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MathematicsFaculty FacultyChairpersonProfessor James Davis College ProfessorRoger Temam William H Boucher ProfessorVladimir Touraev ProfessorsEric Bedford, Grahame Bennett, Hari Bercovici, Richard Bradley, Jiri Dadok, James Davis, Vinay Deodhar, Allan Edmonds, Robert Glassey, Darrell Haile, David Hoff, Michael Jolly, Nets Katz, Paul Kirk, Jee Koh, Michael Larsen, Charles Livingston, Valery Lunts, Russell Lyons, Larry Moss, Kent Orr, Sergey Pinchuk, Jacob Rubinstein, Peter Sternberg, Alberto Torchinsky, Shouhong Wang, Kevin Zumbrun Associate ProfessorsScott Brown, Mihai Ciucu, Marlies Gerber, Elizabeth Housworth, Christopher Judge, Ayelet Lindenstrauss, Michael Mandell, Kevin Pilgrim, Ji-Ping Sha, Bruce Solomon, Matthias Weber, William Wheeler Assistant ProfessorsChristopher Connell, Ciprian Demeter, David Fisher, Patricia Hersh Senior LecturersLinda McKinley, Greg Peters LecturersJeremy Boggess, Andrew Dabrowski, Greg Kattner, Norm Levenberg, Steve McKinley, Will Orrick, Kimberley Polly, John Steele, Tracy Whelan Professors EmeritiGoro Azumaya, Thomas Bagby, Rabi Bhattacharya, John Brothers, Arlen Brown, John Challifour, Ciprian Foias, Victor Goodman, Jan Jaworowski, Andrew Lenard, Morton Lowengrub, Robert MacKenzie, Daniel Maki, Madan Puri, Billy Rhoades, Joseph Stampfli, Maynard Thompson, William Ziemer Academic AdvisingElizabeth Smith, Rawles Hall 115, (812) 855-3171 IntroductionMathematics explores patterns in numbers, space, nature, music, science, and art. Its precise language facilitates qualitative and quantitative analysis of these patterns, and often connects them with completely unexpected areas of application. A math degree prepares a student for a mathematical research career or for graduate or professional school in statistics, medicine, law, or the natural sciences. Mathematicians work in actuarial science, in education, and in a wide array of government and business-related organizations which seek out creative and disciplined analytic thinkers. Mathematics underpins the natural and life sciences, economics, and computer science, and affords dynamic double-degree opportunities. The honors course S212 is available for students of outstanding ability (credit for M211 and consent of department required). Particularly well-prepared students may take the accelerated course M213, which covers the material of M211 and M212 in one semester (consent of department required). Advanced Placement (AP) CreditHigh school students who have taken the CEEB Advanced Placement Program mathematics examinations will be awarded credit as follows: A score of 4 or 5 on the Calculus AB exam earns a student 4 credits of M211. A score of 4 or 5 on the Calculus BC exam earns a student 4 credits of M211 and 4 credits of M212. Special credit will be awarded with the grade of S. Credit by ExaminationCredit for M118, M119, M211, M212, and certain other courses may be obtained by passing an examination administered by the mathematics department with a grade of at least a B. Special credit will be awarded with the grade of S. Major in Mathematics—B.A.RequirementsMajors must complete the following:
For any mathematics course not listed above, the department will determine whether it will count toward the conditions in number 3. At most, one course from mathematics education and history may count toward the five required courses. With approval of the Department of Mathematics, one course outside of mathematics that has significant mathematical content may count toward the five required courses as long as conditions in number 3 are still satisfied. Computer Science A201 or C211 is recommended. RecommendationsIn addition to studying mathematics courses, all majors are strongly encouraged to study in depth another discipline that uses mathematics. Majors are also strongly encouraged to take a computer programming course. Majors interested in professional work or graduate study should take additional mathematics courses at the 300 and 400 levels. Major in Mathematics—B.S.PurposeThe B.S. degree is designed to provide students with an extensive background in mathematics. It provides appropriate training for those students who plan to do graduate work in mathematics (Program I), or in related areas such as astronomy, biology, chemistry, computer science, economics, geology, physics, or psychology (Program II). RequirementsProgram I
Students are strongly encouraged to take at least one 500-level course in mathematics. Program II
Students must also complete the requirements and procedures listed in this bulletin under "General Requirements for Bachelor's Degrees." There is no culture studies requirement. Language RecommendationsStudents preparing for graduate study in mathematics or a science are strongly encouraged to study French, German, or Russian. Students must also complete the requirements and procedures listed in this bulletin under "General Requirements for Bachelor's Degrees." There is no culture studies requirement. Departmental Honors ProgramThe honors program of the Department of Mathematics is designed for students with a wide variety of interests and goals. It offers optimal preparation for graduate study and for a career as a professional mathematician. It can be combined with education courses to lead to certification as a secondary school teacher. It prepares those who wish to apply mathematical methods to other fields. The program also includes courses for honors students who are not majoring in science and mathematics. The program for mathematics majors normally begins with S212. Those who wish to graduate with honors in mathematics are expected to complete courses S303, S311, S312, S403, S413-S414, and at least two 6 credit hour "S" or "M" sequences at the 400 level or above. (S403-S404 may be used to fulfill this requirement.) Students in this program must achieve a minimum grade point average of 3.300 in all of their courses and at least a grade point average of 3.500 in their mathematics courses. Students who successfully complete most of the courses above may petition the department to qualify for departmental honors. Qualified non-honors students may petition the department to take honors mathematics courses beginning with S303 or S311. Minor in MathematicsRequirementsStudents must complete at least 16 credit hours that include M212, S212, or M213 as well as at least three courses at the 300 or 400 level. The average grade must be at least 2.000 with no grade lower than C–. Courses selected for the minor must be approved by the director of undergraduate studies. In particular, the courses K300, K305, and K310 cannot be selected for the minor. RecommendationsM365 is recommended for business and social science majors. M371 is recommended for computer science majors. M311, M312, and M343 are recommended for physics majors. M311 and M343 are recommended for chemistry majors. M311 and M365 are recommended for biology majors. Interdepartmental Major in Mathematics and EconomicsPurposeThe interdepartmental major in Mathematics and Economics is designed to enable students to model economic questions mathematically and to analyze and solve those models. RequirementsStudents must meet the following course requirements. Any course may be replaced by the honors equivalent.
Special Considerations
Actuarial StudiesActuaries use mathematics to determine the financial effect that uncertain future events such as birth, death, retirement, fire, earthquake, accident, and sickness have on insurance and other benefit plans. Actuaries may work for insurance companies, consulting firms, or the benefits departments of general businesses and government agencies. The program of study outlined below combined with a B.A. or B.S. degree in mathematics prepares the graduate for entry into the actuarial profession. The B.S. Program II with a Minor in Economics, or the Interdepartmental Major in Mathematics and Economics, works especially well with actuarial career preparation. For further information, contact the mathematics department's academic advisor in Rawles Hall 115, To advance in the actuarial profession, one must pass a series of highly challenging examinations given by the Society of Actuaries (www.soa.org) or the Casualty Actuarial Society (www.casact.org). Passing these examinations requires discipline and additional study beyond Indiana University course work. Actuaries also must be comfortable with the language and substance of a wide range of mathematics, economics, statistics, and finance/accounting to prepare for these exams. An actuary student should aim to pass at least one, and preferably two, of these examinations before graduation. Actuary students who desire a summer internship may benefit from passing the first examination, Exam P, by the summer following their junior year. Internships help assure strong job placement upon graduation and are strongly encouraged. Additionally, an actuary student should take some VEE (Validation by Educational Experience) accredited courses and must receive a grade of B– or better in these courses to earn VEE credit. A list of VEE accredited courses is available at the Society of Actuaries Web site and includes several of the courses mentioned below. M463 covers most of the material for the three-hour Exam P on probability. Students must take M211, M212, M311, and either M301 or M303 as preparation. Economics E425 covers the syllabus for the 2.5-hour Exam FM on Financial Mathematics, but students must first take E201, E202, and E321 as preparation. Students pursuing actuarial studies may benefit by taking Accounting A200, Computer Science C211, and Economics E471 and E472. For further advice and information, contact the department's academic advisor in Rawles Hall 115, (812) 855-1589. Course DescriptionsIn the following list of courses, the first digit indicates the level of difficulty. The middle digit normally indicates the field of mathematics: x0y, algebra; x1y, analysis; x2y, topology; x3y, geometry; x4y, applied mathematics; x6y, probability and statistics; x7y, numerical analysis; x8y, history and foundations. J010 Introduction to Algebra (2 cr.) P: Consent of department. For Groups students only. A review of pre-algebra mathematics. Topics include operations on integers and rational numbers, exponents, evaluating algebraic expressions, and translating English statements into algebraic equations. Emphasis is on problem solving. Credit may not be applied toward a degree Fulfills no distribution or fundamental skills requirement in the College of Arts and Sciences. M014 Basic Algebra (4 cr.) P: One year of high school algebra. Designed to provide algebraic skills needed for future mathematics courses, such as M118 or M119. Operations with fractions, exponents, linear equations, inequalities, elementary graphs. Credit may not be applied toward a degree in the College of Arts and Sciences, the School of Education, the Kelley School of Business, or the School of Public and Environmental Affairs. I Sem., II Sem, SS. X015 Introductory Algebra (2 cr.) P: One year of high school algebra. Designed to provide basic algebraic skills needed for the study of higher-level algebra courses such as X019 or MO25: linear and quadratic equations, operations on polynomials, graphs of lines. Credit may not be applied toward a degree in the College of Arts and Sciences; the School of Education; the School of Health, Physical Education, and Recreation; the Kelley School of Business; or the School of Public and Environmental Affairs. I Sem., II Sem., SS. M018 Basic Algebra for Finite Mathematics (2 cr.) P: One year of high school algebra. Designed to provide algebraic skills needed for the study of finite mathematics: linear equations and inequalities and their graphs, systems of equations, sets, and basic counting. Credit may not be applied toward a degree in the College of Arts and Sciences; the School of Education; the School of Health, Physical Education, and Recreation; the Kelley School of Business; or the School of Public and Environmental Affairs. I Sem., II Sem., SS. A025 Computer-Based Precalculus (3 cr.) P: Two years of high school algebra or M014, and one year of high school geometry. An algebra course to prepare for M119. Credit may not be applied toward a degree in the College of Arts and Sciences; a grade of C– or higher is needed to satisfy the College of Arts and Sciences mathematics fundamental skills requirement. M025 Precalculus Mathematics (3 cr.) P: Two years of high school algebra or M014, and one year of high school geometry. Designed to prepare students for M119. Algebraic operations; polynomial, exponential, and logarithmic functions and their graphs; conic sections; systems of equations; and inequalities. Credit may not be applied toward a degree in the College of Arts and Sciences; a grade of C– or higher is needed to satisfy the College of Arts and Sciences mathematics fundamental skills requirement. I Sem., II Sem., SS. M026 Trigonometric Functions (2 cr.) P or C: M025 or equivalent. Designed to prepare students for M211. Trigonometric functions; identities. Graphs of trigonometric and inverse trigonometric functions. Credit hours may not be applied toward a degree in the College of Arts and Sciences. I Sem., II Sem., SS. M027 Precalculus with Trigonometry (4 cr.) P: Two years of high school algebra or M014, and one year of high school geometry. This course is designed to prepare students for M211 Calculus. The general content of both M025 and M026 is included, with emphasis placed on exponential, logarithmic, and trigonometric functions at a more sophisticated level and pace. Credit may not be applied toward a degree in the College of Arts and Sciences; and a minimum grade of C– is needed to satisfy the College of Arts and Sciences mathematics fundamental skills requirement. Non-College of Arts and Sciences students should see their advisor about appropriate mathematics selection. I Sem., II Sem. T101 Mathematics for Elementary Teachers I (3 cr.) P: M014, M018 or a score of at least 10 on the Math Skills Assessment Exam. Elements of set theory, counting numbers. Operations on counting numbers, integers, rational numbers, and real numbers. Only open to elementary education majors. I Sem., II Sem. T102 Mathematics for Elementary Teachers II (3 cr.) P: T101; students may enroll concurrently in T102 and T103 with the approval of a School of Education advisor. Sets, operations, and functions. Prime numbers and elementary number theory. Elementary combinatorics, probability, and statistics. Open only to elementary education majors. I Sem., II Sem. T103 Mathematics for Elementary Teachers III (3 cr.) P: T101; students may enroll concurrently in T102 and T103 with the approval of a School of Education advisor. Descriptions and properties of basic geometric figures. Rigid motions. Axiomatics. Measurement, analytic geometry, and graphs of functions. Discussion of modern mathematics. Only open to elementary education majors. I Sem., II Sem. J110 Introductory Problem Solving (2 cr.) P: Two years of high school algebra or permission of department. Emphasizes problem solving and the development of logical reasoning skills. Topics include elementary logic, set theory, measurement of geometric figures, and translating English statements into algebraic equations. Not counted toward any College of Arts and Sciences distribution requirement nor toward the College of Arts and Sciences fundamental skills requirement in mathematics. M110 Excursions into Mathematics (3 cr.) P: One year each of high school algebra and geometry or the equivalent. A course designed to convey the flavor and spirit of mathematics, stressing reasoning and comprehension rather than technique. Not preparatory to other courses; explores topics in the theory of games and in properties of polyhedra. This course does not count toward a major in mathematics. J111 Introduction to College Mathematics I (3 cr.) P: Consent of department. For Groups students only. A review of basic algebra. Not counted toward any College of Arts and Sciences distribution requirement or toward the College of Arts and Sciences fundamental skills requirement in mathematics. J112 Introduction to College Mathematics II (3 cr.) P: Consent of department. For Groups students only. A continuation of J111 that includes functions, exponential functions, and logarithmic functions. Not counted toward any College of Arts and Sciences distribution requirement or toward the College of Arts and Sciences fundamental skills requirement in mathematics. J113 Introduction to Calculus with Applications (3 cr.) N & M P: Consent of department. For Groups students only. A survey of calculus. J113 can count toward the College of Arts and Sciences fundamental skills requirement in mathematics and the College of Arts and Sciences natural and mathematical sciences distribution requirement for Groups students. Credit not given for both J113 and MATH M119 or both J113 and MATH M211 or M215. D116-D117 Introduction to Finite Mathematics I-II (2-2 cr.) P: D116: Two years of high school algebra or M014. D117: Two years of high school algebra or M014, and D116 with a grade of at least C–. D116-D117 is a two-course sequence that satisfies the mathematics fundamental skills requirement in the College of Arts and Sciences. Topics for the course are taken from M118. Any requirement of M118 can also be met by D116 and D117 together. Credit not given for D116 until D116 is completed with a minimum grade of C– and D117 is completed with a passing grade. N & M distribution credit will be given only upon completion of both D116 and D117. Credit given for only one of the following: the sequence D116-D117 or M118 or A118. A118 Finite Mathematics for the Social and Biological Sciences (3 cr.) M118 Finite Mathematics (3 cr.) N & M P: Two years of high school algebra or M014. Sets, counting, basic probability, including random variables and expected values. Linear systems, matrices, linear programming, and applications. Credit given for only one of M118, A118, or the sequence D116-D117. S118 Honors Finite Mathematics (3 cr.) N & M P: Mastery of two years of high school algebra. Designed for students of outstanding ability in mathematics. Covers all material of M118 and additional topics from statistics and game theory. Computers may be used in this course, but no previous experience is assumed. I Sem. M119 Brief Survey of Calculus I (3 cr.) N & M P: Two years of high school algebra or M014. Introduction to calculus. Primarily for students from business and the social sciences. A student cannot receive credit for more than one of the following: M119, M211, or MATH J113. I Sem., II Sem., SS. M120 Brief Survey of Calculus II (3 cr.) N & M P: M119. A continuation of M119 covering topics in elementary differential equations, calculus of functions of several variables and infinite series. Intended for non-physical science students. Credit not given for both M212 and M120. I Sem., II Sem., SS. X201 Transition to Calculus II (1 cr.) P: B or higher in M119. Provides a transition from M119 to M212. Trigonometric functions and their identities (rapid review), limits, derivatives of trigonometric functions, related rates, implicit differentiation, mean value theorem, L'Hospital's rule, Riemann sums, antiderivatives of trigonometric functions. Credit not given for both M211 and X201. II Sem. M211 Calculus I (4 cr.) N & M P: Two years of high school algebra, one year of high school geometry, precalculus math (or its equivalent), and trigonometry; or both M025 and M026. Limits, continuity, derivatives, definite and indefinite integrals, applications. A student may receive credit for only one of the following: M119, M211, J113. Credit not given for both M211 and X201. A combination of M119 and X201 is equivalent to M211 as preparation for M212. The combination of J113 and X201 is not advised as preparation for M212. I Sem., II Sem., SS. M212 Calculus II (4 cr.) N & M P: M119 and X201, or M211. Techniques of integration (by parts, trigonometric substitutions, partial fractions), improper integrals, volume, work, arc length, surface area, infinite series. A student may receive credit for only one of M120 and M212. I Sem., II Sem., SS. S212 Honors Calculus II (4 cr.) N & M P: M211 and consent of mathematics department. Includes material of M212 and supplemental topics. Designed for students of outstanding ability in mathematics. I Sem. M213 Accelerated Calculus (4 cr.) N & M P: Placement by examination. Designed for students with one year of calculus in high school. Review of material covered in M211 followed by an intensive study of all material in M212. Students completing M213 with a final grade of A or B may receive credit for M211. Credit not given for both M213 and M212. M295 Readings and Research (1–3 cr.) Supervised problem solving. Admission only with permission of a member of the mathematics faculty who will act as supervisor. I Sem., II Sem., SS. S299 Honors Reading and Research (1–3 cr.) Supervised problem solving. Admission only with permission of a member of the mathematics faculty who will act as supervisor. I Sem., II Sem., SS. M301 Linear Algebra and Applications (3 cr.) N & M P: M212 or both M211 and CSCI C241. R: M212. Solving systems of linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors. Selection of advanced topics. Applications throughout. Computer used for theory and applications. Credit not given for both M301 and M303. M303 Linear Algebra for Undergraduates (3 cr.) N & M P: M212 or both M211 and CSCI C241. R: M212. Introduction to the theory of real vector spaces. Coordinate systems, linear dependence, bases. Linear transformations and matrix calculus. Determinants and rank. Eigenvalues and eigenvectors. Credit not given for both M301 and M303. I Sem., II Sem., SS. S303 Honors Course in Linear Algebra (3 cr.) N & M P: Consent of department. Honors version of M303. For students with unusual aptitude and motivation. Not open to those who have had M301 or M303. II Sem. K310 (PSY K310) Statistical Techniques (3 cr.) N & M P: M119 or equivalent. Introduction to probability and statistics. Elementary probability theory, conditional probability, independence, random variables, discrete and continuous probability distributions, measures of central tendency and dispersion. Concepts of statistical inference and decision: estimation, hypothesis testing, Bayesian inference, statistical decision theory. Special topics discussed may include regression and correlation, time series, analysis of variance, non-parametric methods. Credit given for only one of the following: PSY or MATH K300, K310; CJUS K300; ECON E370 or S370; SOC S371; or SPEA K300. I Sem., II Sem. M311 Calculus III (4 cr.) N & M P: M212, M213 or consent of department. Elementary geometry of 2, 3, and n-space; functions of several variables; partial differentiation; minimum and maximum problems; multiple integration. I Sem., II Sem., SS. S311 Honors Course in Calculus III (4 cr.) N & M P: M212 or M213, and consent of department. Honors version of M311. For students with unusual aptitude and motivation. Credit not given for both M311 and S311. I Sem. M312 Calculus IV (3 cr.) P: M311. Differential calculus of vector-valued functions, transformation of coordinates, change of variables in multiple integrals. Vector integral calculus: line integrals, Green's theorem, surface integrals, Stokes's theorem. Applications. I Sem., II Sem. S312 Honors Course in Calculus IV (3 cr.) P: M311 or consent of instructor. For students with unusual aptitude and motivation. Credit not given for both M312 and S312. II Sem. M321 Intuitive Topology (3 cr.) N & M P: M212 or consent of instructor. Intuitive description of topology, including networks and maps, topological equivalence, classification of surfaces, spheres with handles, knot theory, Jordan curve theorem, transformations, and fixed-point theorems. II Sem. M330 Exploring Mathematical Ideas (3 cr.) N & M P: M211 or consent of the department. An experimental course to illustrate important ideas in major areas of mathematics, including number theory, group theory, topology, geometry, and probability. Additional topics may include newly emerging fields, such as chaos theory. Does not count toward major requirements. T336 Topics in Euclidean Geometry (3 cr.) N & M P: M212. A study of the central aspects of two-dimensional Euclidean geometry from historical and axiomatic points of view as well as through hands-on and/or computer-based explorations of geometric concepts and constructions. I Sem. M337 Mathematics and Art (3 cr.) N & M P: M118 or M119. We explore connections between mathematics and art, illuminating historical and modern developments in art, mathematics, and computer graphics. Group/ activity-based learning will cover correct drawing/viewing of perspective art, fractal geometry and its relation to art, filings and symmetry in art. Field trip to Indianapolis Museum of Art included. M343-M344 Introduction to Differential Equations with Applications I-II S343 Honors Course in Differential Equations (3 cr.) N & M P: M212 and consent of department. Introduction, with historical examples, first order ordinary differential equations (ODEs) and applications, second order linear ODEs, linear ODEs of higher order, series solutions to linear ODEs, and numerical methods for ODEs. In addition, some theoretical aspects will be studied in detail such as the Picard existence/uniqueness theorem for initial-value problems, convergence of series solutions, and the matrix exponential exp(tA). S344 Honors Course in Differential Equations II (3 cr.) N & M P: S343 or M343, M212, M301 or M303, and consent of the department. Covers the topics of M344, in addition to more theoretical material, which may include topics such as the uniqueness theorem for the inversion of the Laplace transform, introduction to the theory of distributions, derivation of the heat and wave equations, eigenvalues of Sturm-Liouville boundary problems, and oscillation theory applied to special functions. Meets with M344, and the additional material will be incorporated in weekly homework sets. Exams will include some of this additional material. M348 Discrete Mathematical Models (3 cr.) N & M P: M118 or equivalent. Introduction to the development and use of discrete mathematical models in the social, life, and management sciences; emphasis on models involving Markov chains, game theory, graph theory, and evolutionary systems. M353 Discrete Mathematics (3 cr.) P: MATH M212 or permission of instructor. Covers fundamental topics chosen from enumerative combinatorics and graph theory. Possible topics include permutations, combinations, pigeonhole principle, inclusion-exclusion, generating functions, recurrence relations, Pólya theory, spanning trees, Eulerian paths, Ramsey theory, graph coloring, flow problems, Hamiltonian paths and cycles, electrical networks, random graphs. M360 Elements of Probability (3 cr.) P: M212 and M311, which may be taken concurrently. Introduction to theory of probability. Probability models, combinatorial problems, conditional probability and independence, random variables, discrete and continuous distributions, repeated Bernoulli trials, gambler's ruin problems, moments, moment generating functions, law of large numbers, central limit theorem and applications. Credit not given for both M360 and M365. I Sem. M365 Introduction to Probability and Statistics (3 cr.) N & M P: M212. Elementary concepts of probability and statistics. Combinatorics, conditional probability, independence, random variables, discrete and continuous distributions, moments. Statistical inference, point estimation, confidence intervals, test of hypotheses. Applications to social, behavioral, and natural sciences. Credit not given for both M365 and M360. The sequence M365-M366 is not recommended. I Sem., II Sem., SS. M371 Elementary Computational Methods (3 cr.) N & M P: M212. Some computer programming experience is helpful, but not required. Interpolation and approximation of functions, solution of equations, numerical integration and differentiation. Errors, convergence, and stability of the procedures. Students write and use programs applying numerical methods. M380 History of Mathematics (3 cr.) P: M212. Brief study of the development of algebra and trigonometry; practical, demonstrative, and analytic geometry; calculus, famous problems, calculating devices; famous mathematicians and chronological outlines in comparison with outlines in the sciences, history, philosophy, and astronomy. M384 Modal Logic (3 cr.) N & M P: P250, and one 300-level mathematics course, or consent of the instructor. Introduction to modal logic with emphasis on systems of modal logic which apply to philosophy and computer science. Includes epistemic logic, temporal logic, deontic logic, and logics for reasoning about space. Covers the semantics of these systems, and only secondarily will be concerned with the standard results about them. M385 Mathematics from Language (3 cr.) N & M P: M118 or equivalent. Discrete mathematics. Topics in math motivated by linguistics, chosen from formal approaches to syntax and semantics, and from statistical and computational linguistics. M391 Introduction to Mathematical Reasoning (3 cr.) N & M P: M212 or both M211 and CSCI C241. R: M212. Elementary logic, techniques of proof, basic set theory, functions, relations, binary operations, number systems, counting. Bridges the gap between elementary and advanced courses. Recommended for students with insufficient background for 400-level courses and for students in education. Not open to students who have received credit for M403, M413, or M420. Y398 Internship in Professional Practice (1–3 cr.) S/F grading. P: Approval of Department of Mathematics. Professional work experience involving significant use of mathematics or statistics. Evaluation by employer and Department of Mathematics. Does not count toward major requirements. May be repeated once with approval of Department of Mathematics for a total of 6 credits. M403-M404 Introduction to Modern Algebra I-II (3-3 cr.) P: M301 or M303. Study of groups, rings, field extensions, with applications to linear transformations. M403, I Sem.; M404, II Sem. S403-S404 Honors Course in Modern Algebra I-II (3-3 cr.) P: S303. For students of outstanding ability in mathematics. Theory of groups, rings, integral domains, fields, and modules. S403, I Sem.; S404, II Sem. T403 Modern Algebra for Secondary Teachers (3 cr.) P: M301 or M303, and M391. Introduction to the basic concepts of groups, rings, and fields with an emphasis on the theory of equations as it underlies the basic ideas of high school algebra. I Sem. M405 Number Theory (3 cr.) P: M212. Numbers and their representation, divisibility and factorization, primes and their distribution, number theoretic functions, congruences, primitive roots, diophantine equations, quadratic residues, sums of squares. M409 Linear Transformations (3 cr.) P: M301 or M303. The study of linear transformations on a finite dimensional vector space over the complex field. Canonical forms, similarity theory; inner products and diagonalization of normal transformations. M413-M414 Introduction to Analysis I-II (3-3 cr.) P: M301 or M303, and M311, or consent of instructor. Modern theory of real number system, limits, functions, sequences and series, Riemann-Stieltjes integral, and special topics. M413, I Sem.; M414, II Sem. S413-S414 Honors Course in Analysis I-II (3-3 cr.) P: S312 or consent of instructor. Differentiable transformations defined on Euclidean space, inverse and implicit function theorems. Lebesgue integration over Euclidean space and transformation of integrals. Exterior algebra, measure and integration on manifolds. Stokes's theorem. Closed and exact forms. S413, I Sem.; S414, II Sem. M415 Elementary Complex Variables with Applications (3 cr.) P: M311. Algebra and geometry of complex numbers, elementary functions of a complex variable, power series, integrations, calculus of residues, conformal mapping. Application to physics. II Sem. M420 Metric Space Topology (3 cr.) P: M301 or M303. Topology of Euclidean and metric spaces. Limits and continuity. Topological properties of metric spaces, including separation properties, connectedness, and compactness. Complete metric spaces. Elementary general topology. M435 Introduction to Differential Geometry (3 cr.) P: M301 or M303, and M311. An introduction to the geometry of curves and surfaces. Topics will include arc length, torsion, Frenet formulae, metrics, curvatures, and classical theorems in these areas. H435 Summer Institute for Mathematics Teachers (3 cr.) Intensive four-week seminar/workshop designed for middle and high school mathematics teachers. Topics from Euclidean geometry and probability and statistics will be explored using computer software. M441-M442 Introduction to Partial Differential Equations with Applications I-II (3-3 cr.) P: M301 or M303, M311, and M343. Derivation and methods of solution of classical partial differential equations of mathematical physics: heat, wave, and Laplace equations. Separation of variables, Fourier series, Sturm-Liouville theory, special functions, Green's functions, Fourier transform, first order equations, characteristics and special topics. M441, I Sem.; M442, II Sem. H447 Summer Institute in Mathematical Models (1–4 cr.) S/F grading. P: M303, M365. Introduction to mathematical models and computer tools for modeling. Mathematical topics include games, graphs, queues, growth processes, and optimization. Emphasis on small group problem solving and on topics which can be incorporated into the high school curriculum. M447-M448 Mathematical Models and Applications I-II (3-3 cr.) P: M301 or M303, M311, M360 or M365, which may be taken concurrently, or consent of instructor. Formation and study of mathematical models used in the biological, social, and management sciences. Mathematical topics include games, graphs, Markov and Poisson processes, mathematical programming, queues, and equations of growth. M447, I Sem.; M448, II Sem. M451 The Mathematics of Finance (3 cr.) P: M311 and M365. R: M343. Course covers probability theory, Brownian motion, Ito's Lemma, stochastic differential equations, and dynamic hedging. These topics are applied to the Black-Scholes formula, the pricing of financial derivatives, and the term theory of interest rates. M453 Cryptography (3 cr.) N & M P: M301 or M303. The course covers encryption and decryption in secure codes. Topics include cryptosystems and their cryptanalysis, Data Encryption Standard, cryptanalysis, Euclidean algorithm, Chinese remainder theorem, RSA cryptosystem, primality testing, factoring algorithms, EI Gamal cryptosystem, discrete log problem, other public key cryptosystems, signature schemes, hash functions, key distribution and key agreement. M455 Quantum Computing I (3 cr.) P: M118, M211, and M303, or consent of instructor. Covers the interdisciplinary field of quantum information science and aims at senior undergraduate and graduate students majoring in computer science, physics, mathematics, philosophy, and chemistry. Quantum information science is the study of storing, processing, and communicating information using quantum systems. Cross-listed as PHYS P455. Credit given for only one of M455 and PHYS P455. M456 Quantum Computing II (3 cr.) P: M118, M211, M303, and M455, or consent of instructor. Covers the interdisciplinary field of quantum information science and aims at senior undergraduate and graduate students majoring in computer science, physics, mathematics, philosophy, and chemistry. Quantum information science is the study of storing, processing, and communicating information using quantum systems. Cross-listed as PHYS P456. Credit given for only one of M456 and PHYS P456. M463 Introduction to Probability Theory I (3 cr.) P: M301 or M303, and M311. 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. I Sem. S463 Honors Course in Probability Theory I (3 cr.) P: M303 and M311. Honors version of M463. For students of outstanding ability in mathematics. I Sem. M464 Introduction to Probability Theory II (3 cr.) P: M463. Conditional distributions and expectation, linear and nonlinear regression; simple stochastic processes: Poisson process, process with independent increments, random walk, Markov chain with finite state space; information theory. II Sem. S464 Honors Course in Probability Theory II (3 cr.) P: S463 or consent of instructor. Honors version of M464. For students of outstanding ability in mathematics. M466 Introduction to Mathematical Statistics (3 cr.) P: M463 or consent of instructor. Rigorous mathematical treatment of problems in sampling and statistical inference. Possible topics include sufficient statistics, exponential distributions, monotone likelihood ratio, most powerful tests, minimum variance estimates, shortest confidence intervals, linear models, maximum likelihood, simultaneous equations, the relationship of theory to practice. II Sem. M471-M472 Numerical Analysis I-II (3-3 cr.) P: M301 or M303, M311, M343, and knowledge of a computer language such as FORTRAN, C, C++, etc. (Students with other programming backgrounds should consult the instructor.) Interpolation and approximation of functions, numerical integration and differentiation, solution of nonlinear equations, acceleration and extrapolation, solution of systems of linear equations, eigenvalue problems, initial and boundary value problems for ordinary differential equations, and computer programs applying these numerical methods. M471, I Sem.; M472, II Sem. M482 Mathematical Logic (3 cr.) Construction and study of formal mathematical languages. Definitions of, and relationships between, the notions of "truth" and "probability" of a formal sentence. Capabilities and limitations of first-order languages. Alternative formal systems. Introductions to model theory and the decision problem. Additional topics chosen by the instructor. M490 Problem Seminar (3 cr.) P: M301 or M303, M413 (M413 may be concurrent), and consent of the instructor. Introduction to research techniques for advanced undergraduate and beginning graduate students, based on problems from parts of the regular curriculum, such as linear algebra, topology, probability, and analysis. Emphasis will be on problems of both current and historical interest but usually not in the standard literature. M491 Putnam Exam Seminar (1 cr.) P: Approval of the director of undergraduate studies. The Putnam Examination is a national mathematics competition for college undergraduates at all levels of study. It is held in December each year. This problem seminar is designed to help students prepare for the examination. May be repeated twice for credit. S499 Reading for Honors (1–12 cr.; 12 cr. max.) P: Approval of departmental honors committee. I Sem., II Sem., SS.
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Last updated: 18 December 2024 00 32 06
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