Mathematics

Mathematics courses
Mathematics department home page

Director of undergraduate studies: Yair Minsky, 448 DL, 432-4018, yair.minsky@yale.edu

FACULTY OF THE DEPARTMENT OF MATHEMATICS

Professors

Donald Brown, Andrew Casson, Ronald Coifman, Igor Frenkel, Howard Garland, Roger Howe, Peter Jones, Ravindran Kannan, Mikhail Kapranov, Bruce Kleiner, Benoit Mandelbrot (Emeritus), Gregory Margulis, Yair Minsky, Vincent Moncrief, Steven Orszag, David Pollard, Vladimir Rokhlin, Peter Schultheiss (Emeritus), Katepalli Sreenivasan, Gregg Zuckerman

J. W. Gibbs Assistant Professors

Dennis Borisov, Tullia Dymarz, Philip Gressman, Michael Gurski, Jesse Johnson, Triet Le, Yiqiang Li, Matvei Libine, Karin Melnick, Hisham Sati, Dapeng Zhan

Adjunct Professors

Michael Frame, Alex Lubotzky

Operations Research Faculty

Eric Denardo

Statistics Faculty

Andrew Barron, Joseph Chang, John Hartigan

Both the course offerings and the major in Mathematics reflect the many roles of mathematics itself: the language and tool of the sciences, a cultural phenomenon with a rich historical tradition, and a model of abstract reasoning. The Mathematics major provides a broad education in various areas of mathematics in a program flexible enough to accommodate many ranges of interest.

B.A. and B.S. degree programs. The prerequisite for each program is calculus through the level of MATH 120a or b, or the equivalent. Each program normally consists of ten term courses in Mathematics numbered 222 and higher, including MATH 480a or b. Each student is expected to take vector calculus and linear algebra through the level of MATH 230 or of MATH 250a, and one of MATH 222a or b or 225a or b. Beyond this, it is strongly recommended that the major include courses in real analysis (MATH 300b or 301a), in algebra (MATH 350a), and in complex analysis (MATH 310a), together with other courses to provide additional breadth and depth. Students are required to take at least two term courses in each of three of the following five categories: analysis, algebra and number theory, statistics and applied mathematics, geometry and topology, and logic and foundations. Specific courses in each category are listed below.

Each Mathematics major is urged to acquire additional familiarity with the uses of mathematics by taking courses in Applied Mathematics, Computer Science, Engineering and Applied Science, Economics, Operations Research, Physics, Statistics, or other departments. In some instances a limited number of such courses may be counted among the ten courses required for the major in Mathematics, with the approval of the director of undergraduate studies.

A candidate for the B.S. degree must take at least two advanced term courses in the physical sciences, such as CHEM 328a, 332a, 333b, or PHYS 401a, 402b, in addition to the ten term courses required for the B.A. degree. Such courses require the approval of the director of undergraduate studies; written approval is advised.

Any student interested in pursuing further study in pure mathematics should include MATH 301a, 305b, 310a, 350a, 370b, and either 430b or 435b in his or her program, and should consider taking one or more graduate-level courses. Students interested in applications of mathematics should include MATH 300b or 301a, 310a, 350a, and a selection of courses among 241a, 242b, 244a, 246a or b, 251b, 260a, 435b, and CPSC 440b.

The intensive major. Candidates for a degree with an intensive major in Mathematics are expected to include at least two graduate term courses, or equivalent independent study, in their programs. Familiarity with the material of the following courses is prerequisite to graduate courses in each category: algebra: two courses between MATH 350 and 399; analysis: MATH 301a, 305b, 310a; algebraic topology: MATH 301a, 350a; logic and foundations: MATH 270a.

Senior requirement. During the senior year a student majoring in Mathematics must take the senior seminar (MATH 480a or b).

The following members of the department may be consulted by students through their residential college affiliation:

BK	P. Jones	MC	P. Jones

BR	to be announced	PC	to be announced

CC	S. Orszag	SY	to be announced

DC	R. Howe	SM	H. Garland

TD	G. Zuckerman	ES	to be announced

JE	R. Coifman	TC	G. Margulis

Combined B.S./M.S. Degree Program

Students who, by the end of their senior year, complete the requirements of the department for the M.S. in Mathematics will be eligible to receive this degree at their Senior Commencement. Required are: (1) eight term courses numbered 500 or higher, most of which must be completed with grades of B or better; (2) a reading knowledge of mathematical literature in a foreign language of importance for mathematical research (normally French, German, or Russian); (3) satisfactory performance on a general oral examination.

The master's program is in no sense a substitute for the B.A. or B.S. program; rather, it is designed to accommodate a very few exceptional students who, by means of accelerated or independent study, can satisfy the department as to their command of the content of the normal undergraduate program. During spring term of the sophomore year candidates must submit a proposal that foresees this level of achievement by the end of junior year to the director of undergraduate studies. If approved by the department this proposal will be forwarded to the Dean's Office; see "Simultaneous Award of the Bachelor's and Master's Degrees" in chapter III under the heading "Special Arrangements." Students' status and progress will be reviewed before they are permitted to continue in the program in the senior year.

Students take at least two graduate term courses in the junior year (normally courses in algebra or analysis are the first graduate courses taken). The general oral examination covers a list of topics available from the director of graduate studies and will be accepted in lieu of the usual senior oral presentation. Details concerning the requirements for the master's degree may be obtained from the director of graduate studies.

Placement in Courses

The department offers a three-term sequence in calculus, MATH 112a or b, 115a or b, and 120a or b. Students who have not taken calculus at Yale and who wish to enroll in calculus must take the online placement examination found on the department Web page. At the beginning of each term a calculus preregistration session is held in DL 432. To enroll in a calculus course a student must bring the results of the placement exam, as well as other pertinent information such as Advanced Placement test scores, to the preregistration session. Advisers will be on hand to assist each student in enrolling in the appropriate course.

MATH 112a or b is an introductory course that presupposes basic skills in high school algebra, geometry, and trigonometry. Enrolling students are expected to know the basic definitions of the trigonometric functions, synthetic division, factorization, and elementary area and volume formulas of plane and solid geometry. MATH 115a or b presupposes familiarity with the topics covered in MATH 112a or b. MATH 120a or b presupposes familiarity with the topics covered in MATH 115a or b.

MATH 230 is an advanced course in linear algebra and introductory analysis for students with exceptionally strong backgrounds in mathematics. Students who wish to enroll in MATH 230 should consult the instructor for the course. After MATH 115a or b, students with a strong interest in abstract mathematics should consider taking MATH 230.

REQUIREMENTS OF THE MAJOR

Prerequisite: MATH 120a or b or equivalent

Number of courses: B.A.—10 term courses numbered 222 or higher, incl MATH 480a or b; B.S.—same, with 2 addtl courses in physical sciences

Distribution of courses: B.A.—2 courses in each of 3 categories chosen from (a) analysis, (b) algebra and number theory, (c) stat and applied math, (d) geometry and topology, (e) logic and foundations; B.S.—same, with 2 addtl advanced courses in physical sciences approved by DUS

Specific courses required: MATH 230 (counts as 2 courses) or MATH 250a; MATH 222a or b or 225a or b

Substitution permitted: With DUS permission, certain courses in Applied Math, Comp Sci, Engineering & Applied Science, Econ, Operations Research, Physics, Stat

Senior requirement: Senior sem (MATH 480a or b)

Intensive major: 2 grad courses or equivalent independent study counted among the required courses

Unless otherwise indicated, courses in Mathematics do not count toward the natural science requirement for the Class of 2008.

Introductory Courses: MATH 112a or b, 115a or b, 118a or b, 120a or b, 125b, 190a

Analysis: MATH 230 (counts as one term course in this category), 246a or b, 250a, 300b, 301a, 305b, 310a, 315b, 320a, 325b

Statistics and Applied Mathematics: MATH 235b, 241a, 242b, 244a, 246a or b, 251b, 260a, 310a, 330b, 400a; CPSC 201a or b, 365b, 440b

Algebra and Number Theory: MATH 222a or b, 225a or b, 230 (counts as one term course in this category), 244a, 350a, 354b, 360a, 370b, 380a, 381b

Geometry and Topology: MATH 228b, 290b, 430b, 435b

Logic and Foundations: MATH 270a; PHIL 267a, 268b