IU Indianapolis Campus Bulletin 2024-2025

Undergraduate

Lower-Division

• MATH-I 001 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 IU Indianapolis, but is retained in the catalog for historical purposes, as well as equating transfer credit as needed.

• MATH-I 110 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-I 130, MATH-I 136, and STAT-I 301.

• MATH-I 111 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-I 130, MATH-I 136, MATH-I 153, and STAT-I 301. MATH-I 001 (with a minimum grade of C) or placement.

• MATH-M 118 Finite Mathematics (3 cr.) P: MATH-I 111 or MATH-I 110 (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-I 111 or MATH-I 110 (with a minimum grade of C-) or placement. Sets, limits, derivatives, integrals, and applications.

• MATH-I 123 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.

• MATH-I 130 Mathematics for Elementary Teachers I (3 cr.) P: MATH-I 110 or MATH-I 111 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-I 131 Mathematics for Elementary Teachers II (3 cr.) P: MATH-I 130 or MATH-I 136 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-I 132 Mathematics for Elementary Teachers III (3 cr.) P: MATH-I 130 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-I 136 Mathematics for Elementary Teachers (6 cr.) P: MATH-I 110 or MATH-I 111 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-I 136 is a one-semester version of MATH-I 130 and MATH-I 132. Not open to students with credit in MATH-I 130 or MATH-I 132.

• MATH-I 153 College Algebra (3 cr.) P: MATH-I 111 (not MATH-I 110) taken within last 3 terms with a grade of C or better or an appropriate ALEKS placement score taken within last 12 months. MATH-I 153 / MATH-I 154 is a two-semester version of MATH-I 159. Not open to students with credit in MATH-I 159. This course covers college-level algebra and, together with MATH-I 154, provides preparation for MATH-I 165, MATH-I 221, MATH-I 231, and MATH-I 241.

• MATH-I 154 Trigonometry (3 cr.) P: MATH-I 153 with a grade of C or better taken within the last 3 terms. MATH-I 153 / MATH-I 154 is a two-semester version of MATH-I 159. Not open to students with credit in MATH-I 159. This course covers college-level trigonometry and, together with MATH-I 153, provides preparation for MATH-I 165, MATH-I 221, MATH-I 231, and MATH-I 241.

• MATH-I 159 Precalculus (5 cr.) P: MATH-I 111 (not MATH-I 110) 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-I 159 is a one-semester version of MATH-I 153 / MATH-I 154. Not open to students with credit in MATH-I 153 or MATH-I 154. This course covers college-level algebra and trigonometry and provides preparation for MATH-I 165, MATH-I 221, MATH-I 231, and MATH-I 241.

• MATH-I 165 Analytic Geometry and Calculus I (4 cr.) P: MATH-I 159 (or MATH-I 153 and MATH-I 154) 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: MATH-I 159 or MATH-I 153 and MATH-I 154 and consent of instructor. This course covers the same topics as MATH-I 165. 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-I 166 Analytic Geometry and Calculus II (4 cr.) P: MATH-I 165 taken within the last 3 terms with a grade of C- or better. Continuation of MATH-I 165. 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-I 165 (with a minimum grade of A-), and consent of instructor. This course covers the same topics as MATH-I 166. 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-I 171 Multidimensional Mathematics (3 cr.) P: MATH-I 159 or (MATH-I 153 and MATH-I 154) 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-I 190 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-I 221 Calculus for Technology I (3 cr.) P: MATH-I 159 (or MATH-I 153 and MATH-I 154) 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-I 222 Calculus for Technology II (3 cr.) P: MATH-I 221 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-I 231 Calculus for Life Sciences I (3 cr.) P: MATH-I 159 (or MATH-I 153 and MATH-I 154) taken within the last 3 terms with a grade of C or better or an appropriate ALEKS placement 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-I 232 Calculus for Life Sciences II (3 cr.) P: MATH-I 231 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-I 241 Calculus for Data Science I (3 cr.) P: MATH-I 153 and MATH-I 154 or MATH-I 159, 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-I 242 Calculus for Data Science II (3 cr.) P: MATH-I 241 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, partial derivatives, and multiple integrals.

• MATH-I 243 Linear Algebra for Data Science (3 cr.) P: MATH-I 153 and MATH-I 154 with a grade of C or better or MATH-I 159 with a grade of C or better or an appropriate ALEKS placement score taken within last 12 months. This is the third course in a three-course sequence for data science majors. Topics include: vectors, systems of linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, and applications to data science.

• MATH-I 261 Multivariate Calculus (4 cr.) P: MATH-I 165, MATH-I 166 and MATH-I 171 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-I 166 or MATH-S 166 with a minimum grade of B and MATH-I 171 and permission of the instructor. This is an honors level version of third semester calculus (MATH-I 261). 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-I 261.

• MATH-I 266 Ordinary Differential Equations (3 cr.) P: MATH-I 165, MATH-I 166 and MATH-I 171 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-I 276 Discrete Math (3 cr.) P: or C: MATH-I 165. 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-I 290 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-I 300 Logic and the Foundations of Algebra (3 cr.) P: or C: MATH-I 166 and MATH-I 171. MATH-I 276 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-I 321 Elementary Topology (3 cr.) P: MATH-I 261. 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-I 333 Chaotic Dynamical Systems (3 cr.) P: MATH-I 166 or MATH-I 222 or MATH-I 232. 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-I 351 Elementary Linear Algebra (3 cr.) P: MATH-I 166 and MATH-I 171. Not open to students with credit in MATH-I 511. Systems of linear equations, matrices, vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and applications.

• MATH-I 353 Linear Algebra II with Applications (3 cr.) P: MATH-I 351 or MATH-I 511. 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-I 354 Linear Algebra II for Data Science (3 cr.) P: MATH-I 351 or MATH-I 511 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-I 366 Ordinary Differential Equations (3 cr.) P: Prerequisites: MATH-I 165 and MATH-I 166 and MATH-I 171 with a grade of C or better in each course. C: Corequisite: MATH-I 351. 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-I 373 Financial Mathematics (3 cr.) P: MATH-I 261. 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-I 390 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-I 398 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-I 414 Numerical Methods (3 cr.) P: MATH-I 266 and a course in a high-level programming language. 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-I 421 Linear Programming and Optimization Techniques (3 cr.) P: MATH-I 261 and MATH-I 351. 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-I 423 Discrete Modeling (3 cr.) P: MATH-I 266 and MATH-I 351 or MATH-I 511 or consent of instructor. Linear programming, mathematical modeling of problems in economics, management, urban administration, and the behavioral sciences.

• MATH-I 425 Elements of Complex Analysis (3 cr.) P: MATH-I 261. Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; elementary conformal mapping.

• MATH-I 426 Introduction to Applied Mathematics and Modeling (3 cr.) P: MATH-I 266 and PHYS-I 152. 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-I 444 Foundations of Analysis (3 cr.) P: MATH-I 261 and MATH-I 300. 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-I 445 Foundations of Analysis II (3 cr.) P: MATH-I 444. 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-I 453 Beginning Abstract Algebra (3 cr.) P: MATH-I 351 and MATH-I 300. Basic properties of groups, rings,and fields, with special emphasis on polynomial rings.

• MATH-I 454 Galois Theory (3 cr.) P: MATH-I 453. 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-I 456 Introduction to the Theory of Numbers (3 cr.) P: MATH-I 261. 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-I 462 Elementary Differential Geometry (3 cr.) P: MATH-I 351. 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-I 463 Intermediate Euclidean Geometry for Secondary Teachers (3 cr.) P: MATH-I 300. 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-I 490 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-I 491 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-I 492 Capstone Experience (1-3 cr.) By arrangement. Must submit Course Request Form.

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

• 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-M 301 or MATH-M 303 or MATH-I 351 or MATH-I 511) and (MATH-M 311 or MATH-I 261 or MATH-S 261). 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-M 463 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-I 504 Real Analysis (3 cr.) P: MATH-I 444. 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-I 505 Intermediate Abstract Algebra (3 cr.) P: MATH-I 453. Group theory with emphasis on concrete examples and applications. Field theory: ruler and compass constructions, Galois theory, and solvability of equations by radicals.

• MATH-I 510 Vector Calculus (3 cr.) P: MATH-I 261. 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-I 511 Linear Algebra with Applications (3 cr.) P: MATH-I 261. Not open to students with credit in MATH-I 351. Matrices, rank and inverse of a matrix, decomposition theorems, eigenvectors, unitary and similarity transformations on matrices.

• MATH-I 514 Numerical Analysis (3 cr.) P: MATH-I 266 and MATH-I 351 or MATH-I 511, 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-I 518 Advanced Discrete Mathematics (3 cr.) P: MATH-I 266. 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-I 520 Boundary Value Problems of Differential Equations (3 cr.) P: MATH-I 261 and MATH-I 266. Sturm-Liouville theory, singular boundary conditions, orthogonal expansions, separation of variables in partial differential equations, and spherical harmonics.

• MATH-I 522 Qualitative Theory of Differential Equations (3 cr.) P: MATH-I 266 and MATH-I 351. Nonlinear ODEs, critical points, stability and bifurcations, perturbations, averaging, nonlinear oscillations and chaos, and Hamiltonian systems.

• MATH-I 523 Introduction to Partial Differential Equations (3 cr.) P: MATH-I 266 and MATH-I 261 or MATH-I 510. 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-I 525 Introduction to Complex Analysis (3 cr.) P: MATH-I 261 and MATH-I 266. 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-I 526 Principles of Mathematical Modeling (3 cr.) P: MATH-I 266 and MATH-I 426. 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-I 528 Advanced Mathematics for Engineering and Physics II (3 cr.) P: MATH-I 537. Divergence theorem, Stokes' Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.

• MATH-I 530 Functions of a Complex Variable I (3 cr.) P: or C: MATH-I 544. 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-I 531 Functions of a Complex Variable II (3 cr.) P: MATH-I 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 Riemann surfaces, and Picard theorems.

• MATH-I 535 Theoretical Mechanics (3 cr.) P: MATH-I 266 and MATH-I 351 or MATH-I 511 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-I 537 Applied Mathematics for Scientists and Engineers I (3 cr.) P: MATH-I 261 and MATH-I 266. Covers theories, techniques, and applications of partial differential equations, Fourier transforms, and Laplace transforms. Overall emphasis is on applications to physical problems.

• MATH-I 544 Real Analysis and Measure Theory (3 cr.) P: MATH-I 444. 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-I 545 Principles of Analysis II (3 cr.) P: MATH-I 544. Continues the study of measure theory begun in MATH-I 544.

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

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

• MATH-I 549 Applied Mathematics for Secondary School Teachers (3 cr.) P: MATH-I 266 and MATH-I 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. Course is offered on an as needed basis.

• MATH-I 552 Applied Computational Methods II (3 cr.) P: MATH-I 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.

• MATH-I 553 Introduction to Abstract Algebra (3 cr.) P: MATH-I 453. 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-I 554 Linear Algebra (3 cr.) P: MATH-I 351. 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-I 555 Introduction to Biomathematics (3 cr.) P: MATH-I 266, MATH-I 351 (or MATH-I 511), MATH-I 426, 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-I 559 Applied Computational Methods I (3 cr.) P: MATH-I 266 and MATH-I 351 or MATH-I 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.

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

• MATH-I 562 Introduction to Differential Geometry and Topology (3 cr.) P: MATH-I 351 and MATH-I 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, and Gauss-Bonnet and Poincare theorems on vector fields.

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

• MATH-I 567 Dynamical Systems I (3 cr.) P: MATH-I 545 and MATH-I 571. 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-I 571 Elementary Topology (3 cr.) P: MATH-I 444. Topological spaces, metric spaces, continuity, compactness, connectedness, separation axioms, nets, and function spaces.

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

• MATH-I 574 Mathematical Physics I (1-3 cr.) P: MATH-I 530 and MATH-I 545. 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-I 578 Mathematical Modeling of Physical Systems I (3 cr.) P: MATH-I 266, PHYS-I 152, PHYS-I 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.

• MATH-I 581 Introduction to Logic for Teachers (3 cr.) P: MATH-I 351. 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-I 583 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-I 585 Mathematical Logic I (3 cr.) P: MATH-I 351 or an undergraduate proof course; MATH-I 587 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-I 587 General Set Theory (3 cr.) P: MATH-I 351 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-I 588 Mathematical Modeling of Physical Systems II (3 cr.) P: MATH-I 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.

• MATH-I 598 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-I 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.

• MATH-I 612 Methods of Applied Mathematics II (3 cr.) P: MATH-I 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.

• MATH-I 626 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-I 627 Mathematical Formulation of Physical Problems II (3 cr.) P: MATH-I 626. 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-I 626.

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

• MATH-I 667 Dynamical Systems II (3 cr.) P: MATH-I 567. Continuation of MATH-I 567. Topics in dynamics.

• MATH-I 672 Algebraic Topology I (3 cr.) P: MATH-I 572. Continuation of MATH-I 572. Cohomology, homotopy groups, fibrations, and further topics.

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

• MATH-I 674 Mathematical Physics II (3 cr.) P: MATH-I 574. Continuation of MATH-I 574 Mathematical Physics I. Students will learn more advanced notions and theorems of various mathematical theories that have direct applications to physics.

• MATH-I 692 Topics in Applied Mathematics (1-3 cr.) Research topics of current interest in applied mathematics to be chosen by the instructor.

• MATH-I 693 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-I 694 Topics in Differential Equations (1-3 cr.) P: MATH-I 554 and MATH-I 530. Department consent required. Research topics in differential equations related to physics and engineering. Topics of current interest will be chosen by the instructor.

• MATH-I 697 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-I 699 Research Ph.D. Thesis (variable cr.)

• MATH-I 698 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.