MATHEMATICS AND STATISTICS (Div. III)

Chair, Professor CESAR SILVA

Professors: ADAMS*, O. BEAVER, BURGER, R. DE VEAUX, GARRITY*, S. JOHNSON***, LOEPP, MORGAN, SILVA. Associate Professor: DEVADOSS. Assistant Professors: BOTTS*, KLINGENBERG, MILLER, PACELLI, STOICIU*. Visiting Assistant Professor: LOOK. Visiting Lecturer: PEDERSEN.

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 statistics, 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)

Mathematics 104 Calculus II or an equivalent high school course
Mathematics 105 or 106 Multivariable Calculus
Except in unusual circumstances, students planning to major in mathematics should complete the calculus sequence (Mathematics 103, 104, 105/106) before the end of the sophomore year, at the latest.

Applied/Discrete Mathematics/Statistics (one course)

Mathematics 209 Differential Equations and Vector Calculus or
Mathematics 210 Mathematical Methods for Scientists (Same as Physics 210) or
Mathematics 251 Discrete Mathematics or
Statistics 201 Statistics and Data Analysis or
Statistics 231 Statistical Design of Experiments or
a more advanced elective in discrete or applied mathematics or statistics, with prior departmental approval: Mathematics 305, 306, 315, 361, 375, 433, 452, or any Statistics course 300 or above or an appropriate course from another department as listed in the notes below.
Notes: Mathematics 251 is required in Computer Science, and Mathematics 209 is recommended in other sciences, but double majors should understand that no course may count toward both majors.

Core Courses (three courses)

Mathematics 211 Linear Algebra
Mathematics 301 Real Analysis or Mathematics 305 Applied Real Analysis

Mathematics 312 Abstract Algebra or Mathematics 315 Groups and Characters or Mathematics 317 Applied Abstract Algebra

Completion (three courses plus colloquium)

The Senior Major Course is any 400-level course taken in the senior year. In exceptional circumstances, with the prior permission of the department, a student may be allowed to satisfy the Senior Major Course requirement in the junior year, provided that the student has completed three 300-level mathematics courses before enrolling in the Senior Major Course (if it is a statistics seminar, one of the 300-level courses may be replaced by Statistics 231).

Two electives from courses numbered 300 and above or Statistics 231.

Weekly participation as a senior in the Mathematics Colloquium, in which all senior majors present talks on mathematical or statistical 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 and Statistics Department. In any case, at least eight courses must be taken in mathematics and statistics at Williams. These can, with prior permission, include courses taken away. Students with transfer credit should contact the department about special arrangements.

APPLIED MATHEMATICS OR OTHER SCIENCES

Students interested in applied mathematics or other sciences should consider Mathematics 209, 210, 251, 305, 306, 315, 323, 342, 354, 361, 433, or Statistics 201, 231, 346, 442, and additional appropriate courses from outside Mathematics and Statistics, including possibilities such as Chemistry 301, Computer Science 256, Computer Science 361, Economics 255, Physics 201, Physics 202, Physics 210 or more advanced physics courses. Students interested in economics should consult the Economics Department.

BUSINESS AND FINANCE

Students interested in careers in business or finance should consider Mathematics 373, as well as courses in statistics and related areas such as Statistics 101, 201, 231, 331, 344, 346, 442, 443. 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 209 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 341, Statistics 201, 331, 346, 442 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 285, 313, 325, 381, Statistics 201 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/STATISTICS

The degree with honors in Mathematics/Statistics 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 or statistics, 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 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 "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 and Statistics 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 calculus 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 is ordinarily placed in Mathematics 106. A student who receives 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 barred from Mathematics 103 unless they obtain permission from the instructor. A student who receives a 4 or 5 on the Statistics AP examination should consult the department for placement. In any event, students registering for mathematics and statistics 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 mathematics and statistics courses may be used to satisfy the Division III requirement.

Alternate Year Courses

Core courses Mathematics 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 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 Statistics follow the descriptions of Mathematics courses.

Courses Open on a Pass/Fail Basis

Students taking a mathematics or statistics 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.

Study Abroad

Programs like the "Budapest Semester in Mathematics" are recommended for majors who wish to focus on mathematics away. The department, though, normally accommodates students who select other study away programs. Majors typically take their core courses (analysis and algebra) at Williams, and sometimes select courses away which (with prior approval) count as 300-level mathematics or statistics electives. The department offers its core courses in both the fall and the spring to allow students to spend more easily a semester away.

Graduate School Requirements

An increasing number of graduate and professional schools require mathematics and statistics 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.

MATHEMATICS COURSES

STATISTICS COURSE LISTINGS FOLLOW THE MATHEMATICS COURSE LISTINGS.

NOTE: Course Numbers-The first digit of the three-digit label for each course roughly denotes the difficulty level of the course. The middle digit denotes the field (0=calculus/analysis, 1=algebra/number theory, 2=geometry/topology, 3=applied, 4=probability, 5=discrete, 6=computation, 7,8=miscellaneous). The last digit occasionally indicates the level within a field, but most often simply distinguishes amongst the courses in the field. In particular, for example, Math 211 is not "easier" than Math 251; they are simply different mathematical fields (algebra vs. discrete).

There are three types of 300-level courses. There are the core courses: Math 301, 305, 312, 315, 317. There are the "precore" courses, which do not have the core courses as prerequisites. These include Math 313, 316, 327 and others. Finally there are those courses that have a core course as a prerequisite, such as Math 302.