Mathematics Website

MATHEMATICS (Div. III)

Chair, Professor COLIN C. ADAMS

Professors: ADAMS, BERGSTRAND*, O. BEAVER, V. HILL, LENHART, MORGAN, SILVA. Associate Professors: BURGER, R. DE VEAUX, GARRITY, S. JOHNSON. Assistant Professors: CHKHENKELI**, LOEPP*, REITER.

MAJOR

The major in Mathematics is designed to meet two goals: to introduce some of the central ideas in a variety of areas of mathematics, and to develop problem-solving ability by teaching students to combine creative thinking with rigorous reasoning.

Students are urged to consult with the department faculty on choosing courses appropriate to an individualized program of study.

REQUIREMENTS (nine courses plus colloquium)

Calculus (two courses)

104 Calculus II
105 Multivariable Calculus

Applied/Discrete Mathematics (one course)

210 Differential Equations and Vector Calculus or
243 Statistics and Data Analysis or
251 Discrete Mathematics or
a more advanced elective in discrete or applied mathematics, with prior departmental approval: Mathematics 305, 306, 315, 346, 348, 354, 361, 433, 452, or an appropriate course from another department as listed in the notes below.
Note: Mathematics 251 is required in Computer Science, and Mathematics 210 is recommended in other sciences, but double majors should understand that no course may count toward both majors; Mathematics 243 is recommended for students interested in statistics or actuarial science. Physics 210 can be used to satisfy this requirement.

Core Courses (three courses)

211 Linear Algebra
301 Real Analysis or 305 Applied Real Analysis

312 Abstract Algebra or 315 Groups and Characters

Completion (three courses plus colloquium)

Senior Major Course (any 400-level course taken in the senior year or with permission of the department in the junior year)

Two electives from courses numbered 300 or above.

Weekly participation as a senior in the Mathematics Colloquium, in which all senior majors present talks on mathematical topics of their choice.

NOTES

In some cases, an appropriate course from another department may be substituted for one of the electives, with prior permission of the Mathematics Department. In any case, at least eight courses must be taken in Mathematics at Williams. Students with transfer credit should contact the department about special arrangements.

APPLIED MATHEMATICS OR OTHER SCIENCES

Students interested in applied mathematics or other sciences, including economics, should consider Mathematics 210, 243, 244, 251, 305, 306, 315, 342, 346, 348, 354, 361, 433, 442, 448, 452, and additional appropriate courses from outside Mathematics, including possibilities such as Chemistry 301, Computer Science 256, Computer Science 361, Economics 255, Physics 201, Physics 202 or more advanced physics courses.

BUSINESS AND FINANCE

Students interested in careers in business or finance should consider Mathematics 170 or 370, as well as courses in statistics and related areas such as Mathematics 143, 243, 244, 342, 346, 348, 442, 443 and 448. Since these courses address different needs, students should consult with the instructors to determine which seem to be most appropriate for individuals.

ENGINEERING

Students interested in engineering should consider the courses for applied mathematics immediately above, with Mathematics 210 and 305 especially recommended. Williams has exchange and joint programs with good engineering schools. Interested students should consult the section on engineering near the beginning of the Bulletin and the Williams pre-engineering advisor for further information.

GRADUATE SCHOOL IN MATHEMATICS

Students interested in continuing their study of mathematics in graduate school should take Mathematics 301 and 312. Mathematics 302 and 324 are strongly recommended. Many of the 400-level courses would be useful, particularly ones that involve algebra and analysis. Honors theses are encouraged. Reading knowledge of a foreign language (French, German, or Russian) is helpful.

STATISTICS AND ACTUARIAL SCIENCE

Students interested in statistics or actuarial science should consider Mathematics 243, 244, 342, 346, 348, 442, 443, 448 and Economics 255. Additionally, students should consider taking some number of the actuarial exams given by the Society of Actuaries, which can constitute part of an honors program in actuarial studies (see section on honors below).

TEACHING

Students interested in teaching mathematics at the elementary or secondary school level should consider Mathematics 243, 285, 313, 325, 381, and practice as a tutor or teaching assistant. Winter study courses that provide a teaching practicum are also highly recommended.

THE DEGREE WITH HONORS IN MATHEMATICS

The degree with honors in Mathematics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core material and skills, breadth and, particularly, depth of knowledge beyond the core material, ability to pursue independent study of mathematics, originality in methods of investigation, and, where appropriate, creativity in research.

An honors program normally consists of two semesters and a WSP (031) of independent research, culminating in a typed thesis and a presentation. Under certain circumstances, the honors work can consist of coordinated study involving a regular course and one semester plus a WSP (030) of independent study, culminating in a typed "minithesis" and a presentation. At least one semester should be in addition to the major requirements.

An honors program in actuarial studies requires significant achievement on four appropriate examinations of the Society of Actuaries and giving a second colloquium talk. Written work is a possible component.

Highest honors will be reserved for the rare student who has displayed exceptional ability, achievement or originality. Such a student usually will have written a thesis, or pursued actuarial honors and written a minithesis. An outstanding student who writes a minithesis, or pursues actuarial honors and writes a paper, might also be considered.

Prospective honors students are urged to consult with their departmental advisor at the time of registration in the spring of the sophomore or at the beginning of the junior year to arrange a program of study that could lead to the degree with honors. By the time of registration during spring of the junior year, the student must have requested a faculty member to be honors advisor and must have obtained the department's approval of formal admission to the honors program. Such approval depends on both the record and the promise of the applicant. It is conditional on continuing progress.

The recommendation for honors is usually announced at the end of the spring term. Participation in the honors program does not guarantee a recommendation for honors. The decision is based not only on successful completion of the honors program but also on the merit of the student's overall record in mathematics. If the student completes the program during the fall or winter study, the decision may be announced at the beginning of the spring term, conditional on continuing merit.

ADVANCED PLACEMENT

The Mathematics Department attempts to place each student who elects a mathematics course in that course best suited to the student's preparation and goals. The suggested placement in an appropriate mathematics course is determined by the results of the Advanced Placement Examination (AB or BC) if the student took one, and any additional available information. A student who receives a 3, 4, or 5 on the BC examination or a 4 or 5 on the AB examination is ordinarily placed in Mathematics 105. A student who receives a 1 or 2 on the BC examination or a 2 or 3 on the AB examination is ordinarily placed in Mathematics 104. Students who have had calculus in high school, whether or not they took the Advanced Placement Examination, are urged to register for either Mathematics 104 or Mathematics 105. In any event, students registering for mathematics courses are urged to consult with members of the department concerning appropriate courses and placement. In general, students are encouraged to enroll in the most advanced course for which they are qualified: it is much easier to drop back than to jump forward. The department reserves the right to refuse registration in any course for which the student is determined to be over-prepared.

GENERAL REMARKS

Divisional Requirements

All courses may be used to satisfy this requirement.

Alternate Year Courses

Core courses 301, 305, 312, and 315 are normally offered every year. Other 300-level courses may be offered in alternate years. Senior seminars (400-level courses) are normally offered every two to four years. Students should check with the Mathematics Department before planning far into the future.

Course Admission

Courses are normally open to all students meeting the prerequisites. Students with questions about the level at which courses are conducted are invited to consult members of the department.

Course Descriptions

Descriptions of the courses in Computer Science are under that heading. More detailed information on all of the offerings in the department is available in the Informal Guide to Mathematics Courses at Williams that can be obtained at the departmental office.

Courses Open on a Pass/Fail Basis

Students taking a mathematics course on a pass/fail basis must meet all the requirements set for students taking the course on a graded basis.

With the permission of the department, any course offered by the department may be taken on a pass/fail basis. Permission will not be given to mathematics majors to meet any of the requirements of the major or honors degree on this basis. However, with the permission of the department, courses taken in the department beyond those requirements may be taken on a pass/fail basis.

Graduate School Requirements

An increasing number of graduate and professional schools require mathematics as a prerequisite to admission or to attaining their degree. Students interested in graduate or professional training in business, medicine, economics, or psychology are advised to find out the requirements in those fields early in their college careers.