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Mathematics

Chair: Daniel N. Rockmore

Vice Chair: Scott D. Pauls

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 Professors V. Chernov, R. C. Orellana, S. D. Pauls, J. D. Trout; Assistant Professors A. Barnett, S. Elizalde, C. J. Sutton, R. A. Weber; Research Instructors M. G. Mainkar, J. R. Mileti, V. Vatter; Visiting Professor F. Luca; Visiting Assistant Professor A. an Heuf; Lecturer A. Kremer; Research Assistant Professor M. Chu; Research Instructor R. Sadykov; Research Associate D. J. Graham; Adjunct Professor E. Demidenko; Adjunct Associate Professor B. F. Cole.

INTRODUCTORY COURSES

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

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.

MATHEMATICS MAJOR REQUIREMENTS

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 35, 43, or 63;

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

Caveats:

Also acceptable: Mathematics 16, 17

Computer Science 5, 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.

CHOOSING COURSES FOR THE MAJOR

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) Applied mathematics now encompasses a wide expanse of mathematical activity in the sciences, ranging across finance, sociology, psychology, biology, physics, computer science, and engineering. 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, 46, and 50.

We recommend choosing additional courses from among the following: Mathematics 26, 28, 36, 38, 42, 43, 46, 53, 56, 75, 76, 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) Certification as a public school Mathematics teacher is available through partnership with the Education Department. Contact the Education Department for details about course requirements.

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

CULMINATING EXPERIENCE

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, 76, 81, 83, 86, 89, 96, 98.

MINORS IN MATHEMATICS

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. Applied Mathematics for Physical and Engineering Sciences

Prerequisites: Mathematics 3, 8, 13 or 22, Computer Science 5

Courses: Mathematics 23; 46; 50 or 60; 43 or 53 or 76

III. Applied Mathematics for Biological and Social Sciences

Prerequisites: Mathematics 3, 8, 13, 22

Courses: Mathematics 20; 23; 27, 28 or 36; 50 or 53 or 76

THE HONORS PROGRAM IN MATHEMATICS

A student who satisfies the 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 aver age 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 student will give a short presentation of their results. The advisor can then 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.

MODIFIED MAJORS

Modified Major with Mathematics as the primary Department

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 non-prerequisite 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.

THE GRADUATE PROGRAM IN MATHEMATICS

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.

REQUIREMENTS FOR THE MASTER’S DEGREE (A.M.)

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, given on page XXX, 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.

REQUIREMENTS FOR THE DOCTOR’S DEGREE (PH.D.)

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 the opportunity to teach twice in the three years 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.

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