Courses in Mathematics (Division 428)

Elementary Courses. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. All courses require three years of high school mathematics; four years are strongly recommended and more information is given for some individual courses below. Students with College Board Advanced Placement credit and anyone planning to enroll in an upper-level class should consider one of the Honors sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions, and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and mathematical thinking through a single course. They are neither prerequisite nor preparation for any further course.

Each of Math 112, 115, 185, and 195 is a first course in calculus and generally credit can be received for only one course from this list. Math 112 is designed for students of business and the social sciences who require only one term of calculus. It neither presupposes nor covers any trigonometry. The sequence Math 115-116-215 is appropriate for most students who want a complete introduction to calculus. Math 118 is an alternative to Math 116 intended for students of the social sciences who do not intend to continue to Math 215. Math 215 is prerequisite to most more advanced courses in Mathematics. Math 112 does not provide preparation for any subsequent course.

Students planning a career in medicine should note that some medical schools require a course in calculus. Generally 112 or 115 will satisfy this requirement, although most science concentrations require at least a year of calculus. Math 112 is accepted by the School of Business Administration, but Math 115 is prerequisite to concentration in Economics and further math courses are strongly recommended.

The sequences 175-176-285-286, 185-186-285-286, and 195-196-295-296 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-186 covers much of the same material as Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have had a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 195-296 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. The student who completes Math 296 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 195 after one of Math 112 or 115. A list of these and other cases of reduced credit for courses with overlapping material is available from the Department. To avoid unexpected reduction in credit, students should always consult an advisor before switching from one sequence to another. In all cases, a maximum total of 16 credits may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken after the course itself.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in either the regular or Honors sequences. A table explaining the possibilities is available from advisors and the Department. The Department encourages strong students to enter beginning Honors courses in preference to 116 or 215.

Students completing Math 215 may continue either to Math 216 (Introduction to Differential Equations) or to the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217-316 is required for all students who intend to take more advanced courses in mathematics, particularly for those who may concentrate in mathematics. Math 217 both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations.

More detailed descriptions of undergraduate mathematics courses and concentration programs are contained in the brochures Undergraduate Programs and Undergraduate Courses available from the Mathematics Undergraduate Program Office, 3011 Angell Hall, 763-4223.

NOTE: WL:3 for all courses.

A maximum total of 4 credits may be earned in Mathematics courses numbered 110 and below. A maximum total of 16 credits may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken after the course itself.

105. Data, Functions, and Graphs. Students with credit for Math. 103 can elect Math. 105 for only 2 credits. (4). (Excl). (QR/1).

Math 105 is a preparatory class to the calculus sequences. Students who complete 105 are fully prepared for Math 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab. The course prepares students for Math 115

106. Algebra and Analytic Trigonometry (Self-Paced). Students with credit for Math. 103 can elect Math. 106 for only 2 credits. No credit granted to those who have completed or are enrolled in Math 105. (4). (Excl).

Self-study version of Math 105. There are no lectures or sections. Students enrolling in Math 106 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. Math 106 students take the same midterm and final exams as Math 105 students. More detailed information is available from the Math Lab.

110. Pre-Calculus (Self-Study). See Elementary Courses above. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).

Math 110 is a preparatory course for the calculus sequence. Students who complete Math 110 are fully prepared for Math 115. The course is a condensed, half-term version of Math 105 designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students enrolling in Math 110 must visit the Math Lab to complete paperwork and receive course materials. The course covers data analysis by means of functions and graphs. The course prepares students for Math 115.

112. Brief Calculus. See Elementary Courses above. Credit is granted for only one course from among Math. 112, 113, 115, 185 and 195. (4). (N.Excl). (BS).

This is a one-term survey course that provides the basics of elementary calculus. Emphasis is placed on intuitive understanding of concepts and not on rigor. Topics include differentiation with application to curve sketching and maximum-minimum problems, antiderivatives and definite integrals. Trigonometry is not used. This course does not mesh with any of the courses in the other calculus sequences.

115. Calculus I. Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 195. (4). (N.Excl). (BS). (QR/1).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major 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 a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Algebra and Trigonometry) or its self-paced equivalent Math 106. Math 116 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.

116. Calculus II. Math. 115. Credit is granted for only one course from among Math. 116, 186, and 196. (4). (N.Excl). (BS). (QR/2).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major 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 a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. Math 215 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 285.

128. Explorations in Number Theory. High school mathematics through at least Analytic Geometry. (4). (NS). (BS). (QR/1).

This course is intended for non-science concentrators and students in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to take calculus first. Students will be introduced to elementary ideas of number theory, an area of mathematics that deals with properties of the integers. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying 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 a2 + b2 = c2 .

147. Mathematics of Finance. Math. 112 or 115. (3). (Excl). (BS).

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 the calculus. 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. The course is not part of a sequence. Students should possess financial calculators.

176. Dynamical Systems and Calculus. Math. 175 or permission of instructor. (4). (N.Excl). (BS).

The sequence Math 175-176 is a two-term introduction to Combinatorics, Dynamical Systems, and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. 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 term in a group project. UNIX workstations will be a valuable experimental tool in this course and students will run preset lab routines on them using Matlab and MAPLE. The general theme of the course will be discrete-time and continuous-time dynamical systems. Examples of dynamical systems arising in the sciences are used as motivation. Topics include: iterates of functions, simple ordinary differential equations, fixed points, attracting and repelling fixed points and periodic orbits, ordered and chaotic motion, self-similarity, and fractals. Tools such as limits and continuity, Taylor expansions of functions, exponentials, logarithms, eigenvalues, and eigenvectors are reviewed or introduced as needed. There is a weekly computer work-station lab.

186. Honors Analytic Geometry and Calculus II. Permission of the Honors advisor. Credit is granted for only one course from among Math. 114, 116, and 186. (4). (N.Excl). (BS). (QR/1).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major 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. Topics covered include transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor. Math 116 is a somewhat less theoretical course which covers much of the same material. Math 285 is the natural sequel.

196. Honors Mathematics II. Permission of the Honors advisor. (4). (N.Excl). (BS). (QR/1).

The sequence Math 195-196-295-296 is a more intensive Honors sequence than 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. Topics will be chosen from: logarithms and exponentials, sups and infs, sequences and series, techniques of integration, Bolzano-Weierstrass Theorem, uniform continuity and convergence, C[[infinity]] and analytic functions, Weierstrass Approximation Theorem, metric spaces: Rn and C0[a,b], completeness and compactness, topics in linear algebra: vector spaces, linear dependence and bases, matrix operations.

215. Calculus III. Math. 116 or 186. (4). (Excl). (BS). (QR/1).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major 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 a midterm and final exam. 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 software. Math 285 is a somewhat more theoretical course which covers the same material. For students intending to major in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.

216. Introduction to Differential Equations. Math. 215. (4). (Excl). (BS).

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216- 417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math majors and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. This course is not intended for mathematics majors, who should elect the sequence 217-316. Math 286 covers much of the same material in the Honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.

217. Linear Algebra. Math. 215. No credit granted to those who have completed or are enrolled in Math. 417, 419, or 513. (4). (Excl). (BS). (QR/1).

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math majors and other students who have some interest in the theory of mathematics should elect the sequence Math 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. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; Eigenvalues and Eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 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. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.

285. Honors Analytic Geometry and Calculus III. Math. 186 or permission of the Honors advisor. (4). (Excl). (BS).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major 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. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation, maximum-minimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.

286. Honors Differential Equations. Math. 285. (3). (Excl). (BS).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major 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. Topics include first-order differential equations, higher-order linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, Runge-Kutta). If time permits, Picard's Theorem will be proved. Math 216 and 316 cover much of the same material. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 286 is also preparation for more theoretical courses such as Math 451.

289. Problem Seminar. (1). (Excl). (BS). May be repeated for credit with permission.

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 exam. 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.

296. Honors Analysis II. Math. 295. (4). (Excl). (BS).

This course is a continuation of Math 295 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Differential and integral calculus of functions on Euclidean spaces. Students who have successfully completed the sequence Math 195-296 are generally prepared to take a range of advanced undergraduate and graduate courses such as Math 512, 513, 525, 590, and many others.

312. Applied Modern Algebra. Math. 217. (3). (Excl). (BS).

One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken Math 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently organized, the course is broken into four parts. (1) the integers "mod n" and linear algebra over the integers mod p, with applications to error correcting codes; (2) some number theory, with applications to public-key cryptography; (3) polynomial algebra, with an emphasis on factoring algorithms over various fields, and (4) permutation groups, with applications to enumeration of discrete structures "up to automorphisms" (a.k.a. Polya Theory). Math 412 is a more abstract and proof-oriented course with less emphasis on applications. EECS 303 (Algebraic Foundations of Computer Engineering) covers many of the same topics with a more applied approach. Another good follow-up course is Math 475 (Number Theory). Math 312 is one of the alternative prerequisites for Math 416, and several advanced EECS courses make substantial use of the material of Math 312. Math 412 is better preparation for most subsequent mathematics courses.

316. Differential Equations. Math. 215 and 217, or equivalent. Credit can be received for only one of Math. 216 or Math. 316, and credit can be received for only one of Math. 316 or Math. 404. (3). (Excl). (BS).

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. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. Math 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. Math 286 is the Honors version of Math 316. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451.

354. Fourier Analysis and its Applications. Math. 216, 316, or 286. No credit granted to those who have completed or are enrolled in Math. 454. (3). (Excl). (BS).

This course is an introduction to Fourier analysis with emphasis on applications. The course also can be viewed as a way of deepening one's understanding of the 100- and 200-level material by applying it in interesting ways. 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. Math 454 covers some of the same material with more emphasis on partial differential equations. This course is good preparation for Math 451, which covers the theory of calculus in a mathematically rigorous way.

371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering 103 or 104, or equivalent; and Math. 216. (3). (Excl). (BS).

This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571- 572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for 571- 572.

404. Intermediate Differential Equations. Math. 216. No credit granted to those who have completed Math. 286 or 316. (3). (Excl). (BS).

This is a course oriented to the solutions and applications of differential equations. Numerical methods and computer graphics are incorporated to varying degrees depending on the instructor. There are relatively few proofs. Some background in linear algebra is strongly recommended. First-order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods. Math 454 is a natural sequel.

412. Introduction to Modern Algebra. Math. 215 or 285; and 217. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 312. (3). (Excl). (BS).

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 of 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 are equivalent required as background. 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 two particular types of mathematical structures: rings and groups. 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. Math 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an Honors version of Math 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 513, 581, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.

416. Theory of Algorithms. Math. 312 or 412 or CS 303, and CS 380. (3). (Excl). (BS).

Many common problems from mathematics and computer science may be solved by applying one or more algorithms well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness. This course has substantial overlap with EECS 586 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. Math 516 (given infrequently) and EECS 574 and 575 (Theoretical Computer Science I and II) include some topics which follow those of this course.

417. Matrix Algebra I. Three courses beyond Math. 110. No credit granted to those who have completed or are enrolled in 217, 419, or 513. (3). (Excl). (BS).

Many problems in science, engineering, and mathematics are best formulated in terms of matrices rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). 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. Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

419/EECS 400/CS 400. Linear Spaces and Matrix Theory. Four terms of college mathematics beyond Math 110. No credit granted to those who have completed or are enrolled in 217 or 513. One credit granted to those who have completed Math. 417. (3). (Excl). (BS).

Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations. Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more proof-oriented. Math 513 is much more abstract and sophisticated. EECS 400 is the same course. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571

420. Matrix Algebra II. Math. 217, 417 or 419. (3). (Excl). (BS).

This course is intended to sharpen the student's skills in the manipulations and applications of linear algebra. Some proofs are given and the object is to learn to use the theory to solve problems. One previous proof-oriented course is recommended, although not required. Similarity theory, Euclidean and unitary geometry, applications to linear and differential equations, interpolation theory, least squares and principal components, B-splines.

425/Stat. 425. Introduction to Probability. Math. 215. (3). (N.Excl). (BS).

Sections 001 and 002. 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. Math concentrators should be sure to elect sections of the course that are taught by mathematics (not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

Sections 003 and 004. See Statistics 425.

450. Advanced Mathematics for Engineers I. Math. 216, 286, or 316. (4). (Excl). (BS).

Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g., Math 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.

451. Advanced Calculus I. Math. 215 and one course beyond Math. 215; or Math. 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).

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. The material usually covered is essentially that of Ross' book Elementary Analysis: The Theory of Calculus. Chapter I deals with the properties of the real number system including (optionally) its construction from the natural and rational numbers. Chapter II concentrates on sequences and their limits, Chapters III and IV on the application of these ideas to continuity of functions, and sequences and series of functions. Chapter V covers the basic properties of differentiation and Chapter VI does the same for (Riemann) integration culminating in the proof of the Fundamental Theorem of Calculus. Along the way there are presented generalizations of many of these ideas from the real line to abstract metric spaces. 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. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115- 116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, 590, 596, and 597.

452. Advanced Calculus II. Math. 217, 417, or 419; and Math. 451. (3). (Excl). (BS).

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. Topics include (1) partial derivatives and differentiability, (2) gradients, directional derivatives, and the chain rule, (3) implicit function theorem, (4) surfaces, tangent plane, (5) max-min theory, (6) multiple integration, change of variable, etc. (7) Green's and Stokes' theorems, differential forms, exterior derivatives. Math 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. Math 452 is prerequisite to Math 572 and is good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 537, 635)

454. Boundary Value Problems for Partial Differential Equations. Math. 216, 286 or 316. Students with credit for Math. 354, 455 or 554 can elect Math. 454 for 1 credit. (3). (Excl). (BS).

This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both Math 455 and 554 cover many of the same topics but are very seldom offered. Math 454 is prerequisite to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556.

462. Mathematical Models. Math. 216, 286 or 316; and 217, 417, or 419. Students with credit for 362 must have department permission to elect 462. (3). (Excl). (BS).

The focus of this course is the application of a variety of mathematical techniques to solve real-world problems. Students will learn how to model a problem in mathematical terms and use mathematics to gain insight and eventually solve the problem. Concepts and calculations, including use of spreadsheets and programming, are emphasized, but proofs are included. Construction and analysis of mathematical models in the natural or 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. Math 286 and 316 cover some of the dynamics without the applications

471. Introduction to Numerical Methods. Math. 216, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS).

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. Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for Sophomore and Junior engineering students; the sequence Math 571-572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.

475. Elementary Number Theory. (3). (Excl). (BS).

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 cyphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly number-theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. At least three terms of college mathematics are recommended. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. 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. Math 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with Math 412 which stresses the algebraic content. Math 475 may be followed by Math 575 and is good preparation for Math 412. 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. Each of these is devoted to a special subarea of number theory.

476. Computational Laboratory in Number Theory. Prior or concurrent enrollment in Math. 475 or 575. (1). (Excl). (BS).

Students will be provided software with which to conduct numerical explorations. Student will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the Laboratory should boost the student's performance in Math 475 or Math 575. Students in the Lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.

486. Concepts Basic to Secondary Mathematics. Math. 215. (3). (Excl). (BS).

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. 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. This material is covered in the course pack and scattered points in the text book. 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. Prior completion of Math 486 may be of use for some students planning to take Math 312, 412, or 425.

489. Mathematics for Elementary and Middle School Teachers. Math. 385 or 485, or permission of instructor. May not be used in any graduate program in mathematics. (3). (Excl).

This course, together with its predecessor Math 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.

490. Introduction to Topology. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

This course in an introduction to both point-set and algebraic topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, courses containing some group theory (Math 412 or 512) and advanced calculus (Math 451) are desirable although not absolutely necessary. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. Point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, covering spaces. Metric and Euclidean spaces are emphasized. Math 590 is a deeper and more difficult presentation of much of the same material which is taken mainly by mathematics graduate students. Math 433 is a related course at about the same level. Math 490 is not prerequisite for any later course but provides good background for Math 590 or any of the other courses in geometry or topology.

512. Algebraic Structures. Math. 451 or 513 or permission of the instructor. No credit granted to those who have completed or are enrolled in 412. Math. 512 requires more mathematical maturity than Math. 412. (3). (Excl). (BS).

This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 195- 296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p. Math 412 is a substantially lower level course over about half of the material of Math 512. The sequence Math 593- 594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Together with Math 513, this course is excellent preparation for the sequence Math 593- 594.

513. Introduction to Linear Algebra. Math. 412 or permission of instructor. Two credits granted to those who have completed Math. 417; one credit granted to those who have completed Math 217 or 419. (3). (Excl). (BS).

This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course (e.g., Math 451 or 512) is recommended. Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; Eigenvalues and Eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned with applications. The natural sequel to Math 513 is 593. Math 513 is also prerequisite to several other courses: Math 537, 551, 571, and 575, and may always be substituted for Math 417 or 419.

521. Life Contingencies II. Math. 520; or permission of instructor. (3). (Excl). (BS).

This course extends the single decrement and single life ideas of Math 520 to multi-decrement and multiple-life applications directly related to life insurance. The sequence Math 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Concepts and calculation are emphasized over proof. Topics include multiple life models joint life, last survivor, contingent insurance; multiple decrement models disability, withdrawal, retirement, etc.; and reserving models for life insurance. Math 522 is a parallel course covering mathematical models for prefunded retirement benefit programs.

523. Risk Theory. Math. 425. (3). (Excl). (BS).

This course explores the applications of stochastic models to risk business (uncertain financial payments). The emphasis is on concepts and calculations with some proofs. Students should have a good background in probability (at least a B+ in Math 425). Topics include utility theory, application to buying general insurance to reduce risk, compound distribution models for risk portfolios, application of stochastic processes to the ruin problem and to reinsurance.

525/Stat. 525. Probability Theory. Math. 450 or 451; or permission of instructor. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only 1 credit. (3). (Excl). (BS).

This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510-511 are natural sequels.

526/Stat. 526. Discrete State Stochastic Processes. Math. 525 or EECS 501. (3). (Excl). (BS).

This is a course on the theory and applications of stochastic processes, mostly 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. 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; branching processes; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queuing theory; an introduction to Brownian motion; stationary processes and martingales. This course is similar to EECS 502 and IOE 515, although the latter course tends to be somewhat more oriented to applications. The next courses in probability are Math 625 and 626, which presuppose substantial additional background (Math 597).

531. Transformation Groups in Geometry. Math. 215. (3). (Excl). (BS).

This course gives a rigorous treatment of a selection of topics involving the interaction of group theory and geometry. Most students have substantial preparation beyond the formal prerequisite (e.g., Math 512) and are taking concurrently other advanced courses (e.g., Math 490) The content will vary significantly with the instructor. One version includes subgroups of the group of Euclidean motions of R2, crystallographic groups, hyperbolic and projective geometry, and Fuchsian groups. Other possible topics are tilings of the plane, affine geometries, and regular polytopes. This course is not prerequisite for any later course but provides good general background for any course in Topology (590, 591) or Geometry (537, 631-632).

555. Introduction to Functions of a Complex Variable with Applications. Math. 450 or 451. Students with credit for Math. 455 or 554 can elect Math. 555 for one hour credit. (3). (Excl). (BS).

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 principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.

557. Methods of Applied Mathematics II. Math. 556. (3). (Excl). (BS).

Although this is a continuation of Math 556, this course is not really required as a prerequisite. A strong background in complex analysis and linear algebra is essential. There is somewhat less emphasis on proofs than in Math 556. Topics include transform methods for partial differential equations, asymptotic expansions, regular and singular perturbation problems, non-linear stability theory, bifurcations, non-linear evolution equations, and associated phenomena.

561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. Math. 217, 417, or 419. (3). (Excl). (BS).

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; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

566. Combinatorial Theory. Math. 216, 286 or 316; or permission of instructor. (3). (Excl). (BS).

This course is a rigorous introduction to classical combinatorial theory. Concepts and proofs are the foundation, but there are copious applications to modern industrial problem-solving. Permutations, combinations, generating functions and recurrence relations. The existence and enumeration of finite discrete configurations. Systems of representatives, Ramsey's Theorem and extremal problems. Construction of combinatorial designs. There is no real alternative, although there is some overlap with Math 565. Sequels are Math 664-665 and Math 669.

571. Numerical Methods for Scientific Computing I. Math. 217, 419, or 513; and 454 or permission of instructor. (3). (Excl). (BS).

This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 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.

572. Numerical Methods for Scientific Computing II. Math. 217, 419, or 513; and 454 or permission of instructor. (3). (Excl). (BS).

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. 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, sclalar first order hyperbolic equations, symmetric hyberbolic systems. There is no real alternative; Math 471 covers a small part of the same material at a lower level. 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.

575. Introduction to Theory of Numbers I. Math. 451 and 513; or permission of instructor. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (3). (Excl). (BS).

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. A Computational Laboratory (Math 476, 1 credit) will usually be offered as a supplement to this course. 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'. Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Crytography, and transcendence. Math 475 is a non-Honors version of Math 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. 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.

582. Introduction to Set Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

One of the great discoveries of modern mathematics was that essentially every mathematical concept may be defined in terms of sets and membership. Thus Set Theory plays a special role as a foundation for the whole of mathematics. One of the goals of this course is to develop some understanding of how Set Theory plays this role. The analysis of common mathematical concepts (e.g., function, ordering, infinity) in set-theoretic terms leads to a deeper understanding of these concepts. At the same time, the student will be introduced to many new concepts (e.g., transfinite ordinal and cardinal numbers, the Axiom of Choice) which play a major role in many branches of mathematics. The development of set theory will be largely axiomatic with the emphasis on proving the main results from the axioms. Students should have substantial experience with theorem-proof mathematics; the listed prerequisites are minimal and stronger preparation is recommended. No course in mathematical logic is presupposed. 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. 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. Math 582 is not an explicit prerequisite for any later course, but it is excellent background for many of the advanced courses numbered Math 590 and above.

590. Introduction to Topology. Math. 451. (3). (Excl). (BS).

This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590.

592. Introduction to Algebraic Topology. Math. 591. (3). (Excl). (BS).

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. Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Maclane axioms, Brouwer's and Lefschetz' fixed-point theorems, and other topics.

594. Algebra II. Math. 593. (3). (Excl). (BS).

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. Topics include group theory, permutation representations, simplicity of alternating groups for n > 4, Sylow theorems, series in groups, solvable and nilpotent groups, Jordan-Hölder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, transcendence degree.

597. Analysis II. Math. 451 and 513. (3). (Excl). (BS).

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. 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), Hölder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini's Theorem; Radon-Nikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.


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