Undergraduate Courses 105 | 107 | 115 | 116 | 127 | 128 | 147 | 156 | 174 | 175 | 176 | 185 | 186 | 214 | 215 | 216 | 217 | 255 | 256 | 285 | 286 |289 | 295 | 296 | 310 | 312 | 316 | 327 | 351 | 354 | 371 | 385 | 389 | 395 | 396 | 404 | 412 | 416 | 417 | 419 | 420 |422 | 423 | 424 | 425 | 427 | 431 | 433 | 437 | 450 | 451 | 452 | 454 | 462 | 463 | 464 | 465 | 466 | 471 | 472 | 475 | 476 | 481 | 486 | 489 | 490 | 497 | 498 | 512 | 513 | 520 | 521 | 522 | 523 | 524 | 525 | 526 | 528 | 547 | 550 | 555 | 556 | 557 | 558 | 559 | 561 | 562 | 563 | 564 | 565 | 566 | 567 | 571 | 572 | 575 | 582 | 583 | 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 including trigonometry 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: A Preparation for Calculus (3rd edition) (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 107, offered only in the winter term, is a course designed for students no necessarily planning to take calculus. Subsequent Courses: The course prepares students for Math 115 (Calculus I). Back to Top   Math 107 - Mathematics for the Information Age Prerequisites: 3-4 years HS math Frequency: Winter (II) Student Body: First-year students (non-mathematics concentrators) who are not necessarily required to take calculus. Credit: 3 Credits. Recent Texts: Background and Goals: The course will investigate topics relevant to the information age in which we live. An investigation of cryptography and coding methods, including prime numbers, randomness, and data compression will lead to the mathematics of the web. Use of interactive web sites and web data are an integral part of the course. The course will emphasize the representation of mathematical data in graphical , tabular, and symbolic forms and investigate the inferences that can be drawn from these models. Emphasis will be placed on the development of estimation skills, the ability to determine reasonableness of answers, and the ability to find alternative approaches to a problem. Content: Typical topics include cryptography, coding, politics, biological data, populations, chaos, and game theory. Topics will be presented as modules, generally a week or two in length. Alternatives: None. Subsequent Courses: Math 128 or Math 127 could be taken after Math 107. 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. No credit after Math 116, 215 or 216. 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 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. No credit after Math 215 or 216. 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 186 (Honors Calculus II) is a more theoretical course which covers much of the same material. Math 156 (Applied Honors Calculus II) also covers much of the same material using MAPLE and emphasizing applications to science and engineering. 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 course 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. We then study the 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 students who want to engage in mathematical reasoning without having to take calculus first. It is particularly well-suited for non-science concentrators or those who are thoroughly undecided. Students will make use of software 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 115 Frequency: Fall (I), Winter (II) Student Body: First and second year students 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 Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits. Recent Texts: Single Variable Calculus (UM Edition) by 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: Geometry Revisited (1st edition) by Coxeter and Greitzer 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 of 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: None Background and Goals: This course is an alternative to Math 185 as an entry to the honors calculus sequence. The course stresses discovery as a vehicle for learning. Content: This course gives a historical introduction to Cryptology, from ancient times up to modern public key encryption, particularly RSA, and introduces a number of mathematical ideas involved in the development and analysis of codes. Mathematical topics include some enumeration, probability, and statistics, but the bulk of the course is devoted to elementary number theory. Students also work throughout the course on effectively communicating mathematics, both written and orally. Moreover, students will develop rigorous mathematical proof writing skills, and a primary goal of the course is to not only understand how various cryptosystems work, but why. Structure: The course has two components, classroom and computer lab. The classroom component meets three days each week, and is driven by in-class worksheets students complete in small groups. Each worksheet consists of definitions, examples, problems, and mathematical results that students attempt to understand through discussion with their peers and the instructor. As students solve problems from the worksheet, they present their solutions to the rest of the class. In the computer lab, various discovery-based projects allow the students to explore the ideas developed in the classroom and cryptosystems not covered in the worksheets.  No previous experience with computer programming is necessary. 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 - Explorations in Topology and Analysis Prerequisites: Math 175 or permission of instructor Frequency: Winter (II) Student Body: First-year students Credit: 4 Credits Recent Texts: None 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. 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 Frequency: Fall (I) Student Body: First-year students Credit: 4 Credits Recent Texts: Calculus (4th ed.) by Michael Spivak, Calculus and Linear Algebra (UM edition) by Wilfred Kaplan Background and Goals: Most students take calculus in high school, and it may seem that there isn't much new to learn. The goal of this course is to develop the familiar concepts of calculus using a more rigorous and theoretical approach. In particular, with its emphasis on how to use appropriate mathematical language, this course lays a solid foundation for future math courses, and is suitable for students intending to pursue a concentration in mathematics, science, or engineering who desire a more complete understanding of the underpinnings of calculus. Considerable attention is paid to developing problem solving skills. This sequence is not restricted to students enrolled in the LSA Honors Program. Content:Topics covered include functions, graphs, continuity, limits, derivatives, and integrals. Tuesday meetings are usually devoted to introducing linear algebra. Alternatives: Math 115 (Calculus I) is a 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 (4th edition) by Michael Spivak Background and Goals:Most students take calculus in high school, and it may seem that there isn't much new to learn. The goal of this course is to develop the familiar concepts of calculus using a more rigorous and theoretical approach. In particular, with its emphasis on how to use appropriate mathematical language, this course lays a solid foundation for future math courses, and is suitable for students intending to pursue a concentration in mathematics, science, or engineering who desire a more complete understanding of the underpinnings of calculus. Considerable attention is paid to developing problem solving skills. This sequence is not restricted to students enrolled in the LSA Honors Program. This course is a continuation of Math 185. Content: Topics include integral calculus, transcendental functions, infinite sequences and series (including Taylor's series), and - time permitting - some simple applications to elementary differential equations. Tuesdays are mostly devoted to an introduction to linear algebra. 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 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. Recent Texts: Linear Algebra with Applications (4th edition) by 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 courses 214&215 are designed as an alternative to Math 215&216 for students who need more linear algebra and less differential equations background. 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 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 (6th edition) by 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 (respectively, linear algebra) should take Math 216 (respectively, Math 214). 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 (4th edition) by 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. 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 (3rd updated edition) by 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 (6th edition) by 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) is the natural sequel. 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 (9th edition) by 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 of instructor 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 (6th edition) by 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 of instructor 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 (9th edition) by 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 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 (4th edition) by 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 217 requirement. Content: Axioms of the real numbers, completeness and connectedness in the real line. Functions of a real variable, limits and continuity, uniform continuity, extreme and intermediate value theorems, differentiation, integration, the fundamental theorem of calculus, Taylor's theorem with remainder. 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 (4th edition) by Spivak, Linear Algebra by Kunze and Hoffman 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: Sequences and series of functions, power series, uniform convergence, real analytic functions. Vector spaces, bases, linear transformations, dual spaces determinants, traces, eigenvalues, inner-product spaces, spectral theory. Limits and continuity in Euclidean space, derivative as a linear map, Chain rule, inverse/implicit function theorems. Alternatives: none Subsequent Courses: Math 395 (Honors Analysis I) Back to Top Math 310 - Elementary Topics: Explorations in Randomness 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. This will be an inquiry based learning class which does not stress lecturing. Instead, the current offering of the course is a hands-on exploration of probability. Students will experiment throughout the course by working on a range of problems in peer groups inside of class. Content: The notions of probability and randomness are essential aspects of everyday life. In this course, we will uncover a number of mathematical surprises and structures that arise from these seemingly unstructured notions. Topics include basic probability, random walks, conditional probability, expectations, randomized strategies in games, Poisson distribution, and branching processes. Alternatives: none Subsequent Courses: none Back to Top Math 312 - Applied Modern Algebra Prerequisites: Math 215, 255, or 285 and Math 217 Frequency: Fall (I) 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 (3rd edition) by 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. EECS 203 (Discrete Structures) covers some of the same topics with a more applied approach. 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 (9th edition) by 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: Fall (I), Winter (II) Student Body: Sophomores and Juniors Credit: 3 Credits. No credit after 451 Recent Texts: Understanding Analysis by S. 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 - 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: Fall (I), Winter (II) Student Body: Junior and Senior math and non-math concentrators Credit: 3 Credits. No credit after 454 Recent Texts: Fourier Series and Orthogonal Polynomials by Jackson 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 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 by B. 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 software library subroutines. 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: 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. Recent Texts: Mathematics for Elementary School Teachers (4th edition) by Bassarear 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 sequel. Back to Top Math 389 - Explorations in Mathematics Research 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, much laboratory work, and student presentation of partial results and approaches. Please see the course website for more information. Content: Problems for projects are drawn from a wide variety of mathematical areas, pure and applied. Problems are chosen to be accessible to undergraduates. 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 by 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: Inverse/implicit function theorems, immersion/submersion theorems. Quotient and dual spaces, inner product spaces, spectral theory. Metric spaces, basic point-set topology. Integration in Euclidean space, Fubini's theorem, change of variables formula. Topics in linear algebra: tensor products, exterior and symmetric powers, Jordan and rational canonical forms. Alternatives: None Subsequent Courses: Math 396 (Honors Analysis II) Back to Top Math 396 - Honors Analysis II Prerequisites: Math 395 Frequency: Winter (II) Student Body: Sophomores Credit: 4 Credits. Recent Texts: Comprehensive Introduction to Differential Geometry (3rd edition) by Spivak 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: Submanifolds (with or without corners) of Euclidean space, abstract manifolds, tangent and cotangent spaces, immersion/submersion theorems/ Partitions of unity, vector fields and differential forms on manifolds, exterior differentiation, integration of differential forms. Stoke's theorem. deRham cohomology, Riemannian metrics, Hodge star operator and the standard vector calculus versions of Stoke's theorem. 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), Winter (II) 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 by 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: Introduction to Abstract Algebra (7th edition) by McCoy and Janusz 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, 412, or EECS 280 and Math 465, 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 by 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: EECS 574 (Computational Complexity) and 575 (Advanced Cryptography) include some topcis 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 (4th edition) by 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 if pursuing the honors concentration. 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) is also a more theoretical course which covers much of the material of 417 at a deeper level (despite its lower number). Math 513 (Introduction to Linear Algebra) should be elected if pursuing honors and is also taken by some mathematics graduate students. Subsequent Courses: 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 and its Applications (4th Edition) by Strang 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. Subsequent Courses: 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. Comp. I). Back to Top Math 420 - Advanced Linear Algebra Prerequisites: Math 214, 217, 417, or 419 and one of 296, 412, or 451 Frequency: Winter (II) Student Body: Graduate students who pursue, or may pursue, the Mathematics masters program and undergraduate mathematics concentrators. This is a required course for MS degree in mathematics. Credit: 3 Credits Recent Texts: Linear Algebra (2nd Edition) [Hardcover], Kenneth M Hoffman (Author), Ray Kunze (Author) Area: Algebra Background and Goals: This is an introduction to the formal theory of abstract vector spaces and linear transformations. It is expected that students have complete at least one prior linear algebra course. The emphasis is on concepts and proofs with some calculations to illustrate the theory. Students should have significant mathematical maturity, at the level of Math 412 or 451. In particular, students should expect to work with and be tested on formal proofs. Content: Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; inner product spaces, duals and spaces of linear transformations, theory of determinants, eigenvalues and eigenvectors; applications to linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form, least squares, singular value theory. Alternatives: None Subsequent Courses: The natural sequel to Math 420 is Math 593 (Algebra 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 (8th edition) by 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 by 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, the capital asset pricing model, the 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 (2nd edition) by 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 and probability theory to the valuation of life contingencies (death benefits and pensions). Back to Top Math 425 (Stats 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 (8th edition) by 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, joint distributions, expectations, variances, covariances. The culminating results are the Law of Large Numbers and the Central Limit Theorem. Beyond this, different instructors may add additional topics of interest. Alternatives: Math 525 (Probability Theory) is a similar course at a faster pace and with deeper coverage. A stronger mathematical background is helpful for Math 525. Subsequent Courses: Stats 426 (Introduction to Theoretical Statistics) 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: Math 115, Junior standing, or permission of instructor Frequency: Sporadically 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 by 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 (4th edition) by 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 is Math 433 (Intro to Differential Geometry).  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 by 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 437 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 437: Introduction to Differentiable Manifolds Prerequisites: Math 590 and 513 Frequency: Fall (I) Credit: 3 credits Recent Texts: Differential Topology (Guillemin and Pollack), Riemannian Geometry (Do Carmo) Past instructors: J. Lott, A. Uribe Student Body: Mainly graduate students in mathematics Background and Goals: This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses 513 and 590. Its goal is to introduce the basic concepts and results of differential topology and differential geometry.  Content: Manifolds, vector fields and flows, differential forms, Stokes' theorem, Lie group basics, Riemannian metrics, Levi-Civita connection, geodesics Alternatives: Math 433 (Intro to Differential Geometry) is an undergraduate version which covers less material in a less sophisticated way.  Subsequent Courses: Math 635 (Differential Geometry)  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. 1 credit after Math 354, No credit after Math 454. Recent Texts: Advanced Engineering Mathematics (2nd edition) by 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: A thorough understanding of Calculus and one of 217, 312, 412, or permission of instructor. 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 by 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 451 is also a prerequisite for several other courses: Math 575, Math 590, Math 596, and Math 597. Back to Top Math 452 - Advanced Calculus II Prerequisites: Math 217, 417, or 419 (may be concurrent) and Math 451 Frequency: Sporadically Student Body: A majority of mathematics undergraduates with some non-mathematics graduate students Credit: 3 Credits. Recent Texts: Advanced Calculus of Several Variables by 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 good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 437, 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: Partial Differential Equations for Scientists by Farlow Area: Applied/Numerical Analysis 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: None 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) Student Body: Mainly Junior and Senior mathematics concentrators; students from engineering, biology, physics, and medicine Credit: 3 Credits. 1-3 credits after 362, 463 depending on overlap Recent Texts: Mathematical Models in Biology (2nd edition) by Edelstein-Keshet Area: Applied/Numerical Analysis 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, 256, 286 or 316 Frequency: Fall (I) Student Body: Juniors, Seniors, and first year graduate students (half engineering and half LS&A) Credit: 3 Credits. Recent Texts: Mathematical Models in Biology (5th edition) by L. Edelstein -Keshet; Mathematical Biology (3rd edition) by J.D. Murray Area: Applied/Interdisciplinary. 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: Sporadically Student Body: Upper-level undergraduates and graduate students in mathematics, science, and engineering Credit: 3 Credits. Recent Texts: None Area: Applied/Numerical Analysis Background and Goals: Solution of an inverse problem is a central component of fields such as medical tomography, geophysics, non-destructive testing, and control theory. The solution of any practical inverse problem is an interdisciplinary task. Each such problem requires a blending of mathematical constructs and physical realities. Thus, each problem has its own unique components; on the other hand, there is a common mathematical framework for these problems and their solutions. This framework is the primary content of the course. This course will allow students interested in the above-named fields to have an opportunity to study mathematical tools related to the mathematical foundations. Content: The course content is motivated by a particular inverse problem from a field such as medical tomography (transmission, emission), geophysics (remote sensing, inverse scattering, tomography), or non-destructive testing. Mathematical topics include ill-posedness (existence, uniqueness, stability), regularization (e.g. Tikhonov, least squares, modified least squares, variation, mollification), pseudoinverses, transforms (e.g. k-plane, Radon, X-ray, Hilbert), special functions, and singular-value decomposition. Physical aspects of particular inverse problems will be introduced as needed, but the emphasis of the course is investigation of the mathematical concepts related to analysis and solution of inverse problems. Alternatives: None Subsequent Courses: None Back to Top Math 465 - Introduction to Combinatorics Prerequisites:Linear Algebra (one of MATH 214, 217, 296, 417, or 419) or permission of instructor Frequency: Fall (I), Winter (II) Student Body: Upper-level undergraduates and graduate students in mathematics, science, and engineering Credit: 3 Credits. No credit granted to those who have completed or are enrolled in MATH 565 or 566. Recent Texts: Introductory Combinatorics (4th edition) by R. Brualdi Area: Discrete Mathematics Background and Goals: Combinatorics is the study of finite mathematical objects, including their enumeration, structural properties, design, and optimization. Combinatorics plays an increasingly important role in various branches of mathematics and in numerous applications, including computer science, statistics and statistical physics, operations research, bioinformatics, and electrical engineering. This course provides an elementary introduction to the fundamental notions, techniques, and theorems of enumerative combinatorics and graph theory. Content: An introduction to combinatorics, covering basic counting techniques (inclusion-exclusion, permutations and combinations, generating functions) and fundamentals of graph theory (paths and cycles, trees, graph coloring). Additional topics may include partially ordered sets, recurrence relations, partitions, matching theory, and combinatorial algorithms. Alternatives: 565 (offered in the Fall) is significantly more demanding and proof-oriented. Math 566 is even more advanced. Subsequent Courses: Math 565 and 566. Back to Top   Math 466 (EEB 466) - Mathematical Ecology Prerequisites: MATH 217, 417, or 419; MATH 256, 286, or 316; and MATH 450 or 451 Frequency: TBD Student Body: The course is intended for graduate students and advanced undergraduates interested in the mathematical analysis of model of ecological systems. Credit: 3 Credits. Recent Texts: None Area: Applied Background and Goals: This course gives an overview of mathematical approaches to questions in the science of ecology.  Topics include: formulation of deterministic and stochastic population models; dynamics of single-species populations; and dynamics of interacting populations (predation, competition, and mutualism), structured populations, and epidemiology. Emphasis is placed on model formulation and techniques of analysis. Content: Why do some diseases become pandemic? Why do certain species introductions result in widespread invasions? Why do some populations grow while others decline and still others cycle rhythmically? How are all of these phenomena influenced by climate change? These and many other fundamental questions in the science of ecology are intrinsically quantitative and concern highly complex systems. To answer them, ecologists formulate and study mathematical models. This course is intended to provide an overview of the principal ecological models and the mathematical techniques available for their analysis. Emphasis is placed on model formulation and techniques of analysis. Although the focus is on ecological dynamics, the methods we discuss are readily applicable across the sciences. The course presumes mathematical maturity at the level of advanced calculus with prior exposure to ordinary differential equations, linear algebra, and probability. Alternatives: None Subsequent Courses: None Back to Top Math 471 - Introduction to Numerical Methods Prerequisites: Math 216, 256, 286, or 316; Math 217, 417, or 419; and a working knowledge of one high-level computer language Frequency: Fall (I), Winter (II), Summer (IIIb) Student Body: Juniors, Seniors, and Master's level; half Engineering, half LS&A Credit: 3 Credits. No credit after Math 371 Recent Texts: A Friendly Introduction to Numerical Analysis by Bradie Area: Applied/Numerical Analysis Background and Goals: This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Content: Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation. Alternatives: Math 371/Engin. 303 (Numerical Methods) is a less sophisticated version intended principally for Sophomore and Junior engineering students; the sequence Math 571-572 (Numer. Meth. for Sci. Comput. I & II) is mainly taken by graduate students, but should be considered by strong undergraduates.  Subsequent Courses: Math 471 is good preparation for Math 571 (Numer. Meth. for Sci. Comput. I) and Math 572 (Numer. Meth. for Sci. Comput. II), although it is not prerequisite to these Back to Top Math 472 - Numerical Methods with Financial Applications Prerequisites: Math 216, 256, 286, or 316; Math 217, 417, or 419; and a working knowledge of one high-level computer language. Math 425 is recommended. Frequency:  Student Body: concentrators in the Actuarial Mathematics and Financial Mathematics programs Credit: 3 Credits. No credit after Math 371 or 471 Recent Texts: A Friendly Introduction to Numerical Analysis (Bradie) Area: Actuarial & Financial Background and Goals: This is a survey of the basic numerical methods which are used to solve scientific problems. The goals of the course are similar to those of Math 471 but the applications are chosen to be of interest to students in the Actuarial Mathematics and Financial Mathematics programs. Content: Topics may include: Newton's method for non-linear equations, systems of linear equations, numerical integration, interpolation and polynomial approximation, ordinary differential equations, partial differential equations - in particular the Black-Scholes equation, Monte Carlo simulation, and numerical modeling. Alternatives: Math 371/Engin. 303 (Numerical Methods) is a less sophisticated version intended principally for Sophomore and Junior engineering students. Subsequent Courses: none Back to Top Math 475 - Elementary Number Theory Prerequisites: no specific prerequisite Frequency: Winter (II) Student Body: Mainly mathematics undergraduates; some non-mathematics undergraduates and graduate students Credit: 3 Credits. Recent Texts: An Introduction to the Theory of Numbers (Niven, Zuckerman and Montgomery) Area: Number Theory Background and Goals: This is an elementary introduction to number theory, especially congruence arithmetic. Number Theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and ciphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly number-theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. At least three semesters of college mathematics are recommended. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. Content: Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. This material corresponds to Chapters 1-3 and selected parts of Chapter 5 of Niven and Zuckerman. Alternatives: Math 575 (Intro. to Theory of Numbers) moves much faster, covers more material, and requires more difficult exercises. There is some overlap with Math 412 (Introduction to Modern Algebra) which stresses the algebraic content.  Subsequent Courses: Math 475 may be followed by Math 575 (Intro. to Theory of Numbers) and is good preparation for Math 412 (Introduction to Modern Algebra). All of the advanced number theory courses, Math 675, 676, 677, 678, and 679, presuppose the material of Math 575, although a good student may get by with Math 475. Back to Top Math 476 - Computational Laboratory in Number Theory Prerequisites: prior or concurrent enrollment in Math 475 or 575 Frequency: Winter (II) Student Body: undergraduate mathematics concentrators Credit: 1 Credit. Recent Texts: coursepack Area: Number Theory Background and Goals: Intended as a companion course to Math 475 or 575. Participation should boost the student's performance in either of those classes. Students in the Lab will see mathematics as an exploratory science (as mathematicians do).  Content: Students will be provided with software with which to conduct numerical explorations. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Students will gain a knowledge of algorithms which have been developed for number theoretic purposes, e.g. for factoring. Alternatives: none Subsequent Courses: none Back to Top Math 481 - Introduction to Mathematical Logic Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics Frequency: Fall (I) Student Body: Undergraduate mathematics, philosophy, and computer science concentrators plus a few non-math graduate students Credit: 3 Credits. Recent Texts: A Mathematical Introduction to Logic (Enderton) Area: Logic Background and Goals: All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite.  Content: In the first third of the course the notion of a formal language is introduced and propositional connectives ('and', 'or', 'not', 'implies'), tautologies and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability leads to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including non-standard analysis. This material corresponds to Chapter 1 and sections 2.0-2.5 and 2.8 of Enderton.  Alternatives: Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Subsequent Courses: Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics. Back to Top Math 486 - Concepts Basic to Secondary School Mathematics Prerequisites: Math 215 Frequency: Winter (II) Student Body: undergraduate concentrators in the Teaching Certificate Program and "minors" in other teaching programs Credit: 3 Credits. Recent Texts: Mathematics for High School Teachers: An Advanced Perspective (Peressini, Usiskin, Marchisotto and Stanley) Area: Teaching Background and Goals: This course is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade.  Content: Topics covered have included problem solving; sets, relations and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. Alternatives: There is no real alternative, but the requirement of Math 486 may be waived for strong students who intend to do graduate work in mathematics. Subsequent Courses: Prior completion of Math 486 may be of use for some students planning to take Math 312, 412, or 425. Back to Top Math 489 - Math for Elem. and Middle School Teachers Prerequisites: Math 385 or 485 or permission Frequency: Winter (II) Student Body: undergraduates in the Elementary Teaching Certificate Program Credit: 3 Credits. Recent Texts: Elementary Mathematics for Teachers (Parker and Baldridge) Area: Teaching Background and Goals: This course, together with its predecessor Math 385, 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 fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course. Alternatives: There is no alternative course. Subsequent Courses: Math 497 (Topics in Elementary Mathematics) Back to Top Math 490 - Introduction to Topology Prerequisites: Math 351 or 451 or equivalent experience with abstract mathematics Frequency: Winter (II) Student Body: Mathematics, math-education, science and engineering Credit: 3 Credits. Recent Texts: Basic Topology (Armstrong) Area: Geometry/Topology Background and Goals: Topology is the study of a class of interesting spaces, geometric examples of which are knots and surfaces. We focus on those properties of such spaces which don't change if the space is deformed. Much of the course is devoted to understanding particular spaces, such as Moebius strips and Klein bottles. The material in this course has a wide range of applications. Most of the material is theoretical, but it is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. Content: Knots, orientable and non-orientable surfaces, Euler characteristic, open sets, connectedness, compactness, metric spaces. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor.  Alternatives: Math 590 (Intro. to Topology) is a deeper and more difficult presentation of much of the same material. Math 433 (Intro. to Differential Geometry) is a related course at about the same level. Subsequent Courses: Math 490 is not prerequisite for any later course but provides good background for Math 591 (General and Differential Topology) or any of the other courses in geometry or topology. Back to Top Math 497 - Topics in Elementary Mathematics Prerequisites: Math 489 or permission of instructor Frequency: Fall (I) Student Body: undergraduates in the Elementary Teaching Certificate Program Credit: 3 Credits. Recent Texts: coursepack Area: Teaching Background and Goals: This is a required course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class usually meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam. Content: Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from term to term. Alternatives: none Subsequent Courses: none Back to Top Math 498 - Topics in Modern Mathematics Prerequisites: Junior or Senior standing Frequency: Sporadically Student Body: Junior and Senior students from mathematics and other fields Credit: 3 Credits. Recent Texts: An Introduction to Chaotic Dynamical Systems (R. Devaney) Background and Goals: As a topics course, this course will vary greatly from term to term. In one recent offering, the aim of the course was to introduce at an elementary level the basic concepts of the theory of dynamical systems. Content: varies Alternatives: none Subsequent Courses: no specific sequels Back to Top Math 512: Algebraic Structures Prerequisites: Math 296, 412, 451, or permission of instructor Frequency: Fall (I) Credit: 3 Credits. Recent Texts: Algebra by Artin Past instructors: K. Smith Student Body: Mainly undergraduate mathematics concentrators with a few graduate students from other fields Background and Goals:Math 512-513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295-296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Excercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from the Math 185-286 and Math 156-256 sequences are strongly advised to take both Math 217 and some 400-500 level course (for examples, Math 451) prior to attempting Math 512. Content:The course covers basic definitions and properties of groups, fields, and vetor spaces including homomorphisms, isomorphisms, subgroups, and bilinear forms. Further topics are selected from: Sylow theorems; structure theorem for finitely-generated abelian groups; permutation representation; the symmetric and alternating groups; vector spaces over arbitrary fields; spectral theorem; and linear groups. Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course which covers about half of the material of Math 512. The sequence Math 593-594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Subsequent Course: Math 513 (Introduction to Linear Algebra) Back to Top Math 513: Introduction to Linear Algebra Prerequisites: Math 512 Frequency: Winter (II) Credit: 3 credits Recent Texts: Algebra by Artin Past instructors: K.E. Smith Student Body: Mainly undergraduate mathematics concentrators with a few graduate students from other fields Background and Goals: Math 512-513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295-296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Excercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from the Math 185-286 and Math 156-256 sequences are strongly advised to take both Math 217 and some 400-500 level course (for examples, Math 451) prior to attempting Math 512. Content: This course is a continuation of Math 512. It covers basic definitions and properties of rings and modules including quotients, ideals, factorization, and field extensions. Further topics are selected from: representation theory; structure theory of modules over a PID; Jordan canonical form; Galois theory, Nullstellensatz; finite fields; Euclidean, Principal ideal, and unique factorization domains; polynomial rings in one and several variables; and algebraic varieties. Alternatives: None Subsequent Courses: The natural sequel to Math 513 is Math 593 Back to Top Math 520: Life Contingencies I Prerequisites: Math 424 and 425 or permission Frequency: Fall (I) Credit: 3 credits Recent Texts: Actuarial Mathematics (N.L. Bowers et al.) Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics Background and Goals: The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof.  Content: The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. This corresponds to Chapters 3--6 and part of 7 of Bowers.  Alternatives: Math 523 (Risk Theory) is a complementary course covering the application of stochastic process models.  Subsequent Courses: Math 520 is prerequisite to all succeeding actuarial courses. Math 521 (Life Contingencies II) extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520--521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 (Act. Theory of Pensions and Soc. Sec) applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.  Back to Top Math 521: Life Contingencies II Prerequisites: Math 520 or permission Frequency: Winter (II) Credit: 3 credits Recent Texts: Actuarial Mathematics (N.L. Bowers et al.) Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics Background and Goals: This course extends the single decrement and single life ideas of Math 520 to multi-decrement and multiple-life applications directly related to life insurance. The sequence 520--521 covers covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Concepts and calculation are emphasized over proof.  Content: Topics include multiple life models--joint life, last survivor, contingent insurance; multiple decrement models---disability, withdrawal, retirement, etc.; and reserving models for life insurance. This corresponds to chapters 7--10, 14, and 15 of Bowers et al.  Alternatives: Math 522 (Act. Theory of Pensions and Soc. Sec) is a parallel course covering mathematical models for prefunded retirement benefit programs.  Subsequent Courses: none  Back to Top Math 522: Act. Theory of Pensions and Social Security Prerequisites: Math 520 or permission Frequency: Sporadically Credit: 3 credits Recent Texts: Fundamentals of Pension Mathematics (B. Berin), Pension Mathematics (A. Anderson) as a reference Past instructors: Student Body: Undergraduate students of actuarial mathematics Background and Goals: This course develops the mathematical models for pre-funded retirement benefit plans. Concepts and calculation are much more important than proofs.  Content: Mathematical models for (1) retirement income, (2) retiree medical benefits, (3) disability benefits, and (4) survivor benefits. There is some coverage of how accounting theory and practice can be explained by these models and of the U.S. laws and regulations that give rise to the models used in practice.  Alternatives: Math 521 (Life Contingencies II) is a parallel course covering models for insurance rather than retirement benefits.  Subsequent Courses: none  Back to Top Math 523: Risk Theory Prerequisites: Math 425 Frequency: Fall (I), Winter (II) Credit: 3 Recent Texts: Fundamentals of Actuarial Mathematics (Promislow). Past instructors: M. Jonsson, K. Moore Student Body: Undergraduate students of financial and actuarial mathematics Background and Goals: Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance, and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least Junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time, and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models.   Contents: Review of probability; the collective risk model and compound random variables; choices of frequency and severity distributions; deductibles; basic facts on stochastic processes; finite state Markov chains; Poisson processes; compound poisson processes; ruin theory. Alternatives: none  Subsequent Courses: none  Back to Top   Math 525 (Stat. 525): Probability Theory Prerequisites: Math 451 Frequency: Fall (I), Winter (II) Credit: 3 credits Recent Texts:Grimmet and Stirzaker, Probability and Random Processes (required); Ross, Introduction to Probability Models (optional) Past instructors: J. Marker, M. Rudelson, A. Barvinok Student Body: A mix of undergraduate and graduate students, drawn largely from mathematics, statistics, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with Math 425 (Intro. to Probability), but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is the required prerequisite. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.  Content: Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories.  Alternatives: EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 (Intro. to Probability) is a course for students with substantially weaker background and ability.  Subsequent Courses: Math 526 (Discr. State Stoch. Proc.), Stat 426 (Intro. to Math Stat.), and the sequence Stat 510 (Mathematical Statistics I)--Stat 511 (Mathematical Statistics II) are natural sequels.  Back to Top Math 526 (Stat. 526): Discrete State Stochastic Processes Prerequisites: Required: Math 525 or EECS 501 or basic probability theory including: Random variables, expectation, independence, conditional probability. Recommended: Good understanding of advanced calculus covering limits, series, the notion of continuity, differentiation and the Riemann integral ; Linear algebra including eigenvalues and eigenfunctions. Frequency: Varies Credit: 3 credits Required textbook:A First Course in Stochastic Processes, 2nd ed. (Karlin and Taylor) Background and Goals: The theory of stochastic processes is concerned with systems which change in accordance with probability laws. It can be regarded as the 'dynamic' part of statistic theory. Many applications occur in physics, engineering, computer sciences, economics, financial mathematics and biological sciences, as well as in other branches of mathematical analysis such as partial differential equations. The purpose of this course is to provide an introduction to the many specialized treatise on stochastic processes. Most of this course is on discrete state spaces. It is a second course in probability which should be of interest to students of mathematics and statistics as well as students from other disciplines in which stochastic processes have found significant applications. Special efforts will be made to attract and interest students in the rich diversity of applications of stochastic processes and to make them aware of the relevance and importance of the mathematical subtleties underlying stochastic processes. Content: The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queueing theory; an introduction to Brownian motion; stationary processes and martingales. Significant applications will be an important feature of the course. Coursework: weekly or biweekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%. Additional information: Those wishing to discuss the course should contact taoluo@umich.edu. Back to Top Math 528: Topics in Casualty Insurance Prerequisites: Math 217, 417, or 419, or permission Frequency: Sporadically Credit: 3 credits Recent Texts: Past instructors: C. Huntington Student Body: Undergraduate students of actuarial mathematics and insurance majors in Business Background and Goals: Historically the Actuarial Program has emphasized life, health, and pension topics. This course will provide background in casualty topics for the many students who take employment in this field. Guest lecturers from the industry will provide some of the instruction. Students are encouraged to take the Casualty Actuarial Society's Part 3B examination at the completion of the course.  Content: The insurance policy is a contract describing the services and protection which the insurance company provides to the insured. This course will develop an understanding of the nature of the coverages provided, the bases of exposure and principles of the underwriting function, how products are designed and modified, and the different marketing systems. It will also look at how claims are settled, since this determines losses which are key components for insurance ratemaking and reserving. Finally, the course will explore basic ratemaking principles and concepts of loss reserving.  Alternatives: none  Subsequent Courses: none  Back to Top Math 547: Biological Sequence Analysis Prerequisites: Flexible. Basic probability (level of Math/Stat 425) or molecular biology (level of Biology 427) or biochemistry (level of Chem/BioChem 451) or basic programming skills desirable; or permission of instructor. Frequency: Annually; check for semester Credit: 3 credits Recent Texts: Biological Sequence Analysis (R. Durbin, et al.) Past instructors: D. Burns Student Body: Interdisciplinary: mainly Math, Statistics, Biostatistics and Bioinformatics students; also Biology, Biomedical and Engineering students. Background and Goals: Content: Probabilistic models of proteins and nucleic acids. Anaylsis of DNA/RNA and protein sequence data. Algorithms for sequence alignment, statistical analysis of similarity scores, hidden markov models, neural networks, training, gene finding, protein family proviles, multiple sequence alignment, sequence comparison and structure prediction. Analysis of expression array data. Alternatives: Bioinformatics 526 Subsequent Courses: Bioinformatics 551 (Preteome Informatics) Back to Top Math 550: Intro to Adaptive Systems Prerequisites: Math 215, 255, or 285; Math 217, and Math 425 Frequency: Sporadically Credit: 3 credits Recent Texts: Past instructors: C. Simon Student Body: graduate and undergraduate students from many disciplines Background and Goals: This course centers on the construction and use of agent-based adaptive models study phenomena which are prototypical in the social, biological and decision sciences. These models are "agent-based" or "bottom-up" in that t he structure placed at the level of the individuals as basic components; they are "adaptive" in that individuals often adapt to their environment through evolution or learning. The goal of these models is to understand how the structure at the individual or micro level leads to emergent behavior at the macro or aggregate level. Often the individuals are grouped into subpopulations or interesting hierarchies, and the researcher may want to understand how the structure of development of these populations affects macroscopic outcomes.  Content: The course will start with classical differential equation and game theory approaches. It will then focus on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier system and cellular automata. Time permitting, we will discuss more recent developments such as sugarscape and echo.  Alternatives: Complex Systems 510 is the same course.  Subsequent Courses: none  Back to Top Math 555: Intro to Complex Variables Prerequisites: Math 450 or 451 Frequency: Fall (I), Winter (II), Spring (IIIa) Credit: 3 credits Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown); Past instructors: B. Stensones, C. Doering, J. Fornaess Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply princip les to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 1--9 of Churchill.  Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications.  Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.  Back to Top Math 556: Methods of Applied Math I: Applied Functional Analysis Prerequisites: Math 217, 419, or 513; 451 and 555 Frequency: Fall (I) Credit: 3 credits Recent Texts: Applied Functional Analysis (Griffel) Past instructors: P Miller, J Schotland Student Body: Graduate students in matehematics, science, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is an introduction to methods of applied functional analysis. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra,  undergraduate analysis, advanced calculus and complex variables. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content: Topics may vary with the instructor but often include Fourier transform, distributions, Hilbert space, Banach spaces, fixed point theorems, integral equations, spectral theory for compact self-adjoint operators. Alternatives: Math 602 is a more theoretical course covering many of the same topics Subsequent Courses: Math 557 (Methods of Applied Math II), Math 558 (Ordinary Diff. Eq.), Math 656 (Partial Differential Equations) and 658 (Ordinary Differential Equations.)  Back to Top Math 557: Methods of Applied Math II Prerequisites: Math 217, 419, or 513; 451 and 555 Frequency: Winter (II) Credit: 3 credits Recent Texts: Asymptotic Analysis (Murray) Past instructors: C. Doering, P. Federbush Student Body: Graduate students in mathematics, science and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is an introduction to methods of asymptotic analysis including asymptotic expansions for integrals and solutions of ordinary and partial differential equations. The prerequisites include linear algebra, advanced calculus and complex variables. Math 556 is not a prerequisite. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content: Topics include stationary phase, steepest descent, characterization of singularities in terms of the Fourier transform, regular and singular perturbation problems, boundary layers, multiple scales, WKB method. Additional topics depend on the instructor but may include non-linear stability theory, bifurcations, applications in fluid dynamics (Rayleigh-Benard convection), combustion (flame speed). Alternatives: none  Subsequent Courses: Math 656 (Partial Differential Equations) and 658 (Ordinary Differential Equations.)  Back to Top Math 558: Applied Nonlinear Dynamics Prerequisites: Math 450 or 451 Frequency: Sporadically Credit: 3 credits Recent Texts: Nonlinear Ordinary Differential Equations (Jordan and Smith) Past instructors: R. Krasny, C. Doering Student Body: grad students in math, science, and engineering, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is an introduction to dynamical systems (differential equations and iterated maps).  The aim is to survey a broad range of topics in the theory of dynamical systems with emphasis on techniques and results that are useful in applications.  Chaotic dynamics will be discussed. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content: Topics may include: bifurcation theory, phase plane analysis for linear systems, Floquet theory, nonlinear stability theory, dissipative and conservative systems, Poincare-Bendixson theorem, Lagrangian and Hamiltonian mechanics, nonlinear oscillations, forced systems, resonance, chaotic dynamics, logistic map, period doubling, Feigenbaum sequence, renormalization, complex dynamics, fractals, Mandelbrot set, Lorenz model, homoclinic orbits, Melnikov's method, Smale horseshoe, symbolic dynamics, KAM theory, homoclinic chaos Alternatives: Math 404 (Intermediate Diff. Eq.) is an undergraduate course on similar topics Subsequent Courses: Math 658 (Ordinary Differential Equations)  Back to Top Math 559: Topics in Applied Mathematics Prerequisites: Math 451 and Math 419 or equivalent Frequency: Sporadically Credit: 3 credits Recent Texts: Varies Past instructors: Student Body: undergraduate and graduate students in mathematics or science Background and Goals: This course in intended for students with a fairly strong background in pure mathematics, but not necessarily any experience with applied mathematics. Proofs and concepts, as will as intuitions arising from the field of application will be stressed.  Content: This course will focus on a particular area of applied mathematics in which the mathematical ideas have been strongly influenced by the application. It is intended for students with a background in pure mathematics, and the course will develop the intuitions of the field of application as well as the mathematical proofs. The applications considered will vary with the instructor and may come from physics, biology, economics, electrical engineering, and other fields. Recent examples have been: Dynamical Systems, Statistical Mechanics, Solitons, and Nonlinear Waves.  Alternatives: none  Subsequent Courses: Other courses in applied mathematics  Back to Top Math 561 (Bus. Adm. Stat. 518, IOE 510): Linear Programming I Prerequisites: Math 217, 417, or 419 Frequency: Fall (I), Winter (II), and Spring (IIIa) Credit: 3 credits Recent Texts: Linear Optimizations and Extensions: Theory and Algorithms(Fang and Puthenpura) Past instructors: J. Goldberg Student Body: Graduate and undergraduate students from many fields Background and Goals: A fundamental problem is the allocation of constrained resources such as funds among investment possibilities or personnel among production facilities. Each such problem has as it's goal the maximization of some positive objective such as investment return or the minimization of some negative objective such as cost or risk. Such problems are called Optimization Problems. Linear Programming deals with optimization problems in which both the objective and constraint functions are linear (the word "programming" is historical and means "planning" rather that necessarily computer programming). In practice, such problems involve thousands of decision variables and constraints, so a primary focus is the development and implementation of efficient algorithms. However, the subject also has deep connections with higher-dimensional convex geometry. A recent survey showed that most Fortune 500 companies regularly use linear programming in their decision making. This course will present both the classical and modern approaches to the subject and discuss numerous applications of current interest. Content: Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; a lgorithmic complexity; the elipsoid method; scaling algorithms; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to form ulate and solve models developed from more complex case studies and use various computer programs.  Alternatives: Cross-listed as IOE 510.  Subsequent Courses: IOE 610 (Linear Programming II) and IOE 611 (Nonlinear Programming) Back to Top Math 562 (IOE 511, Aero Eng. 577): Continuous Optimization Meth. Prerequisites: Math 217, 417, or 419 Frequency: Fall (I) Credit: 3 credits Recent Texts: Past instructors: Student Body: Background and Goals: Not Available  Content: Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.  Alternatives: Cross-listed as IOE 511.  Subsequent Courses: This is not a prerequisite for any other course. Back to Top Math 563: Advanced Mathematical Methods For the Biological Sciences Prerequisites: Math 217, 417, or 419 and Math 216, 286, or 316 and to have a basic familiarity with partial differential equations as would be gained by the completion of Math 450 or 454 or to have permission of the instructor. Frequency: Winter (II) Student Body: Graduate Students, Math, Science, Engineering and Medical School. (Both the Department of Mathematics and the Program in Bioinformatics have approved this course for cross-listing. Further approval is in process). Credit: 3 Credits. Recent Texts: Math Biology, J. D. Murray Past Instructors: T. Jackson Background and Goals: Mathematical biology is a fast growing and exciting modem application which has gained worldwide recognition. This course will focus on the devrivation, analysis, and simulation of partial differential equations (PDEs) which model specific phenomena in molecular, cellular, and population biology. A goal of this course is to understand how the underlying spatial variability in natural systems influences motion and behavior. Content: Mathematical topics covered include derivation of relevant PDEs from first principle; reduction of PDEs to ODEs under steady state, quasi-state and traveling wave assumptions; solution techniques for PDEs and concepts of spatial stability and instability. These concepts will be introduced within the setting of classical and current problems in biology and the biomedical sciences such as cell motion, transport of biological substances and, biological pattern formation. Above all, this course aims to enhance the interdisciplinary training of advanced undergraduate and graduate students from mathematics and other disciplines by introducing fundamental properties of partial differential equations in the context of interesting biological phenomena. Alternatives: None Subsequent Courses: Back to Top   Math 564: Topics in Mathematical Biology Prerequisites: Math 217, 216 and EECS 183 or equivalent, or permission of instructor Frequency: Winter (II) Credit: 3 credits Recent Texts: none Past instructors: D. Forger Student Body: Background and Goals: Numerical methods have become an essential part of modern biological and medical research.  This course will survey many of these methods and provide students with tools necessary to meet the next generation of challenges in biological research.  Unlike many other courses in numerical methods, we will focus on solving specific problems rather than analysis of techniques.  Content: Will vary, but a recent topics offering included: methods to extract parameters from data, stochastic methods to simulate biochemical networks within cells and neural networks, principal component analysis in large data sets, techniques for simulating biological fluids, time series analysis, model reduction techniques, and optimal perturbations of biological systems.  Alternatives: Subsequent Courses: Back to Top   Math 565: Combinatorics and Graph Theory Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics Frequency: Fall (I) Credit: 3 credits Recent Texts: A Course in Combinatorics (van Lint and Wilson) Past instructors: N. Reading, A. Blass, S. Fomin Student Body: Largely math and EECS grad students with a few math undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program. Background and Goals: This course has two somewhat distinct halves devoted to Graph Theory and Combinatorics. Proofs, concepts, and applications play about an equal role. Students should have taken at least one proof-oriented course. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.   Content: Eulerian and Hamiltonian graphs; tournaments; network flows; graph coloring; the 5-Color Theorem; Kuratowski's Theorem; the Matrix-Tree Theorem; fundamental enumeration principles, bijections, and generating functions; inclusion-exclusion; partially ordered sets; matroids; Ramsey's Theorem. Alternatives: There are small overlaps with Math 566 (Introduction to Algebraic Combinatorics) and Math 416 (Theory of Algorithms). Math 465 has similar content but is significantly less demanding than Math 565. Subsequent Courses: Math 566 (Introduction to Algebraic Combinatorics) Back to Top Math 566: Combinatorial Theory Prerequisites: Math 512, or equivalent experience with abstract algebra Frequency: Winter (II) Credit: 3 credits Recent Texts: Enumerative Combinatorics (Stanley) Past Instructors: M. Skandera, J. Stembridge, S. Fomin Student Body: Undergraduates and graduates from Math, EECS, or IOE Background and Goals: This course is a rigorous introduction to modern algebraic combinatorics, primarily focused on enumeration.  Content: varies considerably with instructor. Topics may include: generating functions (ordinary and exponential); sieve methods; Lagrange inversion; perfect matchings; words and formal languages; group-theoretic enumeration methods; partitions and tableaux; algebraic graph theory. Alternatives: Math 664 (Combinatorial Theory I) occasionally covers similar material in greater depth at a faster pace. Subsequent Courses: Sequels include Math 665 and Math 669. Back to Top Math 567: Introduction to Coding Theory Prerequisites: Math 217, 417, or 419 Frequency: Winter (II) Credit: 3 credits Recent Texts: Introductin to Coding Theory (van Lint) Past instructors: T. Wooley Student Body: Undergraduate math majors and EECS graduate students Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology. Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form. Alternatives: none Subsequent Courses: Math 565 (Combinatorics and Graph Theory) and Math 556 (Methods of Applied Math I) are natural sequels or predecessors. This course also complements Math 312 (Applied Modern Algebra) in presenting direct applications of finite fields and linear algebra. Back to Top Math 571: Numerical Methods for Scientific Computing I Prerequisites: Math 217, 417, 419, or 513 and Math 450, 451, or 454 or permission Frequency: Fall (I) and Winter (II) Credit: 3 credits Recent Texts: A Multigrid Tutorial (Briggs), Introduction to Numerical Linear Algebra and Optimization (Ciarlet) Past instructors: R. Krasny, S. Karni, J. Rauch Student Body: math and engineering grads, strong undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors.  Alternatives: Math 471 (Intro to Numerical Methods) is a survey course in numerical methods at a more elementary level.  Subsequent Courses: Math 572 (Numer Meth for Sci Comput II) covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.  Back to Top Math 572: Numerical Methods for Scientific Computing II Prerequisites: Math 217, 417, 419, or 513 and one of Math 450, 451, or 454 or permission Frequency: Winter (II) Credit: 3 credits Recent Texts: Numerical Solutions of PDE's (Morton and Mayer) Past instructors: S. Karni, P. Smereka Student Body: math and engineering grads, strong undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Graduate students from engineering and science departments and strong undergraduates are also welcome. The course is an introduction to numerical methods for solving ordinary differential equations and hyperbolic and parabolic partial differential equations. Fundamental concepts and methods of analysis are emphasized. Students should have a strong background in linear algebra and analysis, and some experience with computer programming. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program.  Content: Content varies somewhat with the instructor. Numerical methods for ordinary differential equations; Lax's equivalence theorem; finite difference and spectral methods for linear time dependent PDEs: diffusion equations, scalar first order hyperbolic equations, symmetric hyberbolic systems.  Alternatives: There is no real alternative; Math 471 (Intro to Numerical Methods) covers a small part of the same material at a lower level. Math 571 and 572 may be taken in either order.  Subsequent Courses: Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.  Back to Top Math 575: Intro to Theory of Numbers Prerequisites: Math 451 and 513 or permission Frequency: Fall (I) Credit: 3 credits; 1 credit after Math 475 Recent Texts: An introduction to the Theory of Numbers (Niven, Zuckerman, and Montgomery) Past instructors: T. Wooley, H. Montgomery/li> Student Body: Roughly half honors math undergrads and half graduate students Background and Goals: Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and preferably Math 512. Proofs are emphasized, but they are often pleasantly short. Content: Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, application of these ideas to the solution of classical problems such as Fermat's last `theorem'(proved recently by A. Wiles). Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. This material corresponds to Chapters 1--3 and selected parts of Chapters 4, 5, 7, 8, and 9 of Niven, Zuckerman, and Montgomery.  Alternatives: Math 475 (Elementary Number Theory) is a non-honors version of Math 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier.  Subsequent Courses: All of the advanced number theory courses Math 675, 676, 677, 678, and 679 presuppose the material of Math 575. Each of these is devoted to a special subarea of number theory.  Back to Top Math 582: Intro to Set Theory Prerequisites: Math 412 or 451 or equivalent experience with abstract mathematics Frequency: Winter (II) Credit: 3 credits Recent Texts: Elements of Set Theory (H. Enderton) Past instructors: A. Blass, P. Hinman Student Body: undergraduate math (often honors) majors and some grad students Background and Goals: One of the great discoveries of modern mathematics was that essentially every mathematical concept may be defined in terms of sets and membership. Thus Set Theory plays a special role as a foundation for the whole of mathematics. One of the goals of this course is to develop some understanding of how Set Theory plays this role. The analysis of common mathematical concepts (e.g. function, ordering, infinity) in set-theoretic terms leads to a deeper understanding of these concepts. At the same time, the student will be introduced to many new concepts (e.g. transfinite ordinal and cardinal numbers, the Axiom of Choice) which play a major role in many branches of mathematics. The development of set theory will be largely axiomatic with the emphasis on proving the main results from the axioms. Students should have substantial experience with theorem-proof mathematics; the listed prerequisites are minimal and stronger preparation is recommended. No course in mathematical logic is presupposed.  Content: The main topics covered are set algebra (union, intersection), relations and functions, orderings (partial, linear, well), the natural numbers, finite and denumerable sets, the Axiom of Choice, and ordinal and cardinal numbers. Alternatives: Some elementary set theory is typically covered in a number of advanced courses, but Math 582 is the only course which presents a thorough development of the subject.  Subsequent Courses: Math 582 is not an explicit prerequisite for any later course, but it is excellent background for many of the advanced courses numbered 590 and above.  Back to Top Math 583: Probabilistic and Interactive Proofs Prerequisites: Math 412, 451 or permission of instructor Frequency: Winter (II) Credit: 3 credits Recent Texts: none Past instructors: M. Strauss Student Body: Grad students and advanced undergrads in Math, Computer Science and Philosophy. Background and Goals: Can we be convinced that a proof is correct, even if we only check it in three places?  Can a proof convince us that a statement is true, while giving us no aid in convincing anyone else that the statement is true?  The answer to both is affirmative.  How?  Using randomness and interaction, two elements missing from traditional deductive proofs. Why?  Checking a proof in just a few places is useful for checking computer-generated proofs that are too long to read; there are also surprising connections to showing that certain functions cannot be computed or approximated efficiently.  A "zero-knowledge proof" might be used, for example, for a customer to prove to a merchant that the customer is the rightful owner of a credit card, without giving the merchant any ability to prove (fraudulently) that the merchant is the owner of that credit card. Content: Probabilistically-checkable proofs, zero-knowlege proofs, and interactive proofs are studied and their computational, cryptographic, and other advantages discussed.  The course will include a presentation of the necessary background material from mathematics (including probability theory and error-correcting codes) and computer science (including randomized computation).  Motivations and applications in other fields, such as the security of credit card transactions and the philosophical nature of proof and knowledge, are briefly discussed. Alternatives: Subsequent Courses: Back to Top Math 590: Intro to Topology Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits Recent Texts: An Introduction to Topology and Homotopy (Sieradski) Past instructors: M. Brown, A. Wasserman Student Body: math grads, some non-math grads, math undergrads Background and Goals: This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs.  Content: Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics.  Alternatives: Math 490 (Introduction to Topology) is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 591 and Math 592 (General and Differential Topology). Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 (General and Differential Topology) is a more rigorous course covering much of this material and more.  Subsequent Courses: Both Math 591 (General and Differential Topology) and Math 437 (Intro to Differentiable Manifolds) use much of the material from Math 590.  Back to Top Math 591: General and Differential Topology Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits Recent Texts: Topology (Munkres); Differential Topology (Guillemin and Pollack) Past instructors: P. Scott, R. Canary, J. Lott Student Body: mainly math grads, a few math undergrads and non-math grads Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs.  Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces.  Alternatives: none  Subsequent Courses: Math 592 (An Introduction to Algebraic Topology) is the natural sequel.  Back to Top Math 592: An Introduction to Algebraic Topology Prerequisites: Math 591 Frequency: Winter (II) Credit: 3 credits Recent Texts: Elements of Algebraic Topology (Munkres) Past instructors: I. Kriz, P. Scott, R. Canary Student Body: largely math graduate students Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs.  Content: Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, Brouwer's and Lefschetz' fixed-point theorems, and other topics.  Alternatives: none  Subsequent Courses: Math 695 (Algebraic Topology I)  Back to Top Math 593: Algebra I Prerequisites: Math 513 Frequency: Fall (I) Credit: 3 credits Recent Texts: Algebra (Artin) Past instructors: A. Moy, P.J. Hanlon, R.L. Griess, Jr. Student Body: largely math graduate students Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to Math 512 (Algebraic Structures).  Content: Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras.  Alternatives: none  Subsequent Courses: Math 594 (Algebra II) and Math 614 (Commutative Algebra I).  Back to Top Math 594: Algebra II Prerequisites: Math 593 Frequency: Winter (II) Credit: 3 credits Recent Texts: Algebra, A Graduate Course (Isaacs) Past instructors: I.V. Dolgachev, R. Lazarsfeld, R.L. Griess, Jr. Student Body: largely math graduate students Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs.  Content: Topics include group theory, permutation representations, simplicity of alternating groups for n>4, Sylow theorems, series in groups, solvable and nilpotent groups, Jordan-Holder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, transcendence degree.  Alternatives: none  Subsequent Courses: Math 612 (Algebra III), Math 613 (Homological Algebra), Math 614 (Commutative Algebra I) and various topics courses in algebra.  Back to Top Math 596: Analysis I (Complex) Prerequisites: Math 451 Frequency: Fall (I) Credit: 3 credits; 2 credits after Math 555 Recent Texts: Complex Analysis, 3rd ed. (L. Ahlfors) Past instructors: D.M. Burns, Jr., P. Duren Student Body: largely math grad students Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs.  Content: Review of analysis in R^2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem.  Alternatives: Math 555 (Intro to Complex Variables) covers some of the same material with greater emphasis on applications and less attention to proofs.  Subsequent Courses: Math 597 (Analysis II (Real)), Math 604 (Complex Analysis II), and Math 605 (Several Complex Variables).  Back to Top Math 597: Analysis II (Real) Prerequisites: Math 451 and 513 Frequency: Winter (II) Credit: 3 credits Recent Texts: Real Analysis (Bruckert et. al.) Past instructors: D. Barrett, J. Heinonoen, L. Ji, B. Stensones Student Body: largely math graduate students Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs.  Content: Topics include Lebesgue measure on the real line; measurable functions and integration on R; differentiation theory, fundamental theorem of calculus; function spaces, L^p(R), C(K), Holder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini's Theorem; Radon-Nikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.  Alternatives: none  Subsequent Courses: Math 602 (Real Analysis II).