.

• MATH 56100 Projective Geometry (3 cr.) P: 35100. Projective invariants, Desargues' theorem, cross-ratio, axiomatic foundation, duality, consistency, independence, coordinates, and conics.

• MATH 56200 Introduction to Differential Geometry and Topology (3 cr.) P: 35100 and 44500. 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, and Gauss-Bonnet and Poincare theorems on vector fields.

• MATH 56300 Advanced Geometry (3 cr.) P: 30000 or consent of instructor. Topics in Euclidean and non-Euclidean geometry.

• MATH 56700 Dynamical Systems I (3 cr.) P: 54500 and 57100. Covers the basic notions and theorems of the theory of dynamical systems and their connections with other branches of mathematics. Topics covered include fundamental concepts and examples, one-dimensional systems, symbolic dynamics, topological entropy, hyperbolicity, structural stability, bifurcations, invariant measures, and ergodicity.

• MATH 57100 Elementary Topology (3 cr.) P: 44400. Topological spaces, metric spaces, continuity, compactness, connectedness, separation axioms, nets, and function spaces.

• MATH 57200 Introduction to Algebraic Topology (3 cr.) P: 57100. Singular homology theory, Ellenberg-Steenrod axioms, simplicial and cell complexes, elementary homotopy theory, and Lefschetz fixed point theorem.

• MATH 57400 Mathematical Physics I (1-3 cr.) P: 53000 and 54500. Covers the basic concepts and theorems of mathematical theories that have direct applications to physics. Topics to be covered include special functions ODEs and PDEs of mathematical physics, groups and manifolds, mathematical foundations of statistical physics.

• MATH 57800 Mathematical Modeling of Physical Systems I (3 cr.) P: 26600, PHYS 15200, PHYS 25100, 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.

• MATH 58100 Introduction to Logic for Teachers (3 cr.) P: 35100. Logical connectives, rules of sentential inference, quantifiers, bound and free variables, rules of inference, interpretations and validity, theorems in group theory, and introduction to set theory.

• MATH 58300 History of Elementary Mathematics (3 cr.) P: 26100. 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.

• MATH 58500 Mathematical Logic I (3 cr.) P: MATH 35100 or an undergraduate proof course; MATH 58700 recommended. Formal theories for propositional and predicate calculus with study of models, completeness, and compactness. Formalization of elementary number theory; Turing machines, halting problem, and the undecidability of arithmetic.

• MATH 58700 General Set Theory (3 cr.) P: MATH 35100 or equivalent proof course in Linear Algebra. An introduction to set theory, including both so-called "naive" and formal approaches, leading to a careful development using the Zermelo-Fraenkel axioms for set theory and an in-depth discussion of cardinal and ordinal numbers, the Axiom of Choice, and the Continuum Hypothesis.

• MATH 58800 Mathematical Modeling of Physical Systems II (3 cr.) P: 57800. 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.

• MATH 59800 Topics in Mathematics (1 - 6 cr.) By arrangement. Directed study and reports for students who wish to undertake individual reading and study on approved topics.

Graduate

• MATH 61100 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.

• MATH 61200 Methods of Applied Mathematics II (3 cr.) P: 61100. Continuation of theory of linear integral equations; Sturm-Liouville and Weyl theory for second-order differential operators, distributions in n dimensions, and Fourier transforms.

• MATH 62600 Mathematical Formulation of Physical Problems I (3 cr.) P: Advanced calculus or vector calculus, partial differential equations, linear algebra. Nature of applied mathematics, deterministic systems and ordinary differential equations, random processes and partial differential equations, Fourier analysis, dimensional analysis and scaling.

• MATH 62700 Mathematical Formulation of Physical Problems II (3 cr.) P: MATH 62600 Theories of continuous fields, continuous medium, field equations of continuum mechanics, inviscid fluid flow, viscous flow, turbulence. Additional topics to be discussed include application of the theory of dynamical systems, methods for analysis of nonlinear ordinary and partial differential equations, and others. This course is an advancement of topics covered in MATH 62600.

• MATH 64600 Functional Analysis (3 cr.) P: 54600. Advanced topics in functional analysis, varying from year to year at the discretion of the instructor.

• MATH 66700 Dynamical Systems II (3 cr.) P: 56700. Topics in dynamics. Continuation of MATH 56700.

• MATH 67200 Algebraic Topology I (3 cr.) P: 57200. Continuation of 57200; cohomology, homotopy groups, fibrations, and further topics.

• MATH 67300 Algebraic Topology II (3 cr.) P: 67200. continuation of 67200, covering further advanced topics in algebraic and differential topology such as K-theory and characteristic classes.

• MATH 67400 Mathematical Physics II (3 cr.) P: 57400. MATH 67400 is a continuation of MATH 57400, Mathematical Physics I. Students should learn more advanced notions and theorems of various mathematical theories that have direct applications to physics.

• MATH 69200 Topics in Applied Mathematics (1-3 cr.) Research topics of current interest in applied mathematics to be chosen by the instructor.

• MATH 69300 Topics in Analysis (1-3 cr.) P: Department consent required. Research topics in analysis and their relationships to other branches of mathematics. Topics of current interest will be chosen by the instructor.

• MATH 69400 Topics in Differential Equations (1-3 cr.) P: MATH 55400 and MATH 53000. Department consent required. Research topics in differential equations related to physics and engineering. Topics of current interest will be chosen by the instructor.

• MATH 69700 Topics in Topology (1-3 cr.) Research topics in topology and their relationships to other branches of mathematics. Topics of current interest will be chosen by the instructor.

• MATH 69900 Research Ph.D. Thesis (Arr. cr.)

• MATH 69800 Research M.S. Thesis (1 - 6 cr.) Students conduct original research under the direction of a member of the graduate faculty leading to a Masters Thesis. This course is eligible for a deferred grade. Course may be repeated for credit.