Undergraduate Courses 105 | 115 | 116 | 127 | 128 | 147 | 156 | 174 | 175 | 176 | 185 | 186 | 214 | 215 | 216 | 217 | 255 | 256 | 285 | 286 | 288 | 289 | 295 | 296 | 310 | 312 | 316 | 327 | 351 | 354 | 362 | 371 | 385 | 389 | 395 | 396 | 404 | 412 | 416 | 417 | 419 | 420 | 422 | 423 | 424 | 425 | 427 | 431 | 432 | 433 | 450 | 451 | 452 | 454 | 462 | 463 | 464 | 471 | 472 | 475 | 476 | 481 | 486 | 489 | 490 | 497 | 498 | 512 | 513 | 520 | 521 | 522 | 523 | 524 | 525 | 526 | 528 | 531 | 532 | 535 | 537 | 547 | 548 | 550 | 551 | 555 | 556 | 557 | 558 | 559 | 561 | 562 | 563 | 565 | 566 | 567 | 571 | 572 | 575 | 582 | 590 | 591 | 592 | 593 | 594 | 596 | 597 *For a list of current Math textbooks, please click here. Math 105 - Data, Functions and Graphs Prerequisites: 3-4 years HS math Frequency: Fall (I), Winter (II), Summer (IIIb) Student Body: First-year students Credit: 4 Credits. No credit granted for those who have completed any Math course numbered 110 or higher. Recent Texts: Functions Modeling Change (Connally, Hughes-Hallett and Gleason) Background and Goals: Math 105 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who successfully complete 105 are fully prepared for Math 115. Content: This course presents the concepts of precalculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and written (verbal descriptions). The emphasis is on the mathematical modeling of real-life problems using linear, polynomial, exponential, logarithmic, and trigonometric functions. Students develop their reading, writing, and questioning skills in an interactive classroom setting. Alternatives: Math 110 (Pre-Calculus (Self-Paced)) is a condensed half-term version of the same material offered as a self-study course through the Math Lab. Subsequent Courses: The course prepares students for Math 115 (Calculus I). Back to Top Math 115 - Calculus I Prerequisites: 3-4 years HS math including trigonometry Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: First-year students Credit: 4 Credits Recent Texts: Calculus: Single Variable (Hughes-Hallett, et al.) Background and Goals: The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given two uniform midterms and a final exam. Content: The course presents the concepts of calculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and verbal descriptions. Students will develop their reading, writing and questioning skills, as well as their ability to work cooperatively. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and an introduction to integration. The classroom atmosphere is interactive and cooperative. Both individual and team homework is assigned. Alternatives: Math 185 (Honors Calculus I ) is a somewhat more theoretical course which covers some of the same material. Math 175 (Intro to Cryptology) is a non-calculus alternative for students with a good command of first-semester calculus. Math 295 (Honors Mathematics I) is a much more intensive and rigorous course. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions and Graphs). Subsequent Courses: Math 116 (Calculus II) is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking Math 186 (Honors Calculus II). Back to Top Math 116 - Calculus II Prerequisites: Math 115 or AP credit Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: First-year students Credit: 4 Credits Recent Texts: Calculus: Single Variable (Hughes-Hallett, et al.) Background and Goals: The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given two uniform midterms and a final exam. Content: The course presents the concepts of calculus from four points of view: geometric (graphs), numeric (tables), symbolic (formulas), and verbal descriptions. Students will develop their reading, writing and questioning skills, as well as their ability to work cooperatively. Topics include techniques of integration, applications of integration, Taylor series, an introduction to differential equations, and infinite series. The classroom atmosphere is interactive and cooperative. Both individual and team homework is assigned. Alternatives: Math 156 (Applied Honors Calculus II) also covers much of the same material; emphasizing applications to science and engineering. Math 186 (Honors Calculus II) is a somewhat more theoretical course which covers much of the same material. Subsequent Courses: Math 215 (Calculus III) is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking Math 255 (Applied Honors Calculus III) or Math 285 (Honors Calculus III). Back to Top Math 127 - Geometry and the Imagination Prerequisites: 3 years HS math Frequency: Winter (II) Student Body: First-year students and sophomores Credit: 4 Credits. No credit after completing any 200+ level math course except 385, 489, or 497 Recent Texts: Beyond the Third Dimension: Geometry, Computer Graphics and Higher Dimensions (Banchoff) Background and Goals: This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The course is intended for students who want an introduction to mathematical ideas and culture. Emphasis is on conceptual thinking — students will do hands-on experimentation with geometric shapes, patterns and ideas. Content: The section begins with the independence of Euclid’s Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry differs from Euclidean geometry. The last topic is geometry of higher dimensions: coordinization — the mathematician’s tool for studying higher dimensions; construction of higher-dimension analogues of some familiar objects like spheres and cubes; discussion of the proper higher-dimensional analogues of some geometric notions (length, angle, orthogonality, etc.). Alternatives: none Subsequent Courses: This course does not provide preparation for any further study of mathematics. Back to Top Math 128 - Explorations in Number Theory Prerequisites: 3 years HS math Frequency: Fall (I) Student Body: First-year students and sophomores Credit: 4 Credits. No credit after completing any 200+ level math course except 385, 489, or 497. Recent Texts: coursepack Area: Number Theory Background and Goals: This course is intended for non-science concentrators and students in the pre-concentration years with no intended concentration who want to engage in mathematical reasoning without having to take calculus first. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. Content: After studying the factorizations and greatest common divisors, emphasis will shift to the patterns that emerge when the integers are classified according to the remainder produced upon division by some fixed number (congruences). Once some basic tools have been established, applications will be made in several directions. For example, students may derive a precise parameterization of Pythagorean triples. Alternatives: none Subsequent Courses: This course does not provide preparation for any further study of mathematics. Back to Top Math 147 - Introduction to Interest Theory Prerequisites: Math 112 or 115 Frequency: Fall (I), Winter (II) Student Body: First-year students and sophomores Credit: 3 Credits. no credit after completing any 200+ level math course except 385, 489, or 497 Recent Texts: Mathematics of Finance (Zima and Brown) Area: Actuarial & Financial Background and Goals: This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators, should elect Math 424 which covers the same topics but on a more rigorous basis requiring considerable use of calculus. The course is not part of a sequence. Students should possess financial calculators. Content: Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. Alternatives: Math 424 (Compound Interest and Life Ins) covers the same material in greater depth and with a higher level of mathematical content. Subsequent Courses: none Back to Top Math 156 - Applied Honors Calculus II Prerequisites: Score of 4 or 5 on Advanced Placement AB or BC Calculus exam, or permission of instructor Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits. Recent Texts: Single Variable Calculus (Stewart) Background and Goals: Math 156 is part of the applied honors calculus sequence for engineering and science concentrators. The course is an alternative to Math 116 for students with strong mathematics ability. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, and science. Content: Riemann sums, definite integral, fundamental theorem of calculus, applications of integral calculus (e.g. arclength, surface area, work, center of mass, probability density functions), improper integrals, infinite sequences and series, geometric series, alternating series, power series, Taylor series, differential equations, complex numbers. Students are introduced to MAPLE. Alternatives: Math 116 (Calculus II) or Math 186 (Honors Calculus II). Subsequent Courses: Math 255 (Applied Honors Calculus III) is the natural sequel. Back to Top Math 174 - Proofs in Geometry Prerequisites: Permission of honors advisor Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits. No credit granted to those who have completed a 200-level or higher Math course. Recent Texts: none Background and Goals: The course will be very interactive, eliciting suggestions towards proof from the students so that all the problems are eventually solved by a joint effort between the students and the instructor. The format has worked well in the past for honors courses. To enhance the visualization, we plan to develop software for two-dimensional geometric constructions. This software will be able to produce multi-color pictures if geometric configurations. In the long run, such software will save us time in creating problem sets, handouts and perhaps slides. Additional topics may be added depending on the interest and abilities of the students. Content: A good text for the course is already available: the classic "Geometry Revisited" by Coxter and Greitzer, which contains a wonderful exposition of the material and has suitable exercises. As a precursor to the mathematics, the course will use familiar games such as the old game Mastermind where player A has a code which player B has to use. Students will pair off and play the game, with the important additional feature that the guesser must write down what(s) he knows and can deduce after each guess, and therefore motivate his/her next guess. This should help set the mood and instill the idea of analyzing the facts at hand and making logical deductions. After this the course will develop some basic theorems of Euclidean geometry. An example of such a theorem is that the angle bisectors (or medians, or altitudes, or perpendicular bisectors) of a triangle are concurrent. These results are fairly straightforward but exemplify the spirit of the course by providing a good introduction to rigorous proofs, Then we move to some more difficult but beautiful theorems from geometry such as Ceva's theorem, the Euler line, the nine-point circle theorem, Ptolemy's theorem and Morley's theorem. Alternatives: none Subsequent Courses: none Back to Top Math 175 - Introduction to Cryptology Prerequisites: Permission of honors advisor Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits Recent Texts: coursepack; Invitation to Cryptology (Barr) Background and Goals: This course is an alternative to Math 185 as an entry to the honors sequence. Students are expected to have previous experience with the basic concepts and techniques of first-semester calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each semester in a group project. Grades will be based on homework and projects with a strong emphasis on homework. Personal computers will be a valuable experimental tool in this course and students will be asked to learn to program in either BASIC, PASCAL or FORTRAN. Content:This course gives a historical introduction to Cryptology and introduces a number of mathematical ideas and results involved in the development and analysis of secret codes. The course begins with the study of permutation-based codes: substitutional ciphers, transpositional codes, and more complex polyalphabetic substitutions. The mathematical subjects treated in this section include enumeration, modular arithmetic and some elementary statistics. The subject then moves to bit stream encryption methods. These include block cipher schemes such as the Data Encryption Standard. The mathematical concepts introduced here are recurrence relations and some more advanced statistical results. The final part of the course is devoted to public key encryption, including Diffie-Hellman key exchange, RSA and Knapsack codes. The mathematical tools come from elementary number theory. Alternatives: Math 115 (Calculus I), Math 185 (Honors Calculus I), or Math 295 (Honors Mathematics I). Subsequent Courses:Math 176 (Dynamical Systems and Calculus), Math 186 (Honors Calculus II), or Math 116 (Calculus II). Back to Top Math 176 - Dynamical Systems and Calculus Prerequisites: Math 175 or permission of instructor Frequency: Sporadically Student Body: First-year students Credit: 4 Credits Recent Texts: coursepack; Nonlinear Dynamics and Chaos (Strogatz) Background and Goals: Students are expected to have some previous experience with the basic concepts and techniques of calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each semester in a group project. UNIX workstations will be a valuable experimental tool in this course and students will run preset lab routines on them using Matlab and MAPLE. Content: The general theme of the course will be discrete-time and continuous-time dynamical systems. Examples of dynamical systems arising in the sciences are used as motivation. Topics include: iterates of functions, simple ordinary differential equations, fixed points, attracting and repelling fixed points and periodic orbits, ordered and chaotic motion, self-similarity, and fractals. Tools such as limits and continuity, Taylor expansions of functions, exponentials, logarithms, eigenvalues, and eigenvectors are reviewed or introduced as needed. There is a weekly computer work-station lab. Alternatives: Math 116 (Calculus I) or Math 186 (Honors Calculus II) Subsequent Courses: Math 285 (Honors Calculus III) Back to Top Math 185 - Honors Calculus I Prerequisites: Permission of honors advisor; not intended for students receiving a score of 5 on the AB, or 4 or 5 on the BC Advanced Placement exam. Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits Recent Texts: Calculus 5th edition (Stewart) Background and Goals:The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. It is not appropriate for students who have received scores of 5 on the AB, or 4 or 5 on the BC Advanced Placement exam (those students should elect Math 156 or Math 295). It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Content: Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Alternatives: Math 115 (Calculus I) is a somewhat less theoretical course which covers much of the same material. Math 295 (Honors Mathematics I) gives a much more theoretical treatment of much of the same material. Subsequent Courses: Math 186 (Honors Calculus II) is the natural sequel. Back to Top Math 186 - Honors Calculus II Prerequisites: Permission of honors advisor Frequency: Winter (II) Student Body: First-year students Credit: 4 Credits Recent Texts: Calculus 5th edition (Stewart) Background and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Content: Topics covered include transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor. Alternatives: Math 116 (Calculus II) is a somewhat less theoretical course which covers much of the same material. Math 156 (Applied Honors Calculus II) is more application based, but covers much of the same material. Subsequent Courses: Math 285 (Honors Calculus III) is the natural sequel. Back to Top Math 214 - Linear Algebra and Differential Equations Prerequisites: Math 115 and 116 Frequency: Fall (I), Winter (II) Student Body: Engineering students, particularly in Industrial and Operations Engineering Credit: 4 Credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 513. Two credits granted after completing Math 216. Recent Texts: Linear Algebra with Applications (Bretscher) Background and Goals:An introduction to matrices and linear algebra. This course covers the basics needed to understand a wide variety of applications that use the ideas of linear algebra, from linear programming to mathematical economics. The emphasis is on concepts and problem solving. The course is designed as an alternative to Math 216 for students who need more linear algebra and less differential equations background than provided in 216. Content:An introduction to the main concepts of linear algebra… matrix operations, echelon form, solution of systems of linear equations, Euclidean vector spaces, linear combinations, independence and spans of sets of vectors in Euclidean space, eigenvectors and eigenvalues, similarity theory. There are applications to discrete Markov processes, linear programming, and solutions of linear differential equations with constant coefficients. Alternatives: Math 419 (Linear Spaces and Matrix Theory) has a somewhat more theoretical emphasis. Math 217 is a more theoretical course which covers much of the material of Math 214 at a deeper level. Math 513 (Intro. to Linear Algebra) is an honors version of this course. Mathematics concentrators are required to take Math 217 or Math 513. Subsequent Courses: Math 420 (Matrix Algebra II), Math 561 (Linear Programming I), Math 462 (Mathematical Models), Math 571 (Numer. Meth. for Sci. Comput. I). Back to Top Math 215 - Calculus III Prerequisites: Math 116, 156, or 186 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: Sophomores Credit: 4 Credits. Credit is granted for only one course among Math 215, 255, and 285. Recent Texts: Multivariable Calculus 5th edition (Stewart) Background and Goals: The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. Content: Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green’s Theorem and Stokes’ Theorem. There is a weekly computer lab using MAPLE. Alternatives: Math 285 (Honors Calculus III) is a somewhat more theoretical course which covers the same material. Math 255 (Applied Honors Calculus III) is also an alternative. Subsequent Courses: For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217 (Linear Algebra). Students who intend to take only one further mathematics course and need differential equations should take Math 216 (Intro. to Differential Equations). Back to Top Math 216 - Introduction to Differential Equations Prerequisites: Math 116, 156, or 186 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: Sophomore engineering students Credit: 4 Credits. 2 credits granted to those who have completed or are enrolled in Math 214; credit is granted for only one course among Math 216, 256, 286, and 316. Recent Texts: Differential Equations: Computing and Modeling 3rd edition (Edwards and Penney) Background and Goals: For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations: 216-417 (or 419) and 217-316. The sequence 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Mathematics concentrators and other students who have some interest in the theory of mathematics should elect the sequence 217-316. Content: Math 216 is a basic course on differential equations, intended for engineers and other scientists who need to apply the techniques in their work. The lectures are accompanied by a computer lab and recitation section where students have the opportunity to discuss problems and work through computer experiments to further develop their understanding of the concepts of the class. Topics covered include some material on complex numbers and matrix algebra, first and second order linear and non-linear systems with applications, introductory numerical methods, and elementary Laplace transform techniques. Alternatives: Math 286 (Honors Differential Equations) covers much of the same material in the honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 256 (Applied Honors Calculus IV) is also an alternative. Subsequent Courses: Math 404 (Intermediate Diff. Eq.) covers further material on differential equations. Math 217 (Linear Algebra) and Math 417 (Matrix Algebra I) cover further material on linear algebra. Math 371 (Engin. 303 - Numerical Methods) and Math 471 (Intro. to Numerical Methods) cover additional material on numerical methods. Back to Top Math 217 - Linear Algebra Prerequisites: Math 215, 255, or 285 Frequency: Fall (I), Winter (II) Student Body: Sophomore prospective mathematics concentrators Credit: 4 Credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 513. Recent Texts: Linear Algebra and Its Applications (Lay) Background and Goals: For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations: 216-417 (or 419) and 217-316. The sequence 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Mathematics concentrators and other students who have some interest in the theory of mathematics should elect the sequence 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Content: The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; Eigenvalues and Eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Alternatives: Math 214, 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. Subsequent Courses: The intended course to follow Math 217 is Math 316 (Differential Equations). Math 217 is also prerequisite for Math 312 (Applied Modern Algebra), Math 412 (Introduction to Modern Algebra) and all more advanced courses in mathematics. Back to Top Math 255 - Applied Honors Calculus III Prerequisites: Math 156, or permission of instructor Frequency: Winter (II) Student Body: First-year students Credit: 4 Credits. Credit is granted for only one course among Math 215, 255, and 285. Recent Texts: Multivariable Calculus 5th edition (Stewart) Background and Goals: Math 255 is part of the applied honors calculus sequence for engineering and science concentrators. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, or science. Content: Analytic geometry of lines and planes using vector notation, parametric representation of curves and surfaces, multivariable calculus, line surface and volume integrals, vector fields, Green’s theorem, Stokes’ theorem, divergence theorem, applications (e.g. electromagnetic fields, fluid dynamics). MAPLE will be used throughout. Alternatives: Math 215 (Calculus III) or Math 285 (Honors Calculus III). Subsequent Courses: Math 256 (Applied Honors Calculus IV) Back to Top Math 256 - Applied Honors Calculus IV Prerequisites: Math 255 Frequency: Fall (I) Student Body: Sophomores Credit: 4 Credits. Credit is granted for only one course among Math 216, 256, 286, and 316. Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima) Background and Goals: Math 256 is part of the applied honors calculus sequence for engineering science concentrators. Applications and concepts receive equal treatment. Theorems are stated precisely and are derived, but technical details are omitted. Examples are given to illustrate the theory. Critical thinking and class participation are encouraged. The goal is to provide students with a solid background needed for subsequent courses in mathematics, engineering, or science. Content: Linear algebra, matrices, systems of differential equations, initial value problems, qualitative theory of dynamical systems (e.g. equilibria, phase space, stability, bifurcations), nonlinear equations, numerical methods. Optional: boundary value problems. Students will learn to use MATLAB for computer simulations. Alternatives: Math 216 (Intro. to Differential Equations) or Math 286 (Honors Differential Equations) Subsequent Courses: Many upper-level courses Back to Top Math 285 - Honors Calculus III Prerequisites: Math 156, 176 or 186 or permission Frequency: Fall (I) Student Body: Sophomores and first-year students with suitable background Credit: 4 Credits. Credit is granted for only one course among Math 215, 255, and 285. Recent Texts: Multivariable Calculus 5th edition (Stewart) Background and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Content: Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation, maximum-minimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green’s Theorem and Stokes’ Theorem. Additional topics may be added at the discretion of the instructor. Alternatives: Math 215 (Calculus III) is a less theoretical course which covers the same material. Math 255 (Applied Honors Calc. III) is an applications-oriented honors course which covers much of the same material. Subsequent Courses: Math 216 (Intro. to Differential Equations), Math 286 (Honors Differential Equations) or Math 217 (Linear Algebra). Back to Top Math 286 - Honors Differential Equations Prerequisites: Math 285 or permission Frequency: Winter (II) Student Body: Sophomores Credit: 3 Credits. Credit is granted for only one course among Math 216, 256, 286, and 316. Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima) Background and Goals: The sequence Math 185-186-285-286 is an introduction to calculus at the honors level. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Content: Topics include first-order differential equations, higher-order linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, Runge-Kutta). If time permits, Picard’s Theorem will be proved. Alternatives: Math 216 (Intro. to Differential Equations) and Math 316 (Differential Equations) cover much of the same material. Math 256 (Applied Honors Calculus IV) is also an alternative. Subsequent Courses: Math 471 (Intro. to Numerical Methods) and/or Math 572 (Numer. Meth. for Sci. Comput. II) are natural sequels in the area of differential equations, but Math 286 is also preparation for more theoretical courses such as Math 451 (Advanced Calculus I). Back to Top Math 288 - Math Modeling Workshop Prerequisites: Math 216, 256, 286, or 316, and Math 217, 417, or 419 Frequency: Sporadically Student Body: Junior and Senior mathematics concentrators Credit: 1 Credit. Recent Texts: none Background and Goals: This course is designed to help students understand more clearly how techniques from other undergraduate mathematics courses can be used in concert to solve real-world problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used. Content: During the weekly workshop students will be presented with real-world problems on which techniques of undergraduate mathematics offer insights. They will see examples of (1) how to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can’t be solved, and (4) how to present the solution to others. Students will have a chance to use the skills developed by participating in the U-M Undergraduate Math Modeling Meet. Alternatives: Math 462 (Mathematical Models) is a formal course in mathematical modeling. Subsequent Courses: This course may be repeated for credit. Back to Top Math 289 - Problem Solving Prerequisites: permission Frequency: Fall (I), Winter (II) Student Body: Junior and Senior mathematics concentrators Credit: 1 Credit. Recent Texts: none Background and Goals: One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to honors students. This course is excellent preparation for the Putnam competition. Content: Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students. Alternatives: none Subsequent Courses: This course may be repeated for credit. Back to Top Math 295 - Honors Mathematics I Prerequisites: permission of honors advisor Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits. Recent Texts: Calculus (M. Spivak) Background and Goals: Math 295-296-395-396 is the most theoretical and demanding honors calculus sequence. The emphasis is on concepts, problem solving, as well as the underlying theory and proofs of important results. It provides an excellent background for advanced courses in mathematics. The expected background is high school trigonometry and algebra (previous calculus is not required, but helpful). This sequence is not restricted to students enrolled in the LS&A Honors program. Math 295 and 296 may be substituted for any Math 451 requirement. Math 296 and 395 may be substituted for any Math 513 requirement. Content: Real functions, limits, elementary topology of the real line, continuous functions, derivatives, indefinite and definite integrals. Alternatives: Math 156 (Applied Honors Calc II), Math 175 (Intro. to Cryptology) and Math 185 (Honors Calculus I) are alternative honors courses. Subsequent Courses: Math 296 (Honors Mathematics II) Back to Top Math 296 - Honors Mathematics II Prerequisites: Math 295 Frequency: Winter (II) Student Body: First-year students Credit: 4 Credits. Recent Texts: Calculus (Spivak) Background and Goals: Math 295-296-395-396 is the most theoretical and demanding honors calculus sequence. The emphasis is on concepts, problem solving, as well as the underlying theory and proofs of important results. It provides an excellent background for advanced courses in mathematics. The expected background is high school trigonometry and algebra (previous calculus is not required, but helpful). This sequence is not restricted to students enrolled in the LS&A Honors program. Content: Infinite series, power series, vector spaces, structure of linear maps, duality, eigenvalues, normed vector spaces, higher-dimensional derivatives (Chain rule, inverse/implicit function theorems).  Alternatives: none Subsequent Courses: Math 395 (Honors Analysis I) Back to Top Math 310 - Elementary Topics: Math Games and Theory of Games Prerequisites: Sophomore standing and one previous university math course Frequency: Winter (II) Student Body:   Credit: 3 Credits. Recent Texts: none Background and Goals: The Elementary Topics course may focus on any one of several topics. The material is presented at a level appropriate for sophomores and juniors without extensive coursework in math. The current offering of the course focuses on game theory. Content: Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the structure of a variety of two person games of strategy: tic tac toe, tic tac toe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games. Alternatives: none Subsequent Courses: none Back to Top Math 312 - Applied Modern Algebra Prerequisites: Math 215, 255, or 285 and Math 217 Frequency: Sporadically Student Body: Sophomore and Junior mathematics and computer science concentrators Credit: 3 Credits. 1 credit after Math 412 Recent Texts: A Concrete Introduction to Higher Algebra (Childs) Background and Goals: One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken Math 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. Content: Sets and functions, relations and graphs, rings, Boolean algebras, semigroups, groups, and lattices. Applications from areas such as switching, automata, and coding theory, and may include finite and minimal state machines, algebraic decompositions of logic circuits, semigroup machines, binary codes, and series and parallel decomposition of machines. Alternatives: Math 412 (Introduction to Modern Algebra) is a more abstract and proof-oriented course with less emphasis on applications and is better preparation for most pure mathematics courses. Math 567 is a more advanced course on coding theory. Subsequent Courses: Math 312 is one of the alternative prerequisites for Math 416 (Theory of Algorithms), and several advanced EECS courses make substantial use of the material of Math 312. Another good follow-up course is Math 475 (Elementary Number Theory). Back to Top Math 316 - Differential Equations Prerequisites: Math 215, 255, or 285; and Math 217 Frequency: Fall (I), Winter (II) Student Body: A mix of undergraduate mathematics, science, and economics concentrators Credit: 3 Credits. Credit is granted for only one course among Math 216, 256, 286, and 316. Recent Texts: Elementary Differential Equations and Boundary Value Problems (Boyce and DiPrima) Background and Goals: This is an introduction to differential equations for students who have studied linear algebra (Math 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. Content: First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. Alternatives: Math 216 covers somewhat less material without presupposing linear algebra and with less emphasis on theory. Math 286 (Honors Differential Equations) is the honors version of Math 316. Subsequent Courses: Math 471 (Intro. to Numerical Methods) and/or Math 572 (Numer. Meth. For Sci. Comput. III) are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451 (Advanced Calculus I). Back to Top Math 327 - Evolution of Mathematical Concepts Prerequisites: Math 116 or Math 186 Frequency: Sporadically Student Body: Juniors and seniors interested in mathematics and the history of science Credit: 3 Credits. Recent Texts: none Background and Goals: This course examines the evolution of major mathematical concepts from mathematical and historical points of view. The course's goal is to throw light on contemporary mathematics by retracing the history of some of the major mathematical discoveries. Content: This course follows the evolution of three mathematical ideas in geometry, analysis and algebra. Typical choices of subject are: Euclid's parallel postulate and the development of non-Euclidean geometries, the notions of limit and infinitesimals, and the development of the theory of equations culminating with Galois theory. Alternatives: none Subsequent Courses: none Back to Top Math 351 - Principles of Analysis Prerequisites: Math 215 and 217 or permission of instructor Frequency: Sporadically Student Body: Sophomores and juniors Credit: 3 Credits. No credit after 451 Recent Texts: Understanding Analysis (Abbott) Area: Analysis Background and Goals: The course content is similar to that of Math 451, but Math 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Mathematics. Content: Analysis of the real line, rational and irrational numbers, infinity - large and small, limits, convergence, infinite sequences and series, continuous functions, power series, and differentiation. Alternatives: Math 451 (Advanced Calculus I) covers similar topics while assuming more background than 351. Subsequent Courses: none Back to Top Math 354 - Fourier Analysis and its Applications Prerequisites: Math 216, 256, 286, or 316 Frequency: Sporadically Student Body: Junior and Senior math and non-math concentrators Credit: 3 Credits. No credit after 454 Recent Texts: Fourier and Laplace Transforms (Beerends, et al.) Area: Analysis Background and Goals: This course is an introduction to Fourier analysis with emphasis on applications. The course also can be viewed as a way of deepening one's understanding of the 100-and 200-level material by applying it in interesting ways. Content: This is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications (e.g. signal processing, Fourier optics), and applications in other branches of mathematics (e.g. partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary. Alternatives: Math 454 (Bound Val. Probs. for Part. Diff. Eq.) covers some of the same material with more emphasis on partial differential equations. Subsequent Courses: This course is good preparation for Math 451 (Advanced Calculus I), which covers the theory of calculus in a mathematically rigorous way. Back to Top Math 362 - Applications of Calculus and Linear Algebra Prerequisites: Math 215 and Math 216, 256, 286, or 316; or 217; or Math 396 Frequency: Sporadically Student Body: Junior and Senior math and non-math concentrators Credit: 3 Credits. Credit after 462 only with advisor approval Recent Texts: Mathematics in Medicine and the Life Sciences (Hoppensteadt and Peskin) Area: Applied/NA Background and Goals: This is an introduction to applied mathematics, making use of calculus and linear algebra. While the specific content is very much instructor dependent, the course will always focus on applications of mathematics in the natural or social sciences. Any knowledge required to understand the applications is taught in the course. The goal is to deepen the students' understanding of calculus and linear algebra and motivate them to pursue mathematics further. Content: Instructor dependent. Examples of suitable topics are mathematical biology or mathematical fluid dynamics. No background in the specific application is needed, but often some high-school level knowledge of physics will be helpful or even needed (when in doubt, check with the instructor!) Students will often use computers in this course, but absolutely no previous experience with computers is assumed. Alternatives: Math 462 (Mathematical Models) or Math 463 (Math Modeling in Biology) are similar courses at a somewhat more advanced level. Subsequent Courses: This course is not a specific prerequisite for any subsequent course. Back to Top Math 371 (Engin 303) - Numerical Methods Prerequisites: Engin 101; and Math 216, 255, 286, or 316 Frequency: Fall (I), Winter (II) Student Body: Sophomore, Junior, and Senior engineering students Credit: 3 Credits. No credit after Math 471. Recent Texts: A Friendly Introduction to Numerical Analysis (Bradie) Area: Applied/NA Background and Goals: This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Content: Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear  equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Alternatives: Alternatives: Math 471 (Numerical Analysis) provides a more in-depth study of the same topics, with a greater emphasis on analyzing the accuracy and stability of the numerical methods. Math 571 (Numerical Linear Algebra) is a detailed study of the solution of systems of linear equations and eigenvalue problems, with some emphasis on large-scale problems. Math 572 (Numerical Methods for Differential Equations) covers numerical methods for both ordinary and partial differential equations. (Math 571 and 572 can be taken in either order). Subsequent Courses: This course is basic for many later courses in science and engineering. It is good background for 571-572. Back to Top Math 385 - Math for Elementary School Teachers Prerequisites: One year each of HS algebra and geometry Frequency: Fall (I), Spring (IIIa) of even-numbered years Student Body: Undergraduate concentrators in the Teaching Certificate Program Credit: 3 Credits. No credit after 485 Recent Texts: Elementary Mathematics for Teachers Complete Package (Parker and Baldridge) Area: Teaching Background and Goals: This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Content: Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory) and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. Alternatives: none Subsequent Courses: Math 489 (Math for Elem. and Middle Sch. Teach.) is the required sequel. Back to Top Math 389 - Explorations in Mathematics Prerequisites: None formally; see instructor beforehand. Frequency: Winter (II) Student Body: Undergraduates Credit: 3 Credits. Recent Texts: none Background and Goals: The course is designed to show you how new mathematics is actually created: how to take a problem, make models and experiment with them, and search for underlying structure. The format involves little formal lecturing, and much more laboratory work and student presentation of partial results and approaches. Content: Problems for projects are drawn from a wide variety of mathematical areas, pure and applied. Problems are chosen to be accessible to undergraduates. Please see the course website for more information. Alternatives: none Subsequent Courses: After this course students should be ready for a variety of courses and research experiences. Back to Top Math 395 - Honors Analysis I Prerequisites: Math 296 Frequency: Fall (I) Student Body: First-year students and Sophomores Credit: 4 Credits. Recent Texts: Analysis on Manifolds (Munkres) Background and Goals: This course is a continuation of the sequence Math 295-296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Content: Structure of bilinear forms, tensor products, metric spaces, function spaces, topology of vector spaces, higher-dimensional integrals, change of variables formula, partitions of unity. Alternatives: none Subsequent Courses: Math 396 (Honors Analysis II), Math 512 (Algebraic Structures), Math 525 (Probability Theory) Back to Top Math 396 - Honors Analysis II Prerequisites: Math 395 Frequency: Winter (II) Student Body: Sophomores Credit: 4 Credits. Recent Texts: Analysis on Manifolds (Munkres) Background and Goals: This course is a continuation of Math 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Content: Geometry on manifolds, differential forms, vector fields, Stokes' theorem, deRham cohomology. Alternatives: none Subsequent Courses: Students who have successfully completed the sequence Math 295-396 are generally prepared to take a range of advanced undergraduate and graduate courses such as Math 512 (Algebraic Structures), Math 513 (Intro. to Linear Algebra), Math 525 (Probability Theory), Math 590 (Intro. to Topology), and many others. Back to Top Math 404 - Intermediate Differential Equations Prerequisites: Math 216, 256, 286, or 316 Frequency: Fall (I) sporadically Student Body: undergraduate and graduate students from engineering and LS&A Credit: 3 Credits. Recent Texts: Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering (Strogatz) Area: Applied/NA Background and Goals: This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs. Content: Geometric representation of solutions, autonomous systems, flows and evolution, linear systems and phase portraits, nonlinear systems, local and global behavior, linearization, stability, conservation laws, periodic orbits. Applications: free and forced oscillations, resonance, relaxation oscillations, competing species, Zeeman's models of heartbeat and nerve impulse, chaotic orbits, strange attractors. Alternatives: Math 558 (Applied Nonlinear Dynamics) covers some of the same material at a faster pace and includes additional topics. Subsequent Courses: Math 454 (Boundary Value Problems for Partial Differential Equations) is a natural sequel. Back to Top Math 412 - Introduction to Modern Algebra Prerequisites: Math 215, 255, or 285, and Math 217 Frequency: Fall (I), Winter (II) Student Body: Mainly undergraduate mathematics concentrators with some graduate students from other departments Credit: 3 Credits. 1 credit after Math 312 Recent Texts: Abstract Algebra: an Introduction (T. Hungerford) Area: Algebra Background and Goals: This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions and the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. Math 217 or equivalent required as background. Content: The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of particular types of mathematical structures: groups, rings and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied. Alternatives: Math 312 (Applied Modern Algebra) is a somewhat less abstract course which replaces some of the material on rings and fields of Math 412 with additional applications to areas such as switching and coding theory. Subsequent Courses: A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics : Math 416 (Theory of Algorithms), Math 451 (Advanced Calculus I), Math 475 (Elementary Number Theory), Math 575 (Intro. to Theory of Numbers), Math 513 (Introduction to Linear Algebra), Math 481 (Intro. to Mathematical Logic), and Math 582 (Intro. to Set Theory). All of these courses will extend and deepen the student's grasp of modern abstract mathematics. Back to Top Math 416 - Theory of Algorithms Prerequisites: Math 312 or 412 or EECS 203, and EECS 281 or permission of instructor Frequency: Sporadically Student Body: Largely computer science concentrators with a few graduate students from other fields Credit: 3 Credits. Recent Texts: Algorithm Design (Kleinberg and Tardos) Background and Goals: Many common problems from mathematics and computer science may be solved by applying one or more algorithms- well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas. Content: Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness. Alternatives: This course has substantial overlap with EECS 586 (Design and Analysis of Algorithms)- more or less depending on the instructors. In general, Math 416 will put more emphasis on the analysis aspect in contrast to design of algorithms. Subsequent Courses: Math 516 (Topics in Theory of Algorithms) (given infrequently) and EECS 574 (Computational Complexity) and 575 (Advanced Cryptography) include some topics which follow those of this course. Back to Top Math 417 - Matrix Algebra I Prerequisites: Three mathematics courses beyond Math 110 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: largely engineering and science students, both undergraduate and graduate Credit: 3 Credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 513. Recent Texts: Linear Algebra with Applications (Bretscher) Area: Algebra Background and Goals: Many problems in science, engineering, and mathematics are best formulated in terms of matrices - rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (honors). Content: Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, Eigenvalues and Eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations. Alternatives: Math 419 (Lin. Spaces and Matrix Thy.) is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (Linear Algebra) (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 (Introduction to Linear Algebra) is an honors version of this course, which is also taken by some mathematics graduate students. Mathematics concentrators are required to take Math 217 or Math 513. Subsequent Courses: Math 420 (Matrix Algebra II) is the natural sequel but this course serves as prerequisite to several courses: Math 452 (Advanced Calculus II), Math 462 (Mathematical Models), Math 561 (Linear Programming I), and Math 571 (Numer. Meth. for Sci. Comput. I). Back to Top Math 419 - Linear Spaces and Matrix Theory Prerequisites: 4 mathematics courses beyond Math 110 Frequency: Fall (I), Winter (II) Student Body: largely engineering graduate students and undergraduates; some mathematics undergraduates Credit: 3 Credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 513. Recent Texts: Linear Algebra with Applications (Bretscher) Area: Algebra Background and Goals: Math 419 covers much of the same ground as Math 417 (Matrix Algebra I) but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary.  Content: Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations. Alternatives: Math 417 (Matrix Algebra I) is less rigorous and theoretical and more oriented to applications. Math 217 (Linear Algebra) is similar to Math 419 but slightly more proof-oriented. Math 513 (Introduction to Linear Algebra) is much more abstract and sophisticated. Mathematics concentrators are required to take Math 217 or Math 513. Subsequent Courses: Math 420 (Matrix Algebra II) is the natural sequel but this course serves as prerequisite to several courses: Math 452 (Advanced Calculus II), Math 462 (Mathematical Models), Math 561 (Linear Programming I) , and Math 571 (Numer. Meth. for Sci. Comput. I). Back to Top Math 420 - Matrix Algebra II Prerequisites: Math 214, 217, 417, or 419 Frequency: Varies Student Body: largely engineering and economics graduate students and undergraduates; some mathematics undergraduates Credit: 3 Credits. Recent Texts: Linear Algebra and its Applications 3rd edition (G. Strang) Area: Algebra Background and Goals: This course is designed to improve the student's understanding of the fundamental results of linear algebra and to provide an introduction to some if its most important concepts. Content: Gauss elimination and LDU-factorization. Vector spaces and dimension. Orthogonality, including least-squares approximations, Gram-Schmidt orthogonalization and QR-factorization. Review of determinants. Eigenvalue decompositions and Jordan canonical forms-applications to difference and differential equations. Positive-definite and semi-definite matrices. Singular value decompositions, pseudoinverses, and optimal (minimal norm) solutions. Alternatives: none Subsequent Courses: Math 571 (Numerical Methods for Scientific Computing I). Back to Top Math 422 (BE 440) - Risk Management and Insurance Prerequisites: Math 115, junior standing and permission of instructor Frequency: Winter (II) Student Body: Junior and senior mathematics concentrators; some business undergraduates Credit: 3 Credits. Recent Texts: Introduction to Risk Management and Insurance (Dorfman) Area: Actuarial & Financial Background and Goals: This course is designed to allow students to explore the insurance mechanism as a means of replacing uncertainty with certainty. A main goal of the course is to explain, using mathematical models from the theory of interest, risk theory, credibility theory and ruin theory, how mathematics underlies many important individual and societal problems. Content: We will explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the consumer's needs are equally important. In addition, there are different options available (e.g. in social insurance programs) that offer the opportunity of discussing alternative approaches. This course may be used to satisfy the LS&A upper-level writing requirement. Alternatives: none Subsequent Courses: none Back to Top Math 423 - Mathematics of Finance Prerequisites: Math 217, Math 425 and EECS 183 or equivalents Frequency: Fall (I), Winter (II) Student Body: Junior and senior mathematics concentrators; some business undergraduates Credit: 3 Credits. Recent Texts: Mathematics for Finance: An Introduction to Financial Engineering (Capinksi and Zastawniak) Area: Actuarial & Financial Background and Goals: This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary. Content: Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods and interest rate models. Alternatives: none Subsequent Courses: Math 523 (Risk Theory) and Math 623 (Computational Finance). Back to Top Math 424 - Compound Interest and Life Insurance Prerequisites: Math 215, 255, or 285 Frequency: Fall (I), Winter (II) Student Body: undergraduate and graduate students in the Actuarial Mathematics Program, or students interested in exploring the concepts underlying the theory of interest. Credit: 3 Credits. Recent Texts: Mathematical Interest Theory (Daniel and Vaaler) Area: Actuarial & Financial Background and Goals: This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for some of the professional actuarial exams. Content: The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the professional examinations; the material covered will depend somewhat on the instructor. Alternatives: Math 424 is required for students concentrating in actuarial mathematics; others may take Math 147 (Introduction to Interest Theory), which deals with the same techniques but with less emphasis on continuous growth situations.  Subsequent Courses: Math 520 (Life Contingencies I) applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions). Back to Top Math 425 (Stat 425) - Introduction to Probability Prerequisites: Math 215, 255, or 285 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb) Student Body: about 80% undergraduate mathematics, engineering, and computer science concentrators with a few graduate students Credit: 3 Credits. Recent Texts: A First Course in Probability 7th edition (S. Ross) Area: Analysis Background and Goals: This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. Content: Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. Alternatives: Math 525 (Probability Theory) is a similar course for students with stronger mathematical background and ability. Subsequent Courses: Stat 426 (Intro. to Theoretical Stat.) is a natural sequel for students. Math 423 (Mathematics of Finance) and Math 523 (Risk Theory) include many applications of probability theory. Back to Top Math 427 - Retirement Plans and Other Employee Benefits Prerequisites: Junior standing or permission of instructor Frequency: Winter (II) Student Body: Mainly Actuarial Mathematics students, but also some non-mathematics students Credit: 3 Credits. Recent Texts: Pension Planning: Pensions, Profit-Sharing, and Other Deferred Compensation Plans (Allen et al.) Area: Actuarial & Financial Background and Goals: An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. This course is certified for satisfaction of the upper-level writing requirement. Content: Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course. Alternatives: none Subsequent Courses: Math 521 (Life Contingencies II) and/or Math 522 (Act. Theory of Pensions and Soc. Sec.) (which can be taken independently of each other) provide more in-depth examination of the actuarial techniques used in employee benefit plans. Back to Top Math 431 - Topics in Geometry for Teachers Prerequisites: Math 215 Frequency: Fall (I) Student Body: mainly teaching certificate candidates; a few other mathematics and engineering students Credit: 3 Credits. Recent Texts: Euclidean and Non-Euclidean Geometry (Greenberg) Area: Teaching Background and Goals: This course is a study of the axiomatic foundations of Euclidean and non-Euclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Content: Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tesselations; triangle groups; finite, hyperbolic, and taxicab non-Euclidean geometries. Alternatives: Alternative geometry courses at this level are Math 432 (Projective Geometry) and Math 433 (Intro to Differential Geometry).  Subsequent Courses: Although it is not strictly a prerequisite, Math 431 is good preparation for Math 531 (Transformation Groups in Geom.). Back to Top Math 432 - Projective Geometry Prerequisites: Math 215, 255, or 285 Frequency: sporadically Student Body:  Credit: 3 Credits. Recent Texts: Projective Geometry 2nd edition (H.M.S. Coxeter) Area: Geometry/Topology Background and Goals: The primary goal of the course is to introduce students to non-standard concepts of geometry. A secondary goal is to provide students with experience with proofs and to make them more comfortable with abstract notions. Concepts and proofs are emphasized; there is almost no calculational component. Of course additional background is helpful, but the course is entirely self-contained.  Content: Content will vary with the instructor; one recent version included the following topics. Combinatorial Geometries: incidence structures, geometric lattices, affine and linear subspaces, graphic matroids; Projective Spaces: axioms, duality, harmonic relation, Pappas' and Desargues' Theorems, projectivities, conics; Projective Planes: finite fields, constructing and coordinatizing projective planes. Another version provided examples of the application of linear algebra to geometric problems.  Alternatives: The experience with proofs in Math 432 can be useful in other proof-oriented courses such as Math 412 (Introduction to Modern Algebra) and Math 451 (Advanced Calculus I). Subsequent Courses: none Back to Top Math 433 - Introduction to Differential Geometry Prerequisites: Math 215, 255, or 285; and Math 217 Frequency: Fall (I) Student Body: Half undergraduate mathematics concentrators, half graduate students from EECS and physics Credit: 3 Credits. Recent Texts: Elements of Differential Geometry (Millman and Parker) Area: Geometry/Topology Background and Goals: This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Content: Curves and surfaces in three-space using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem. Minimal surfaces.  Alternatives: Math 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513) and advanced multivariable calculus (452). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Subsequent Courses: Math 635 (Differential Geometry) and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course. Back to Top Math 450 - Advanced Mathematics for Engineers I Prerequisites: Math 215, 255, or 285, and Math 216, 256, 286, or 316 Frequency: Fall (I), Winter (II), Summer (IIIb) Student Body: Undergraduate mathematics and engineering students; engineering graduate students Credit: 4 Credits. No credit after Math 454. Recent Texts: Advanced Engineering Mathematics (Greenberg) Area: Analysis Background and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout. Content: Topics covered include: Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed. Alternatives: This course overlaps with 454 and, to a much lesser extent, with 555. The coverage of PDEs in 450 is not as in-depth as 454; for example, in 450 coverage of special functions is reduced to the simplest Bessel functions. Those students needing a more thorough discussion of PDEs and boundary-value problems should take 454. On the other hand, 450 should provide a broader introduction to applied methods. Subsequent Courses: Math 555 (Complex Variables) and Math 556 (Methods of Applied Math I) are graduate-level courses that further develops both the theory and applications covered in 450. Back to Top Math 451 - Advanced Calculus I Prerequisites: Math 285, or Math 215 and one subsequent course Frequency: Fall (I), Winter (II), Spring (IIIa) Student Body: Half undergraduate mathematics concentrators, half science and engineering graduate students Credit: 3 Credits. No credit after 351. Recent Texts: Elementary Analysis: The Theory of Calculus (K. Ross) Area: Analysis Background and Goals: This course has two complementary goals: (1) a rigorous development of the fundamental ideas of Calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Content: Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions. Alternatives: There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. Math 351 covers similar topics with much less rigor. Subsequent Courses: The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 (Advanced Multivariable Calculus) is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575 (Intro. to Theory of Numbers), Math 590 (Intro. to Topology), Math 596 (Analysis I) (Complex)), and Math 597 (Analysis II (Real)). Back to Top Math 452 - Advanced Calculus II Prerequisites: Math 217, 417, or 419 (may be concurrent) and Math 451 Frequency: Winter (II) Student Body: A majority of mathematics undergraduates with some non-mathematics graduate students Credit: 3 Credits. Recent Texts: Analysis on Manifolds (Munkres), Advanced Calculus of Several Variables (Edwards) Area: Analysis Background and Goals: This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Content: Topics include (1) partial derivatives and differentiability, (2) gradients, directional derivatives, and the chain rule, (3) implicit function theorem, (4) surfaces, tangent plane, (5) max-min theory, (6) multiple integration, change of variable, etc. (7) Green's and Stokes' theorems, differential forms, exterior derivatives. Alternatives: none Subsequent Courses: Math 452 is prerequisite to Math 572 and is good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 537, 635). Back to Top Math 454 - Boundary Value Problems for Partial Differential Equations Prerequisites: Math 216, 256, 286, or 316 Frequency: Fall (I), Winter (II), Spring (IIIa) Student Body: Some mathematics undergraduates, but more non-mathematics graduate students Credit: 3 Credits. 1 credit after Math 354. No credit after Math 450. Recent Texts: Applied Partial Differential Equations (Haberman) Area: Applied/NA Background and Goals: This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of initial-value and boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Content: Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; eigenfunction expansions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Laplace's equation and harmonic functions, including the maximum principle. As time permits, additional topics will be selected from: Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis; dispersive wave equations; the method of stationary phase; the method of characteristics. Alternatives: Both Math 455 (Bound Val. Probs. and Complex Anal.) and Math 554 (Advanced Math for Engineers) cover many of the same topics but are very seldom offered.  Subsequent Courses: Math 454 is prerequisite to Math 571 (Numer. Meth. for Sci. Comput. I) and Math 572 (Numer. Meth. for Sci. Comput. II). Although it is not a formal prerequisite, it is good background for Math 556 (Methods of Applied Math I). Back to Top Math 462 - Mathematical Models Prerequisites: Math 216, 256, 286, or 316, and Math 217, 417, or 419 Frequency: Winter (II) sporadically Student Body: Mainly Junior and Senior mathematics concentrators; students from engineering, biology, physics, and medicine Credit: 3 Credits. 1-3 credits after 362 depending on overlap Recent Texts: Mathematical Models (R. Haberman) Area: Applied/NA Background and Goals: The focus of this course is the application of a variety of mathematical techniques to solve real-world problems. Students will learn how to model a problem in mathematical terms and use mathematics to gain insight and eventually solve the problem. Concepts and calculations, using applied analysis and numerical simulations, are emphasized. Content: Construction and analysis of mathematical models in physics, engineering, economics, biology, medicine, and social sciences. Content varies considerably with instructor. Recent versions: Use and theory of dynamical systems (chaotic dynamics, ecological and biological models, classical mechanics), and mathematical models in physiology and population biology. Alternatives: Students who are particularly interested in biology should considered Math 463 (Math Modeling in Biology). Subsequent Courses: any higher-level course in differential equations Back to Top Math 463 - Math Modeling in Biology Prerequisites: Math 217, 417 or 419; and 216, 286 or 316 Frequency: Fall (I) Student Body: Juniors, seniors, and Master's level; half Engineering and half LS&A Credit: 3 Credits. Recent Texts: Mathematical Models in Biology (Keshet-Edelstein); Mathematical Biology (Murray) Area: Applied/NA Background and Goals:It is widely anticipated that Biology and Medicine will be the premier sciences of the 21st century. The complexities of the biological sciences make interdisciplinary involvement essential and the increasing use of mathematics in biology is inevitable as biology becomes more quantitative. Mathematical biology is a fast growing and exciting modern application of mathematics that has gained worldwide recognition. In this course, mathematical models that suggest possible mechanisms that may underlie specific biological processes are developed and analyzed. Another major emphasis of the course is illustrating how these models can be used to predict what may follow under currently untested conditions. The course moves from classical to contemporary models at the population, organ, cellular, and molecular levels. The goals of this course are: (1) Critical understanding of the use of differential equation methods in mathematical biology and (2) Exposure to specialized mathematical and computations techniques which are required to study ordinary differential equations that arise in mathematical biology. By the end of this course students will be able to derive, interpret, solve, understand, discuss, and critique discrete and differential equation models of biological systems. Content: This course provides an introduction to the use of continuous and discrete differential equations in the biological sciences. Biological topics may include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions and receptor-ligand binding, biological oscillators, and an introduction to biological pattern formation. There will also be discussions of current topics of interest such as Tumor Growth and Angiogenesis, HIV and AIDS, and Control of the Mitotic Clock. Mathematical tools such as phase portraits, bifurcation diagrams, perturbation theory, and parameter estimation techniques that are necessary to analyze and interpret biological models will also be covered. Approximately one class period each week will be held in the mathematics computer laboratory where numerical techniques for finding and visualizing solutions of differential and discrete systems will be discussed. Alternatives: none Subsequent Courses: Math 563 Back to Top Math 464 - Inverse Problems Prerequisites: Math 217, 417, or 419, and Math 216, 256, 286, or 316 Frequency: Sporadic