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*Chair: Thomas R. Shemanske*

*Vice Chair: Marcia J. Groszek*

*Professors M. Arkowitz, P. Doyle, C. S. Gordon, M. J. Groszek, C. D. Lahr, C. B. Pomerance, D. Rockmore, T. R. Shemanske, D. I. Wallace, D. L. Webb, D. P. Williams, P. Winkler; Associate Professor J. D. Trout; Assistant Professors A. Barnett, V. V. Chernov, R. C. Orellana, S. D. Pauls, C. J. Sutton, R. A. Weber; Research Instructors R. Daileda, S. Elizalde, R. K. Hladky, M. Ionescu; Visiting Assistant Professors G. W. Kiralis, G. Leibon; Adjunct Associate Professors B. F. Cole, E. Demidenko.*

The three courses Mathematics 3, 8, and 13 provide a coherent three-term sequence in calculus. Mathematics 3 and 8 cover the basic calculus of functions of a single variable, as well as vector geometry and calculus of scalar-valued functions of several variables. In addition, these two courses are prerequisite for many advanced courses in Mathematics and Computer Science. Mathematics 13 covers the basic calculus of vector-valued functions of several variables. Mathematics 11 is a special version of Mathematics 13 for first-year students with two terms of advanced placement. Most students planning advanced work in mathematics or the physical sciences will need a fourth course in calculus, Mathematics 23. Students interested in physical and engineering sciences are encouraged to consider Mathematics 33. Students with two terms of advanced placement credit who possibly are interested in a mathematics major or minor should consider Mathematics 17 as an option in their first term. Mathematics 17, “An Introduction to Mathematics Beyond Calculus�?, is a course designed for students interested in learning about some of the aspects of mathematics not usually encountered in the first years of mathematical studies. Topics change from year to year but may include aspects of combinatorics, algebra, analysis, number theory, geometry, and/or topology. Students planning to take upper-level mathematics courses are strongly encouraged to take Mathematics 22 or 24 (linear algebra) early in their curriculum.

A student wishing to devote only two to three terms to the study of mathematics is encouraged to choose among courses 3, 5, 6, and 10 (as well as 1 and 2 if his or her background indicates this is desirable). The combination of Mathematics 3 and 6 will introduce the student to the ideas and applications of the differential and integral calculus as well as to several branches of modern mathematics. Mathematics 5 is a topics and sometimes interdisciplinary course. Recent topics include “Chance,�? “The World According to Mathematics,�? “Pattern,�? “Geometry in Art and Architecture,�? “A Matter of Time,�? “Applications of Calculus to Medicine and Biology,�? “Music and Computers,�? and “Geometry and the Imagination.�? Mathematics 10 covers the fundamental concepts of statistics.

The major in mathematics is intended both for students who plan careers in mathematics and related fields, and for those who simply find mathematics interesting and wish to continue its study. The content of the major is quite flexible, and courses may be selected largely to reflect student interests. Students who major in mathematics have an opportunity to participate in activities that bring them in close contact with a faculty member - for example, through a small seminar or through an independent research project under the direction of a faculty member. In addition to regular course offerings, a student with specialized interests, not reflected in our current course offerings, often arranges for an independent reading course. Proposals for independent activities should be directed to the Departmental Advisor to Mathematics Majors.

In general, the mathematics major requires the student to pass eight mathematics or computer science courses beyond prerequisites. At least six of the required eight courses must be mathematics, and at least four of these courses must be taken at Dartmouth. In addition, a student must fulfill the College’s requirement for a culminating experience in the major (see below). Additional requirements for honors are described below in a separate section.

Students are encouraged to take Mathematics 22/24 as soon as feasible, since not only is it an explicit prerequisite to many upper-division courses, but also the level of mathematical sophistication developed in Mathematics 22/24 will be presumed in many upper-division courses for which Mathematics 22/24 is not an explicit prerequisite.

Prerequisite Courses: Mathematics 3; 8; 13; 22 or 24

Requirements: To complete the major, it is necessary to complete successfully at least eight courses in addition to the prerequisites, as well as a culminating experience (which may or may not be part of the eight major courses). These eight courses must include:

1. (Algebra) Mathematics 31 or 71;

2. (Analysis) At least one of Mathematics 33, 35, 43, or 63;

3. Six additional Mathematics/Computer Science courses numbered 20 or

above.

Caveats:

Also acceptable: Mathematics 15.3, 16

Computer Science 5, 14, 15, 16, 18

Not acceptable: Mathematics 97

Computer Science 97

At most two Computer Science courses may be used. The culminating experience requirements are described in a separate section below.

While the student interested only in a general exposure to mathematics may choose their major courses subject only to the constraints above, those with more focused interests (pure mathematics, applied mathematics, and mathematics education), will want to consider the course recommendations below.

A.) (Pure Mathematics) For students interested in pure mathematics, Mathematics 24 is preferable to Mathematics 22 as prerequisite.

We recommend that the following courses be included among the eight courses needed for the major:

(Algebra) Mathematics 71 and 81;

(Analysis) Mathematics 63, and 43 or 73;

(Topology/Geometry) Mathematics 54, and at least one of 74, 32, 42 or 72.

Students planning to attend graduate school should take substantially more than the minimum requirements for the major. In particular, such students are strongly urged to take both Mathematics 43 and 73; moreover, undergraduates with adequate preparation are encouraged to enroll in graduate courses.

B.) (Applied Mathematics) Students interested in applied mathematics, especially those considering graduate school in applied mathematics or any of the sciences, are advised to take Mathematics 23, 20 or 60, and 50.

We recommend choosing additional courses from among the following: Mathematics 16, 26, 28, 30, 33, 36, 38, 40, 42, 43, 53, 56, 70, 83, 88.

We do not make any specific recommendations concerning the choice of Mathematics 22 versus 24 as prerequisite and the choices for requirements (1) Algebra and (2) Analysis; these choices depend on the interest of the student.

All students planning to attend graduate school should take substantially more than the minimum requirements for the major. In particular, undergraduates with adequate preparation are encouraged to enroll in graduate courses.

C.) (Mathematics Education) Students who are considering a career in teaching should pay close attention to the recommendations of the National Council of Teachers of Mathematics (NCTM). The NCTM has endorsed a series of recommendations for a suggested course of study for those people interested in teaching mathematics at the secondary level. In general, their recommendations (www.nctm.org) are for a vigorous course of study. At the moment, these recommendations far exceed the requirements for obtaining a teaching certificate, but indicate the direction in which the NCTM hopes that educators will proceed. Highly qualified teachers in the elementary and secondary schools are of vital national importance, and these guidelines should be carefully considered. Dartmouth courses that closely fit the recommendations of the NCTM are (in addition to the prerequisites): Mathematics 20 or 60; 23 or 36; 25 or 75; 28, 38 or 68; 31 or 71; 32 or 42 or 72; 35 or 43 or 63; 50

The Department will accept any of the following in satisfaction of the requirement of a culminating experience:

1. Submission of an Honors thesis acceptable for honors or high honors.

2. Satisfactory completion of any graduate course in mathematics.

3. Satisfactory completion of a one-term independent research project (subject to approval by the advisor to majors).

4. Satisfactory completion of an advanced undergraduate course from among: Mathematics 56, 68, 69, 70, 72, 73, 74, 75, 81, 83, 89, 98.

The following minors are available to all students who are not majoring in mathematics and who do not have a modified major with the Mathematics Department. For each minor, the prerequisites and required courses are listed below. Approval of a minor can be obtained through the Department’s Advisor to Mathematics Majors.

I. Mathematics

*Prerequisites:* Mathematics 3, 8, 13, 22

*Courses:* Mathematics 31 or 71; Mathematics 33 or 35 or 43 or 63; plus two other Mathematics courses numbered 20 or above. Computer Science 5 and Mathematics 5 or 10 or 15.3 or 16 are also acceptable.

II. Statistics

*Prerequisites:* Mathematics 3, 8, 13, 22

*Courses:* Mathematics 20, 30, 50, 70

III. Applied Mathematics for Physical and Engineering Sciences

*Prerequisites:* Mathematics 3, 8, 13, 15.3 or 23, Computer Science 5

*Courses:* Mathematics 20; 22; 33; 16 or 26 or 40 or 56; 43 or 50 or 53

IV. Applied Mathematics for Biological and Social Sciences

*Prerequisites:* Mathematics 3, 8, 13, 22

*Courses:* Mathematics 20; 16 or 15.3 or 23; 36; 30 or 50

A student who satisfies requirements of the College for admission to the Honors Program and is interested in doing independent work is strongly encouraged to participate in the departmental Honors Program. Students who successfully complete the Honors Program will have their degrees conferred with ‘Honors’ or ‘High Honors’ in mathematics; high honors is awarded only if the student submits a written thesis. Interested students should read this section of the *ORC* carefully and consult the Department Advisor to Mathematics Majors. This program can be especially important to those who contemplate graduate work in mathematics or a related field.

*Admission:* Admission to the Honors Program requires a general College average of B, and a B average in the Mathematics Department at the time of admission *and at the time of graduation.* Moreover, a B+ average is required in the work of the Honors Program. The B average in the Department is computed as follows: Courses prerequisite to the major and undergraduate research courses (Mathematics 97) are not counted, but *all other courses* titled (or cross-listed with) mathematics which the student has taken are counted, whether or not these courses form part of the student’s formal major. In the case of a modified major, this average may include courses outside the Mathematics Department. The B+ average required in the work of the Honors Program is defined to be a grade of B+ given by the faculty advisor on the research project. Questions about this requirement should be directed to the Departmental Advisor to Mathematics Majors.

*Requirements:* Under the supervision of a faculty member, the student must complete an independent research project or thesis beyond what is required as part of a course. Often the subject of the project or thesis will be motivated by concepts or the content of an advanced seminar or course in which the student has participated, and, typically, the project or thesis will be completed over a period of three terms. The student should consult with his/her prospective faculty advisor and submit to the Advisor to Mathematics Majors a brief written proposal of the project that has the written approval of the faculty advisor. The Advisor to Majors will then review the student’s proposal and the courses that have been selected for the Honors major. Approval of the proposal and course selection constitutes formal admission into the Honors Program. This procedure should be completed by the beginning of fall term of the student’s senior year. The student may then register for (at most two terms of) Mathematics 97, Undergraduate Research.

In the first week of the student’s final term in residence, the student must register with his/her faculty advisor for ‘Honors Thesis/Project Supervision.’ This is not an official College course; rather, it represents a declaration of intent to the Department that the student wishes to be considered for honors at the time of graduation. Forms for this purpose are available from the Advisor to Majors. *No student who has failed to file this intent form with the Advisor to Majors will be considered for honors in the major.*

After the thesis is completed and submitted to the faculty advisor, the advisor can offer a recommendation for honors or high honors on behalf of the student; this recommendation must be ratified by a vote of the Department faculty.

*Prerequisite:* Same as mathematics major plus some additional prerequisites from modifying major (subject to approval of Advisor to Majors).

*Requirements:* An algebra and an analysis course that satisfy the requirements of the mathematics major, together with four additional courses that normally count towards the major in mathematics, including one course that satisfies the culminating experience requirement (choice subject to approval of Advisor to Majors). Subject to the approval of the Advisor to Majors, the algebra course can be replaced by one of the following courses: Mathematics 28, 38, 39, 54, 69, 89.

Four additional courses from the secondary department selected with the approval of the Advisor to Majors and the secondary department. In particular, these ten nonprerequisite courses must form a coherent unit that renders the modified major academically more valuable than an abbreviated major together with a minor in the secondary department.

Dartmouth College offers a program of graduate study leading to the Ph.D. degree in mathematics. This program is designed to meet the need for mathematicians who are highly qualified in both teaching and scholarship. The College provides an environment in which a doctoral candidate can pursue professional study in mathematics and prepare to be an effective teacher.

With rare exceptions, the A.M. in mathematics is offered only to those enrolled in the Ph.D. program. Normally the requirements for the A.M. must be fulfilled within two years after entering and enrolling as a graduate student in the Mathematics Department at Dartmouth. In addition to the general College requirements for the master’s degree, the requirement is departmental certification in algebra, analysis, topology, and one other area.*

Note (1): Continuation in the program for a second year is contingent on a review of a student’s work by the Mathematics Graduate Program Committee, the review to take place early in the spring term of the first year.

Note (2): The general College requirements referred to above are three terms in residence at Dartmouth and credit in eight courses of graduate quality; these courses may sometimes, up to a limit of four, be replaced by approved research or special study.

The requirements for the Ph.D. degree in mathematics are as follows:

1. Departmental certification in algebra, analysis, topology, and one other area.*

2. Admission to Ph.D. candidacy by the departmental Graduate Program Committee as a result of its second review, which takes place at the end of the spring term of the second year of graduate study. This review will take account of all the relevant information that the Graduate Program Committee can gather, such as the student’s record in courses and seminars, the student’s performance during the certification process, and an estimate of the student’s ability to write an acceptable thesis.

3. Demonstration of a reading knowledge of a foreign language normally chosen from French, German, and Russian. The Graduate Program Committee will monitor students’ progress in its annual review.

4. Completion of a doctoral thesis of acceptable quality, and its defense in an oral examination.

5. Preparation for the teaching seminar through such activities as tutoring in the years before admission to candidacy, completion of the teaching seminar, and teaching at least one course (or the equivalent) for each year following admission to candidacy. This requirement is met by receiving credit for Mathematics 107 once during each year preceding admission to candidacy, credit for Mathematics 147, and credit for Mathematics 149 once during each year following admission to candidacy. The Graduate Program Committee may approve substitutions subject to the minimum requirements: each student must earn credit for Mathematics 107 at least once, credit for Mathematics 147, and credit for Mathematics 149 at least twice.

*Course Numbering System:* For most courses numbered 20 or above, the last digit in the course number indicates the field of mathematics as follows: probability and statistics, 0; algebra, 1; geometry, 2; analysis, 3; topology, 4; number theory, 5; applications, 6; combinatorics, 8; logic and foundations, 9.

Course Prerequisites: In all cases in which a prerequisite to a course is listed, the honors or advanced placement equivalent of that course may be substituted. For example, wherever Mathematics 13 appears as a prerequisite, Mathematics 14 will serve.

05F, 06F: 10, 11

The course description is given under Mathematics 2. *This course is open only to students having the permission of the Department. Dist: QDS.* The staff.

06W, 07W: 9, 10

Mathematics 1-2 is a two-term sequence. Its purpose is to cover the calculus of Mathematics 3, the standard introduction to calculus, and, at the same time, to develop proficiency in algebra. The sequence is specifically designed for first-year students whose manipulative skill with the techniques of secondary-school algebra is inadequate for Mathematics 3. The objective is to introduce and develop algebraic techniques as they are needed to study the ideas of calculus. The techniques will be taught in class, and the students will be required to practice by solving many drill problems for homework. There will be tutorial-help sessions.

Mathematics 1 will include the concepts of function and graph and the basic ideas and applications of differential and integral calculus, at least as they pertain to polynomial functions. In the second course, Mathematics 2, the study of calculus will be continued so that by the end of the sequence the students will have been introduced to the algebra and calculus of the exponential and logarithm functions and the trigonometric functions and to differential equations.

Prerequisite: Mathematics 1, or permission of the Department. *Dist: QDS.* The staff.

05F: 9L, 10, 1106W: 9L06F: 9L, 1107W: 9L

This course is the basic introduction to calculus. Students planning to specialize in mathematics, computer science, chemistry, physics, or engineering should elect this course in the fall term. Others may elect it in the winter.

A study of polynomials and rational functions leads to the introduction of the basic ideas of differential and integral calculus. The course also introduces exponential, logarithmic, and trigonometric functions. The emphasis throughout is on fundamental ideas and problem solving.

Mathematics 3 is open to all students who have had intermediate algebra and plane geometry. No knowledge of trigonometry is required. The lectures are supplemented by problem sessions. *Dist: QDS.* Elizalde, Ionescu, Lahr (fall), Trout (winter).

06W: 10A07W: Arrange

This course will establish the relevance of calculus to medicine. It will develop mathematical tools extending the techniques of introductory calculus, including some matrix algebra and solution techniques for first order differential equations. These methods will be used to construct simple and elegant models of phenomena such as the mutation of HIV, spread of infectious disease, and biological disposition of drugs and inorganic toxins, enzyme kinetics and population growth.

Prerequisite: Mathematics 3. Note: This is a second-term calculus course, but it does not cover the same material as Mathematics 8, and does not serve as a prerequisite for Mathematics 13. There is a version of this course suitable for major credit: see Mathematics 27. *Dist: QDS*. Wallace.

05F: 10A06W: 1006F, 07W: Arrange

In 05F, *3D Computer Animation and the Math that Drives It (*Identical to College Course 1 and Computer Science 3)*.* This introductory, interdisciplinary course will focus on the relationship between mathematics and computer animation. Students will get an overview of the history of computer animation along with the theory, practice and technology behind it. Students will come away with a well-rounded understanding of computer animation, 3D computer animation concepts and the mathematics behind Maya and other 3D computer animation software programs.

This course will provide an artistic perspective for mathematics students and a mathematics perspective for art students. It will be hands on, in that students will complete computer-animated exercises using Maya 3D animation software while learning how to create custom features in Maya using the Maya Embedded Programming Language (MEL). Classes will consist of lectures, discussions, technical demonstrations, and animation screenings.

*Dist: TAS. Satisfies the Interdisciplinary requirement (Class of 2004 and earlier)*. Loeb, Rockmore.

In 06W, *The World According to Mathematics.* Near the beginning of the seventeenth century, a young French mathematician, Rene Descartes, dreamed of a world that could be approached rationally through logical computation. What had been the important ideas of mathematics that led him to believe that his dreams could indeed become reality? To what extent can the intellectual history of the world since Descartes be described as a continuous program to achieve his dream? To what extent has the dream been realized?

In this course we will explore these questions by learning the mathematics that will allow us to appreciate what is involved in a variety of specific cases. Intended for students whose major interests lie outside of mathematics and the sciences, the course assumes that students have only a background in high school mathematics. Computers will be introduced and used throughout the course.

*Dist: QDS.* Lahr.

06S: 1106X, 07S: Arrange

This course is designed for students whose interests lie outside the physical sciences. The course includes an introduction to sets and logic, elementary counting techniques, an introduction to probability, and topics in matrix algebra including the solution of systems of equations and matrix inverses. Illustrative examples and problems will be chosen from the social, managerial, and biological sciences. Computing may be used to illustrate concepts and solve problems. No background in computer programming is assumed. The course is appropriate for the student who plans to take no advanced courses in mathematics. *Dist: QDS.* The staff.

*Consult special listing*

05F: 11, 1206W, 06S: 1106F: 11, 1207W, 07S: 11

This course is a sequel to Mathematics 3 and is appropriate for students who have successfully completed an AB calculus curriculum in secondary school. Roughly half of the course is devoted to topics in one-variable calculus: techniques of integrations, areas, volumes, trigonometric integrals and substitutions, numerical integration, sequences and series including Taylor series.

The second half of the course generally studies scalar valued functions of several variables. It begins with the study of vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength). The rest of the course is devoted to studying different calculus of functions of several variables. Topics include limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems including the use of Lagrange multipliers.

Prerequisite: Mathematics 3 or equivalent. *Dist: QDS.* Arkowitz, Kiralis (fall), Gordon (winter), Pauls (spring).

05F, 06F: 9L

Sections of Mathematics 8 for students who have been invited by the Department Chair based on their record in high school or exceptional work in Mathematics 3. *Dist: QDS.* Hladky.

06S: 1107S: Arrange

An introduction to the basic concepts of statistics. Topics include elementary probability theory, descriptive statistics, the binomial and normal distributions, confidence intervals, basic concepts of tests of hypotheses, chi-square tests, nonparametric tests, normal theory t-tests, correlation, and simple regression. Packaged statistical programs will be used. Because of the large overlap in material covered, no student may receive credit for more than one of the courses Economics 10, Government 10, Mathematics 10, Psychology 10, Social Sciences 10, or Sociology 10 except by special petition. *Dist: QDS.* Leibon.

05F, 06F: 9L, 11

This course can be viewed as equivalent to Mathematics 13, but is designed especially for first-year students who have successfully completed a BC calculus curriculum in secondary school. In particular, as part of its syllabus it includes most of the multivariable calculus material present in Mathematics 8.

Topics include vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength), limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems. It continues with multiple integration, vector fields, line integrals, and finishes with a study of Green’s and Stokes’ theorem. *Dist: QDS.* Daileda, Shemanske.

05F: 9L06W: 1006F, 07W: Arrange

Section(s) of Mathematics 11 for students who have been invited by the Department Chair based on their record in secondary school. *Dist: QDS.* Groszek (fall), Weber (winter).

05F: 1206W: 1106S, 06F:

1207W: 1107S: 12

This course is a sequel to Mathematics 8 and provides an introduction to calculus of vector-valued functions. Topics include differentiation and integration of parametrically defined functions with interpretations of velocity, acceleration, arclength and curvature. Other topics include iterated, double, triple and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green’s theorem, curl and divergence, and Stokes’ theorem.

Prerequisite: Mathematics 8 or equivalent. Note: First-year students who have received two terms on the BC exam generally should take Mathematics 11 instead. On the other hand, if the student has had substantial exposure to multivariable techniques, they are encouraged to take a placement exam during orientation week to determine if placement into Mathematics 13 is more appropriate. *Dist: QDS.* Sutton (fall), Orellana (winter), the staff (spring).

06W, 07W: 11

Sections of Mathematics 13 for students who have done satisfactory work in Mathematics 9 or by invitation or approval of Department Chair based on exceptional work in Mathematics 8. *Dist: QDS.* The staff.

05F, 06F: 11

The two parts of Mathematics 15, referred to as 15.1 and 15.2, are a two-term introduction to the mathematics related to physics for students intending advanced work in physical science, engineering or applied mathematics. We advise students to take Physics 13 concurrently with Mathematics 15.1 if possible.

First-order differential equations and integration techniques, second-order constant-coefficient differential equations, vectors, lines and planes, dot product and Euclidean geometry, vector functions of one variable, line integrals, matrix algebra and linear equations.

Prerequisite: Credit for Mathematics 3, Physics 13 concurrently, or the equivalent, or permission of the instructor. Mathematics 15.1 can be substituted for Mathematics 8 as a prerequisite for any course or program. *Dist: QDS.* Rockmore.

06W, 07W: 11

The two parts of Mathematics 15, referred to as 15.1 and 15.2, are a two-term introduction to the mathematics related to physics for students intending advanced work in physical science, engineering, or applied mathematics.

Differentiable functions of several variables, gradient fields, general chain rule for differentiable functions, max-min techniques, iterated and multiple integrals, centroids and moments, improper integrals, curvilinear coordinates, divergence and curl of a vector field, Green’s theorem, conservative fields, Gauss’s theorem, Stokes’ theorem, Taylor and Fourier series.

Prerequisite: Mathematics 15.1, Physics 14 concurrently, or permission of the instructor. Mathematics 15.2 can be substituted for Mathematics 13 as a prerequisite for any course or program. *Dist: QDS.* Hladky.

06S: 1107S: Arrange

This course introduces one of the fundamental tools of modern business planning and an exciting area of current mathematical and computer science research. The course begins with a discussion of the kinds of problems to which linear programming applies, followed by an introduction to the simplex algorithm and duality and shadow prices. After a discussion of some pitfalls of the simplex algorithm, the course turns to the revised simplex method, the solution of general linear programming problems, the general theory of duality and feasibility, a discussion of the applications of linear programming to the efficient allocation of scarce resources, and such problems as production scheduling and inventory. The course will close with topics selected from applications to matrix games, connections with geometry, connections to optimal matchings, network flows and transportation problems or the nature and implications of interior point methods in linear programming.

Prerequisite: Mathematics 6 or 8 or equivalent knowledge of matrix algebra and permission of instructor. *Dist: TAS.* Elizalde.

05F: 1006F: Arrange

Gives prospective Mathematics majors an early opportunity to delve into topics outside the standard calculus sequence. Specific topics will vary from term to term, according to the interests and expertise of the instructor. Designed to be accessible to bright and curious students who have mastered BC Calculus, or its equivalent. This course counts toward the Mathematics major, and is open to all students, but enrollment may be limited, with preference given to first-year students.

Prerequisite: Mathematics 8, or placement into Mathematics 11. *Dist: QDS.* Orellana.

05F: 1106W: 1006F: 1107W: 10

This course integrates discrete mathematics with algorithms and data structures, using computer science applications to motivate the mathematics. It is designed to be taken simultaneously with Computer Science 15 or18. However, students who are unable to complete it in this way may take it after Computer Science 15 or 18 but before Computer Science 25.

The course introduces counting techniques and number theory, with an emphasis on the application to RSA public key cryptography. It covers logic and proofs, including mathematical induction. Relationships among recursive algorithms, recurrence relations, and mathematical induction are discussed with particular attention to trees as a recursive data structure. Issues of expected running time for algorithms and the technique of “hashing�? data files for quick recovery of information guides the discussion of probability through independent trials, experiments, and expected values.

Prerequisite: Concurrent enrollment in Computer Science 15 or 18 or completion of Computer Science 15 or 18. *Dist: QDS.* Pomerance (fall).

05F: 10A06X, 06F, 07S: Arrange

Basic concepts of probability are introduced in terms of finite probability spaces and stochastic processes having a finite number of outcomes on each experiment. The basic theory is first illustrated in terms of simple models such as coin tossing, random walks, and casino games. Also included are Markov chain models and their applications in the social and physical sciences. The computer will be used to suggest and motivate theoretical results and to study applications in some depth. There is an honors version of this course: see Mathematics 60.

Prerequisite: Mathematics 8. *Dist: QDS.* Ionescu.

05F: 206S: 1006X, 06F: Arrange

This course presents the fundamental concepts and applications of linear algebra with emphasis on Euclidean space. Significant goals of the course are that the student develop the ability to perform meaningful computations and to write accurate proofs. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. Applications may be drawn from areas such as optimization, statistics, biology, physics, and signal processing.

Students who plan to take either Mathematics 63 or Mathematics 71 are strongly encouraged to take Mathematics 24.

Prerequisite: Mathematics 8. *Dist: QDS.* Daileda (fall), Williams (spring).

05F: 1006W: 206S: 1006F, 07W, 07S: Arrange

This course is a survey of important types of differential equations, both linear and nonlinear. Topics include the study of systems of ordinary differential equations using eigenvectors and eigenvalues, numerical solutions of first and second order equations and of systems, and the solution of elementary partial differential equations using Fourier series.

Prerequisite: Mathematics 13. *Dist: QDS.* Barnett (fall), Chernov (winter), Daileda (spring).

06W, 06S: 1007W, 07S: Arrange

This course is an introduction to the fundamental concepts of linear algebra in abstract vector spaces. The topics and goals of this course are similar to those of Mathematics 22, but with an additional emphasis on mathematical abstraction and theory.

(Mathematics 24 can be substituted for Mathematics 22 as a prerequisite for any course or program.)

*Dist: QDS.* Wallace (winter), Weber (spring).

06F: Arrange

This course is a survey of the elementary arithmetic of the integers (prime numbers, factorization, congruences, diophantine equations) with some historical study of important problems. There is an honors section of this course: see Mathematics 75. Offered in alternate years.

Prerequisite: Mathematics 13. *Dist: QDS.*

05F, 06F: 12

A study and analysis of important numerical and computational methods for solving engineering and scientific problems. The course will include methods for solving linear and nonlinear equations, doing polynomial interpolation, evaluating integrals, solving ordinary differential equations, and determining eigenvalues and eigenvectors of matrices. The student will be required to write programs and run them on the computer.

Prerequisite: Mathematics 23, and Computer Science 5 or 14. *Dist: QDS.*

06W: 10A07W: Arrange

This course will prepare students to read the technical literature in mathematical biology, epidemiology, pharmacokinetics, ecological modeling and related areas. Topics include systems of nonlinear ordinary differential equations, equilibria and steady state solutions, phase portraits, bifurcation diagrams, and some aspects of stability analysis. Emphasis is placed on the student’s ability to analyze phenomena and create mathematical models. This interdisciplinary course is open to mathematics majors, biology majors, and students preparing for a career in medicine.

Prerequisite: Mathematics 22. Note: Students without the mathematical prerequisites can take this course as Mathematics 4: no student may take both Mathematics 4 and 27 for credit, and only Mathematics 27 is eligible to count towards the major in mathematics. *Dist: QDS.* Wallace.

06W: 1207W: Arrange

Beginning with techniques for counting - permutations and combinations, inclusion-exclusion, recursions, and generating functions - the course then takes up graphs and directed graphs and ordered sets, and concludes with some examples of maximum-minimum problems of finite sets. Topics in the course have application in the areas of probability, statistics, and computing.

Prerequisite: Mathematics 8, or Mathematics 3 and 6. *Dist: QDS.* Orellana.

07S: Arrange

Several approaches that formalize the notion of computability are presented. The equivalence of these formalizations is discussed, as well as Church’s Thesis, which claims that these formalizations capture the intuitive notion of computability.

Universal machines and undecidable problems are discussed. The course concludes with a study of recursive and recursively enumerable sets, and an introduction to relative computability and degrees of unsolvability. Offered in alternate years.

Prerequisite: Mathematics 22, or Computer Science 49, or permission of the instructor. *Dist: QDS.* The staff.

05F: 12

This course provides an introduction to the most common model used in statistical data analysis. Simple linear regression, multiple regression, and analysis of variance are covered, as well as statistical model-building strategies. Regression diagnostics, analysis of complex data sets and scientific writing skills are emphasized. Methods are illustrated with data sets drawn from the health, biological, and social sciences. Computations require the use of a statistical software package such as STATA. Offered in alternate years.

Prerequisite: Mathematics 10, another elementary statistics course, or permission of the instructor. *Dist: TAS.* Cole.

05F: 206X, 06F: Arrange

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography. *As a result of the variable syllabus, this course may not serve as an adequate prerequisite for Mathematics 81. Students who contemplate taking Mathematics 81 should consider taking Mathematics 71 instead of this course.*

Prerequisite: Mathematics 22. *Dist: QDS.* The staff.

06W: 2A

Topics in intuitive geometry and topology, for example: how to turn a sphere inside out; knots, links, and their invariants; polyhedra in 2, 3, and 4 dimensions; the classification of surfaces; curvature and the Gauss-Bonnet theorem; spherical and hyperbolic geometry; Escher patterns and their quotients; the shape of the universe. Offered in alternate years.

Prerequisite: Mathematics 22 or 24. *Dist: QDS.* Doyle.

06S: 1007S: Arrange

This course provides an overview of the mathematical tools used for analyzing common problems in science and engineering. Particular attention will be given to problems involving linear operators. Topics include partial differential equations, Fourier analysis, linear spaces and operators (in particular, matrix operators), and the calculus of variations. Both analytical and numerical methods will be covered.

Prerequisite: Mathematics 23. *Dist: TAS.* Hladky.

06W: 1207W: Arrange

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Some applications of the theory may be presented. Mathematics 63 presents similar material, but from a more sophisticated point of view. *This course may not serve as an adequate prerequisite for either Mathematics 73 or 83. Students who contemplate taking one of these two advanced courses should consider taking Mathematics 63 instead of this course.*

Prerequisite: Mathematics 13 and permission of the instructor, or Mathematics 22. *Dist: QDS.* Lahr.

06W: 9L07W: Arrange

The mathematical methods of Mathematics 6, or Mathematics 20, and Mathematics 13 are extended and applied to the study of mathematical models developed for use in such fields as anthropology, biology, economics, sociology, psychology, and linguistics. The role that mathematical models play in scientific study is discussed, as well as the methods by which a model may be validated. Students interested in further study of these and related topics are referred to the courses listed in the Mathematics and Social Sciences program.

Prerequisite: Mathematics 6 or 20, and 13. *Dist: TAS.* The staff.

06S: 1207S: Arrange

The theory of graphs has roots in both practical and recreational mathematics. Today there are major applications of graph theory in management science (operations research) and computer science. This course is a survey of the theory and applications of graphs. Topics will be chosen from among connectivity, trees, and Hamiltonian and Eulerian paths and cycles; isomorphism and reconstructability; planarity, duality, and genus; independence and coloring problems, including interval graphs, interval orderings and perfect graphs, color-critical graphs and the four-color theorem; matchings; network flows, including applications to matchings, higher connectivity, and transportation problems; matroids and their relationship with optimization.

Prerequisite: Mathematics 22 (or Computer Science 25 and permission of the instructor). *Dist: QDS.* Elizalde.

05F: 9L

This course begins with a brief treatment of sentential logic and then concentrates on first-order logic. Both proof theory and model theory are taken up. The course ends with a proof of the Gödel incompleteness theorem. Connections with the philosophy of mathematics are discussed. There is an honors version of this course: see Mathematics 69. Offered in alternate years.

Prerequisite: one of Mathematics 22, 28, and 29, or Philosophy 10 by permission of the instructor. *Dist: QDS.* The staff.

07S: Arrange

This is a continuation of Mathematics 20 (60). The course studies probability models chosen from queueing theory, genetics, statistical physics, and gambling. Additional probability concepts such as continuous probability and stochastic processes will be discussed in the context of these models. Offered in alternate years.

Prerequisite: Mathematics 13 and 20, or permission of the instructor. *Dist: QD*S.

06W: 2

This course will cover curves and surfaces in Euclidean 3-dimensional space. Topics include curvature and torsion of curves, the Frenet-Serret equations, Gaussian and mean curvature of surfaces, geodesics and parallel transport, isometries and Gauss’s Theorem Egregium, the Riemann Curvature tensor. One or more of the following topics will be studied if time permits: vector fields, tangent bundles, hypersurfaces, connections, and curvature. Offered in alternate years.

Prerequisite: Mathematics 22 or permission of the instructor, and Mathematics 23. *Dist: QDS.* Hladky.

06S: 1107S: Arrange

This course covers the differential and integral calculus of complex variables including such topics as Cauchy’s theorem, Cauchy’s integral formula and their consequences; singularities, Laurent’s theorem, and the residue calculus; harmonic functions and conformal mapping. Applications will include two-dimensional potential theory, fluid flow, and aspects of Fourier analysis.

Prerequisite: Mathematics 13. *Dist: QDS.* Gordon.

06W: 1207W: Arrange

This course introduces the theory of continuous probability and the theory of statistical inference. Continuous random variables, density functions, expectations, and moment-generating functions are introduced. In addition, special continuous distributions such as the normal, the uniform, and the gamma are discussed. The continuous probability concepts are used to introduce statistical inference, which includes the theory of estimation and the theory of hypothesis testing. Analysis of data and applications are included. The computer will be used whenever possible in applications of the theory.

Prerequisite: Mathematics 13 and 20, or permission of the instructor. *Dist: QDS.* Barnett.

*Not offered in the period from 05F through 07S*

A mathematical model that displays evolution in time is called a dynamical system. Time itself may be thought of as varying continuously or non-continuously.

The behavior of a model may be highly regular, chaotic, or something in between. The purpose of this course is to explore such phenomena in detail in the context of difference equations as well as ordinary and partial differential equations. Offered in alternate years.

Prerequisite: Mathematics 23. *Dist: QDS*.

06X: Arrange

This course begins with the definitions of topological space, open sets, closed sets, neighborhoods, bases and subbases, closure operator, continuous functions, and homeomorphisms. The course will study constructions of spaces including subspaces, product spaces, and quotient spaces. Special categories of spaces and their interrelations will be covered, including the categories defined by the various separation axioms, first and second countable spaces, compact spaces, and connected spaces. Subspaces of Euclidean and general metric spaces will be among the examples studied in some detail.

Prerequisite: Mathematics 13 and 22. *Dist: QDS.* Trout.

*Not offered in the period from 05F through 07S*

This course introduces the student to the concepts of modern numerical analysis. The main emphasis will be on developing effective numerical methods to solve problems in ordinary and partial differential equations. Other topics will be chosen from optimization, approximation, Fourier Transform, and Monte Carlo methods. The specific content will depend in part on the instructor. Offered in alternate years.

Prerequisite: Mathematics 33 and Computer Science 5 or 14, or permission of the instructor. *Dist: QDS.*

06S: 11

This course is a more theoretical introduction to probability theory than Mathematics 20. In addition to the basic content of Mathematics 20, the course will include other topics such as continuous probability distributions and their applications. Offered in alternate years.

Prerequisite: Mathematics 13, or permission of the instructor. *Dist: QDS.* Ionescu.

06W: 1207W: Arrange

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Students may not take both Mathematics 35 and 63 for credit.

Prerequisite: Mathematics 22 or 24, or Mathematics 13 and permission of the instructor. *Dist: QDS.* Gordon.

07S: Arrange

This introductory course presents mathematical topics that are relevant to issues in modern physics. It is mainly designed for two audiences: mathematics majors who would like to see modern physics and the historical motivations for theory in their coursework, and physics majors who want to learn mathematics beyond linear algebra and calculus. Possible topics include (but are not limited to) introductory Hilbert space theory, quantum logics, quantum computing, symplectic geometry, Einstein’s theory of special relativity, Lie groups in quantum field theory, etc. No background in physics is assumed.

Prerequisites: Mathematics 24, or Mathematics 22 and permission of the instructor. *Dist: QDS.*

05F: 9L

This course covers the use of abstract algebra in studying the existence, construction, enumeration, and classification of combinatorial structures. The theory of enumeration, including both Polya Theory and the Incidence Algebra, and culminating in a study of algebras of generating functions, will be a central theme in the course. Other topics that may be included if time permits are the construction of block designs, error-correcting codes, lattice theory, the combinatorial theory of the symmetric group, and incidence matrices of combinatorial structures. Offered in alternate years.

Prerequisite: Mathematics 28 and 31, or Mathematics 71, or permission of the instructor. *Dist: QDS*. Elizalde.

07W: Arrange

This course begins with a study of relational systems as they occur in mathematics. First-order languages suitable for formalizing such systems are treated in detail, and several important theorems about such languages, including the compactness and Lowenheim-Skolem theorems, are studied. The implications of these theorems for the mathematical theories being formulated are assessed. Emphasis is placed on those problems relating to first-order languages that are of fundamental interest in logic. Offered in alternate years.

Prerequisite: experience with mathematical structures and proofs, as offered by such courses as Mathematics 71, 54, or 24; or permission of the instructor. *Dist: QDS.*

07S: Arrange

This course will be a continuation of the study of the theory of statistical inference that was begun in Mathematics 50. Topics will include the mathematical development of normal theory t-tests and nonparametric tests for means and medians, tests for variances, chi-square tests, and an introduction to the theory of the analysis of variance and regression analysis. Offered in alternate years.

Prerequisite: Mathematics 50. *Dist: QDS*.

05F: 1006F: Arrange

The sequence Mathematics 71 and 81 is intended as an introduction to abstract algebra. Mathematics 71 develops basic theorems on groups, rings, fields, and vector spaces.

Prerequisite: Mathematics 22 or 24. *Dist: QDS.* Pomerance.

07S: Arrange

This course develops one or more topics in geometry based on the algebraic background of Mathematics 71. These topics are usually chosen from among geometric algebra (*i.e.,* the properties of Euclidean, affine, and projective spaces under transformations of the classical groups), algebraic geometry, differential geometry, and the classical geometry of real and complex projective spaces. Offered in alternate years.

Prerequisite: Mathematics 31 or 71. *Dist: QDS.*

06S: 1207S: Arrange

This course develops aspects of the general theory of differentiation and integration in Euclidean space. Primary topics include the Implicit and Inverse Function Theorems, differential forms, and Stokes’ Theorem.

Prerequisite: Mathematics 63. In general, Mathematics 35 is not an acceptable prerequisite; however in consultation with the instructor, Mathematics 35 together with some outside reading should be adequate preparation for the course. *Dist: QDS.* Pauls.

06S: 10

This course will study a single area of topology in some detail. Possible topics are: (1) Combinatorial Topology: simplexes, simplicial complexes, subcomplexes; joins, stars, links and subdivisions of complexes; piecewise linear and simplicial maps, the simplicial approximation theorem; the topology of simplicial complexes; classification of surfaces; the edge-path group; examples will include graphs, the nerve of a family of sets, and combinatorial manifolds. (2) Introductory Algebraic Topology: the fundamental group, covering spaces, combinatorial group theory, mod 2 homology groups, the Euler characteristic, the Lefschetz fixed point theorem. Offered in alternate years.

Prerequisite: Mathematics 31/71 and 54, or 31/71, 35/63 and permission of the instructor. *Dist: QDS.* Webb.

06S: 12

Offered in alternate years.

Prerequisite: Mathematics 31 or 71. *Dist: QDS.* Pomerance.

06W: 1007W: Arrange

This course is the second term of the basic algebra sequence begun in Mathematics 71. While the content of this course varies somewhat from year to year, the topics treated will usually be chosen from among permutation groups, Sylow theory, factorization theory in commutative rings, Galois theory, modules, Wedderburn-Artin theory of semi-simple rings, Noetherian rings, integral extensions, and Dedekind domains.

Prerequisite: Mathematics 71. In general, Mathematics 31 is not an acceptable prerequisite; however, in consultation with the instructor, Mathematics 31 together with some outside reading should be adequate preparation for the course. *Dist: QDS.*

Shemanske.

06S: 11

From its beginnings in the eighteenth century, Fourier analysis has branched in many directions that are central to applied mathematics. The core of the course consists of the main ideas of one-dimensional Fourier analysis of both periodic and non-periodic phenomena, coupled with an introduction to Lebesgue integration sufficient for understanding the contemporary foundations of the subject. Additional topics are drawn from such areas as signal processing, probability limit laws, and number theory. Offered in alternate years.

Prerequisite: Mathematics 63. In general, Mathematics 35 is not an acceptable prerequisite, however, in consultation with the instructor, Mathematics 35 together with some outside reading should be adequate preparation for the course. *Dist: QDS.* Trout.

All terms: Arrange

Advanced undergraduates occasionally arrange with a faculty member a reading course in a subject not occurring in the regularly scheduled curriculum.

*Not offered in the period from 05F through 07S*

From time to time a section of Mathematics 88 may be offered in order to provide an advanced course in a topic which would not otherwise appear in the curriculum. Consult the advisor to majors for details about topics to be covered. *Dist: QDS.*

06W: 2

A study of selected topics in logic, such as model theory, set theory, recursive function theory, or undecidability and incompleteness. Offered in alternate years.

Prerequisite: Mathematics 39 or 69. *Dist: QDS.* Groszek.

All terms except summer: Arrange

Open only to students who are officially registered in the Honors Program. Permission of the adviser to majors and thesis adviser required. This course does not serve for major credit nor for distributive credit, and may be taken at most twice.

*Not offered in the period from 05F through 07S*

This course satisfies the college’s requirement for a culminating experience. The topic of the seminar will be chosen by the instructor. After an introduction by the instructor, students will prepare and present short talks on various aspects of the topic in order to develop and refine their ability to present mathematics orally. Each student will then make a formal oral presentation and prepare a written report on a topic chosen by the student and instructor. Students will prepare drafts of their report for feedback from the seminar participants and revise their work in light of this feedback. Students also doing an Honors Project may submit their project in lieu of the final written report.

A qualified honors major may apply to the course instructor for permission to elect a graduate course. This listing covers 100-level offerings for 2005 fall through 2006 spring only. Courses marked with an asterisk (*) are not offered in this period.

05F: Arrange. Arkowitz.

05F: Arrange. Chernov.

06W: Arrange. Daileda.

Tutoring or assisting with teaching under the supervision of a faculty member.

06S: Arrange. Webb.

06S: Arrange. Sutton.

06W: Arrange. Ionescu.

06W: Arrange. Barnett.

05F: Arrange. Winkler.

06S: Arrange. Weber.

05F: Arrange. Chernov.

Advanced graduate students may elect a program of supervised reading continuing the topics of their course work.

Advanced graduate students may, with the approval of the advisor to graduate students, engage in an independent reading program.

A seminar to help prepare graduate students for teaching. (This course does not count toward the general College requirements for the master’s degree.)

A graduate student may, with the approval of the advisor to graduate students, engage in an independent study project. Groups of graduate students may, for example, prepare joint work including reading and informal seminars aimed at mastering a certain topic.

Teaching under the supervision of a faculty member.

Research under the guidance of a staff member.

Research under the guidance of the student’s thesis advisor. Open to candidates for the Ph.D. degree.

Advanced graduate students may, with the approval of the advisor to graduate students, engage in an independent research project.

(Footnotes)

* The syllabus for certification in each area is available from the Mathematics Department.