Undergraduate Courses
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 563
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 582  583  590  591
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 596  597
*For a list of current Math textbooks, please click here.
Math 105  Data, Functions and Graphs
 Prerequisites: 34 years HS math including trigonometry
 Frequency: Fall (I), Winter (II), Summer (IIIb)
 Student Body: Firstyear 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, HughesHallett 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 reallife 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).
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Math 107  Mathematics for the Information Age
 Prerequisites: 34 years HS math
 Frequency: Winter (II)
 Student Body: Firstyear students (nonmathematics 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.
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Math 115  Calculus I
 Prerequisites: 34 years HS math including trigonometry
 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
 Student Body: Firstyear students
 Credit: 4 Credits. No credit after Math 116, 215 or 216.
 Recent Texts: Calculus: Single Variable (HughesHallett, et al.)
 Background and Goals: The sequence Math 115116215 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 reallife 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 noncalculus alternative for students with a good command of firstsemester 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).
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Math 116  Calculus II
 Prerequisites: Math 115 or AP credit
 Frequency: Fall (I), Winter (II), Spring (IIIa), Summer (IIIb)
 Student Body: Firstyear students
 Credit: 4 Credits. No credit after Math 215 or 216.
 Recent Texts: Calculus: Single Variable (HughesHallett, et al.)
 Background and Goals: The sequence Math 115116215 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).
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Math 127  Geometry and the Imagination
 Prerequisites: 3 years HS math
 Frequency: Winter (II)
 Student Body: Firstyear 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 nonEuclidean 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 handson 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 higherdimension analogues of some familiar objects like spheres and cubes; discussion of the proper higherdimensional 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.
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Math 128  Explorations in Number Theory
 Prerequisites: 3 years HS math
 Frequency: Fall (I)
 Student Body: Firstyear 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 wellsuited for nonscience 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.
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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
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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: Firstyear 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.
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Math 174  Proofs in Geometry
 Prerequisites: Permission of honors advisor
 Frequency: Fall (I)
 Student Body: Firstyear students
 Credit: 4 Credits. No credit granted to those who have completed a 200level 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 twodimensional geometric constructions. This software will be able to produce multicolor 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 ninepoint circle theorem, Ptolemy's theorem and Morley's theorem.
 Alternatives: none
 Subsequent Courses: none
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Math 175  Introduction to Cryptology
 Prerequisites: Permission of honors advisor
 Frequency: Fall (I)
 Student Body: Firstyear 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 inclass 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 discoverybased 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).
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Math 176  Explorations in Topology and Analysis
 Prerequisites: Math 175 or permission of instructor
 Frequency: Winter (II)
 Student Body: Firstyear 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 discretetime and continuoustime 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, selfsimilarity, 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 workstation lab.
 Alternatives: Math 116 (Calculus I) or Math 186 (Honors Calculus II)
 Subsequent Courses: Math 285 (Honors Calculus III)
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Math 185  Honors Calculus I
 Prerequisites: Permission of honors advisor
 Frequency: Fall (I)
 Student Body: Firstyear 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.
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Math 186  Honors Calculus II
 Prerequisites: Permission of honors advisor
 Frequency: Winter (II)
 Student Body: Firstyear 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.
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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).
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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 115116215 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).
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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 problemsolving 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 nonlinear 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.
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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 problemsolving 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 R^{n}; 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.
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Math 255  Applied Honors Calculus III
 Prerequisites: Math 156, or permission of instructor
 Frequency: Winter (II)
 Student
Body: Firstyear 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.
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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 upperlevel courses.
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Math 285  Honors Calculus III
 Prerequisites: Math 156, 176 or 186 or permission of instructor
 Frequency: Fall (I)
 Student
Body: Sophomores and firstyear 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 185186285286 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, maximumminimum 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 applicationsoriented 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).
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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 185186285286 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 firstorder differential equations, higherorder linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, RungeKutta). 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).
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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.
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Math 295  Honors Mathematics I
 Prerequisites: permission of honors advisor
 Frequency: Fall (I)
 Student
Body: Firstyear students
 Credit: 4 Credits.
 Recent Texts: Calculus (4th edition) by M. Spivak
 Background
and Goals: Math 295296395396 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)
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Math 296  Honors Mathematics II
 Prerequisites: Math 295
 Frequency: Winter (II)
 Student
Body: Firstyear students
 Credit: 4 Credits.
 Recent Texts: Calculus (4th edition) by Spivak, Linear Algebra by Kunze and Hoffman
 Background
and Goals: Math 295296395396 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, innerproduct 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)
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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 handson 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
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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, prooforiented 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
realworld 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 prooforiented 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 followup course is Math 475 (Elementary Number Theory).
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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:
Firstorder equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvectoreigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higherorder 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).
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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 nonEuclidean
geometries, the notions of limit and infinitesimals, and the development
of the theory of equations culminating with Galois theory.
 Alternatives:
None
 Subsequent
Courses: None
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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
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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 nonmath 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 100and 200level 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.
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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 nonlinear equations, ordinary
differential equations, polynomial approximations. Other topics
may include discrete Fourier transforms, twopoint boundaryvalue
problems, and MonteCarlo methods.
 Alternatives:
Math 471 (Numerical Analysis) provides
a more indepth 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 largescale 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 571572.
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Math 385  Math for Elementary School Teachers
 Prerequisites:
One year each of HS algebra and geometry
 Frequency:
Fall (I), Spring (IIIa) of evennumbered 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 precalculus 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.
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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.
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Math 395  Honors Analysis I
 Prerequisites:
Math 296
 Frequency:
Fall (I)
 Student
Body: Firstyear 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 295296 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 pointset 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)
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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 295396 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.
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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.
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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.
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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 welldefined 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 pseudoprimality testing (in connection with coding questions). Algorithm types such as divideandconquer, 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.
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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 problemsolving, 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).
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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 prooforiented 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, innerproduct spaces; unitary, selfadjoint,
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 prooforiented. 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).
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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).
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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 upperlevel writing requirement.
 Alternatives:
None
 Subsequent
Courses: None
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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, BlackScholes Analysis, numerical methods and interest rate models.
 Alternatives:
none
 Subsequent
Courses: Math 523 (Risk Theory) and Math 623 (Computational
Finance).
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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 reallife 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).
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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.
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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 nonmathematics students
 Credit:
3 Credits.
 Recent Texts: Pension Planning: Pensions, ProfitSharing, 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 upperlevel 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 indepth examination
of the actuarial techniques used in employee benefit plans.
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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 NonEuclidean Geometry (4th edition) by Greenberg
 Area: Teaching
 Background
and Goals: This course is a study of the axiomatic foundations
of Euclidean and nonEuclidean 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
nonEuclidean geometries.
 Alternatives:
Alternative geometry courses at this level is Math 433 (Intro to Differential Geometry).
 Subsequent
Courses: None
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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 3space 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 threespace using calculus. Curvature
and torsion of curves. Curvature, covariant differentiation, parallelism,
isometry, geodesics, and area on surfaces. GaussBonnet 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.
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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, LeviCivita 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)
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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 indepth 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 boundaryvalue 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 graduatelevel courses that further develops
both the theory and applications covered in 450.
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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
115116, 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.
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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 nonmathematics 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) maxmin
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).
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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 nonmathematics 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 initialvalue
and boundaryvalue problems for secondorder 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 onedimensional 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 inputoutput systems, analysis of data smoothing and
filtering, signal processing, timeseries 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).
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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. 13 credits after 362, 463 depending on overlap
 Recent Texts: Mathematical Models in Biology (2nd edition) by EdelsteinKeshet
 Area: Applied/Numerical Analysis
 Background
and Goals: The focus of this course is the application
of a variety of mathematical techniques to solve realworld 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 higherlevel course in differential equations.
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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 receptorligand 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
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Math 464  Inverse Problems
 Prerequisites:
Math 217, 417, or 419 and Math 216, 256, 286, or 316
 Frequency:
Sporadically
 Student
Body: Upperlevel 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, nondestructive 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 abovenamed 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 nondestructive testing. Mathematical topics include illposedness (existence, uniqueness, stability), regularization (e.g. Tikhonov, least squares, modified least squares, variation, mollification), pseudoinverses, transforms (e.g. kplane, Radon, Xray, Hilbert), special functions, and singularvalue 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
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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: Upperlevel 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 (inclusionexclusion, 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 prooforiented. Math 566 is even more advanced.
 Subsequent
Courses: Math 565 and 566.
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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 singlespecies 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
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Math 471  Introduction to Numerical Methods
 Prerequisites:
Math 216, 256, 286, or 316; Math 217, 417, or 419; and a working
knowledge of one highlevel 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 nonlinear 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, 2point 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 571572 (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
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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 highlevel 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 nonlinear equations, systems of linear equations, numerical integration, interpolation and polynomial approximation, ordinary differential equations, partial differential equations  in particular the BlackScholes 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
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Math 475  Elementary Number Theory
 Prerequisites:
no specific prerequisite
 Frequency:
Winter (II)
 Student
Body: Mainly mathematics undergraduates; some nonmathematics 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 numbertheoretic 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 selfcontained, 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 13 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.
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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
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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 nonmath 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 firstorder 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 firstorder 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 nonstandard analysis. This
material corresponds to Chapter 1 and sections 2.02.5 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.
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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.
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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 precalculus 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 problemsolving
strategies are used throughout the course.
 Alternatives:
There is no alternative course.
 Subsequent
Courses: Math 497 (Topics in Elementary Mathematics)
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Math 490  Introduction to Topology
 Prerequisites:
Math 351 or 451 or equivalent experience with abstract mathematics
 Frequency:
Winter (II)
 Student
Body: Mathematics, matheducation, 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 wellsuited for developing
intuition and giving convincing proofs which are pictorial or
geometric rather than completely rigorous.
 Content:
Knots, orientable and nonorientable 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.
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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 twocourse sequence Math
385489. Topics are chosen from geometry, algebra, computer programming,
logic, and combinatorics. Applications and problemsolving are
emphasized. The class usually meets three times per week in recitation
sections. Grades are based on class participation, two onehour
exams, and a final exam.
 Content:
Selected topics in geometry, algebra, computer programming,
logic, and combinatorics for prospective and inservice elementary,
middle, or juniorhigh school teachers. Content will vary from
term to term.
 Alternatives:
none
 Subsequent
Courses: none
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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
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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 512513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295296 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 prooforiented mathematics and be prepared to work hard. Students from the Math 185286 and Math 156256 sequences are strongly advised to take both Math 217 and some 400500 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 finitelygenerated 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 593594 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)
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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 512513 is one of the more abstract and difficult sequences in the undergraduate program. It is frequently elected by students who have completed the 295296 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 prooforiented mathematics and be prepared to work hard. Students from the Math 185286 and Math 156256 sequences are strongly advised to take both Math 217 and some 400500 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
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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 36 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 multidecrement
and multiplelife applications directly related to life insurance
and pensions. The sequence 520521 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.
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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 multidecrement and
multiplelife applications directly related to life insurance.
The sequence 520521 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 modelsjoint life, last survivor,
contingent insurance; multiple decrement modelsdisability,
withdrawal, retirement, etc.; and reserving models for life insurance.
This corresponds to chapters 710, 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
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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
prefunded 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
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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
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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.
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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.
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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
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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)
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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 agentbased adaptive models study phenomena which are prototypical
in the social, biological and decision sciences. These models
are "agentbased" or "bottomup" 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
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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 19 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.
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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 selfadjoint 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.)
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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 nonlinear stability theory, bifurcations, applications
in fluid dynamics (RayleighBenard convection), combustion (flame
speed).
 Alternatives:
none
 Subsequent
Courses: Math 656 (Partial Differential Equations) and 658
(Ordinary Differential Equations.)
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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,
PoincareBendixson 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)
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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
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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 higherdimensional
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:
Crosslisted as IOE 510.
 Subsequent
Courses: IOE 610 (Linear Programming II) and IOE 611 (Nonlinear
Programming)
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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, quasiNewtonian 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:
Crosslisted as IOE 511.
 Subsequent
Courses: This is not a prerequisite for any other course.
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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 crosslisting. 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, quasistate 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:
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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:
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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 prooforiented 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 5Color Theorem; Kuratowski's Theorem; the MatrixTree Theorem; fundamental enumeration principles, bijections, and generating functions; inclusionexclusion; 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)
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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; grouptheoretic 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.
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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 errorcorrecting 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 errorcorrecting 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, ReedMuller, and ReedSolomon.
Introduction to decoding starting with syndrome decoding and covering
weight enumerator polynomials and the MacWilliams 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.
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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 2point 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, GaussSeidel 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.
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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.
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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, padic numbers, elliptic
curves, Diophantine approximation, fast multiplication and factorization,
Public Key Cryptography, and transcendence. This material corresponds
to Chapters 13 and selected parts of Chapters 4, 5, 7, 8, and
9 of Niven, Zuckerman, and Montgomery.
 Alternatives:
Math 475 (Elementary Number
Theory) is a nonhonors 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.
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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 settheoretic 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 theoremproof 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.
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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 computergenerated proofs that are too long to read; there are also surprising connections to showing that certain functions cannot be computed or approximated efficiently. A "zeroknowledge 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:
Probabilisticallycheckable proofs, zeroknowlege 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 errorcorrecting 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:
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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 nonmath grads, math undergrads
 Background
and Goals: This is an introduction to topology with an emphasis
on the settheoretic 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.
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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 nonmath
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.
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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, EilenbergSteenrod axioms, Brouwer's and Lefschetz'
fixedpoint theorems, and other topics.
 Alternatives:
none
 Subsequent
Courses: Math 695 (Algebraic Topology I)
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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).
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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, JordanHolder 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.
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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, CauchyRiemann 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).
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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; RadonNikodym Theorem, conditional
expectation, signed measures, introduction to Fourier transforms.
 Alternatives:
none
 Subsequent
Courses: Math 602 (Real Analysis II).
