Yale University Graduate Program in Applied Mathematics

General Information

Fields of Study

The Graduate Program in Applied Mathematics comprises mathematics and its applications to a range of areas, to the mathematical sciences (including computer science and statistics), and to the other sciences and engineering. Topics covered by the program include classical and modern applied harmonic analysis, linear and nonlinear partial differential equations, numerical analysis, scientific computing and applications, discrete algorithms, combinatorics and combinatorial optimization, graph algorithms, geometric algorithms, discrete mathematics and applications, statistical theory and applications, probability theory and applications, information theory, econometrics, financial mathematics, statistical computing, and applications of mathematical and computational techniques to fluid mechanics, combustion, and other scientific and engineering problems.

Requirements for the Ph.D. in Applied Mathematics

All students are required to: (1) complete 12 term courses (including reading courses) at the graduate level, at least two with Honors grades; (2) pass a qualifying examination on their general applied mathematical knowledge (in algebra, analysis, and probability and statistics) by the end of their second year; (3) submit a dissertation prospectus; (4) participate in the instruction of undergraduates; (5) be in residence for at least three years; and (6) complete a dissertation that clearly advances understanding of the subject it considers. The normal time for completion of the Ph.D. program is four years.

Requirement (1) normally includes four core courses in each of the methods of applied analysis, numerical computation, algorithms, and probability; these should be taken during the first year. The qualifying examination is normally taken by the end of the third term and will test knowledge of the core courses as well as more specialized topics. The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)-(5) and obtaining an adviser.

Master's Degrees

M. Phil. The Master of Philosophy is awarded en route to the Ph.D. The minimum general requirements for this degree are that a student shall have completed all requirements for the Ph.D. except the prospectus and dissertation. Students will not generally have satisfied the requirements for the Master of Philosophy until after two years of study, except where graduate work done before admission to Yale has reduced the student's graduate course work at Yale. In no case will the degree be awarded for less than one year of residence in the Yale Graduate School.

M.S. The Master of Science in Applied Mathematics is offered only in the Master's Degree Program in Applied Mathematics, decribed as follows:

Master's Degree Program. Students may also be admitted to a terminal master's degree program. This program is normally completed in one year, but a part-time program may be spread over as many as four years. To qualify for the M.S., the student must pass eight graduate-level courses. Courses taken as part of the M.S. program must be preapproved by the director of graduate studies to ensure that a suitable distribution of topics is covered. The Master's Degree Program does not offer financial aid.

Honors Requirement for the Ph.D.

To meet the minimum Graduate School quality requirement for the Ph.D., students must achieve the grade of Honors in at least one full-year or two full-term graduate courses, taken after matriculation in the Graduate School and during the nine-month academic year. The Honors requirement must be met in courses other than those concerned exclusively with dissertation research and preparation.

A student who has not met the Honors requirement at the end of the fourth term of full-time study will not be permitted to register for the fifth term. In exceptional circumstances, the director of graduate studies may petition the degree committee, through the appropriate dean, that a student who has not met the Honors requirement be permitted to continue study. Such a petition should be made before the end of the fourth term of study in time to be considered by the degree committee at its meeting that term.

Financial Support for the Ph.D. student

Yale provides a stipend as well as full tuition and medical coverage for the first four years of the Ph.D. program in Applied Mathematics. The university offers the preceding support to each student for the first three terms of the four-year program. During the summers and beginning with the fourth term, the student is expected to have found an adviser with research funds to support the student's stipend, tuition and medical coverage.

Teaching Requirement

Each student is expected to have taught as a teaching fellow in two undergraduate courses by the end of the fourth term.

Required Examinations / Admissions

All applicants are required to submit official results of the Graduate Record Examination (GRE) General Test, which is administered in the United States and abroad by the Educational Testing Service (ETS). This examination should be taken no later than the November testing date. The GRE Subject Test is encouraged.

Additionally, the Test of English as a Foreign Language (TOEFL) is required of all international applicants whose native language is not English. This requirement is waived only for international applicants who have successfully attended, for at least two years, a university where English is the primary language of instruction. The Educational Testing Service also administers the TOEFL.

Applicants whose native language is not English must present evidence of proficiency in English by satisfactorily completing the Test of English as a Foreign Language (TOEFL), which is administered by ETS. This examination should be taken no later than the November testing date. The TOEFL requirement is waived only for applicants who have successfully attended for at least two years a university where English is the language of instruction.

Students who do not demonstrate sufficient proficiency in English may be restested and/or asked to take courses in English for speakers of other languages. A high level of proficiency in English is required in order for students to serve as teaching fellows and fulfill the teaching requirement.

Official score reports must be submitted for all required examinations. Ask the Educational Testing Service to report your scores to the Yale Graduate School.

Program materials and additional information concerning degrees offered and admissions requirements are available upon request to the Office of Admissions, Graduate School of Arts and Sciences, Yale University, PO Box 208323, New Haven CT 06520-8323. Or contact them by email at <http://www.yale.edu/graduateschool/admissions>.

Undergraduate Applied Mathematics at Yale

Director of undergraduate studies: Daniel A. Spielman, AKW 201, 436-1264; senior coordinator: Andrew Barron, 24 Hillhouse Ave., 432-0634, andrew.barron@yale.edu

Mathematical models are widely used throughout science and engineering in fields as diverse as physics, bioinformatics, robotics, image processing, and economics. Despite the broad range of applications, there are a few essential techniques used in addressing most problems. The Applied Mathematics major provides a foundation in these mathematical techniques and trains the student to use these skills in a substantive field of application.

The interdisciplinary major permits a great deal of flexibility in design. It is intended to appeal to students who wish to study the more mathematical aspects of science or engineering as well as those whose primary interest is in mathematics and statistics and who wish to become acquainted with applications. Core courses are drawn from Computer Science, Mathematics, Statistics, and Engineering and Applied Science. Courses applying mathematics may be drawn from participating programs in Astronomy, the biological sciences (Biology, Molecular Biophysics and Biochemistry), Chemistry, Computer Science, Economics, the various programs in engineering (Biomedical, Chemical, Electrical, Environmental, and Mechanical Engineering, as well as Applied Physics), Geology and Geophysics, Mathematics, Operations Research, Physics, and Statistics. The Applied Mathematics degree program requires a three-course concentration in a field in which mathematics is used.

Students may pursue a major in Applied Mathematics as one of two majors and can thereby equip themselves with mathematical modeling skills while being fully engaged in a field of application. In this case, the concentration requirement of the Applied Mathematics program is flexible in order to recognize the contribution of the other major. A two-course overlap is permitted in satisfying the requirements of the two majors.

Prerequisite and introductory courses. Multivariable calculus and linear algebra are required and should be taken before or during the sophomore year. This requirement may be satisfied by MATH 120a or b and MATH 222a or b or 225b, or equivalents. Computer programming skills are also required and may be acquired by taking ENAS 130b or CPSC 112a or b.

Details of individual programs must be worked out in consultation with the director of undergraduate studies, whose signed permission is required.

Requirements of the major for the B.A. degree program. The program requires eleven term courses beyond the prerequisites, including the senior seminar, comprising a coherent program:

1. A course in differential equations (ENAS 194a or b or MATH 246a or b)

2. A course in probability (STAT 241a or 238a)

3. A course in data analysis (STAT 361a or 230b)

4. A course in discrete mathematics (AMTH 244a or CPSC 202a)

5. A selection of at least three of the following: (a) a course in optimization (AMTH 237a) or linear programming (AMTH 235a); (b) stochastic processes (STAT 251b); (c) statistics (STAT 242b); (d) a course in algorithms (CPSC 365b) or numerical computation (ENAS 440a or CPSC 440b); (e) graphs and networks (AMTH 462a); (f) game theory (ECON 156b); (g) a course in signals, dynamics, applications of differential or difference equations, or time series (AMTH 342a, ENAS 391a, 397b, EENG 310a, 436b, 450a, or ECON 163b) or in applied functional analysis (AMTH 260b); (h) a course in image or vision analysis (EENG 445a or CPSC 475b) or information theory (AMTH 364b)

6. At least three courses in a field of concentration involving the application of mathematics to that field, at least two of which are advanced courses. Programs in science, engineering, computer science, statistics, economics, and operations research are natural sources of concentration. Alternatively, when two majors are undertaken, if the second major is in a participating program, then, recognizing that there can be an overlap of two courses, the student may take for the remaining course an additional choice relevant to the Applied Mathematics major such as listed in point 5 above or for the B.S. degree below. Details of a student's program to satisfy the concentration requirement must be worked out in consultation with, and with the approval of, the director of undergraduate studies

7. The senior seminar (AMTH 490b)

The B.S. degree program. In addition to the courses indicated for the B.A. major, the B.S. degree, which totals fourteen term courses beyond the prerequisites, must also include:

1. Topics in analysis (MATH 300b)

2. An additional course selected from the list in item 5 above

3. Another course numbered 300 or higher from the list above, or a mathematics, applied mathematics, statistics, or quantitative computer science or engineering course numbered 300 or higher, subject to the approval of the director of undergraduate studies

Alternatively, students may petition to receive a B.S. in Applied Mathematics by fulfilling the B.A. requirements in Applied Mathematics and the B.S. requirements in another program.

REQUIREMENTS OF THE MAJOR

Prerequisites:  MATH 120a or b and MATH 222a or b or 225b, or equivalents; ENAS 130b or CPSC 112a or b

Number of courses:  B.A. degree program—eleven term courses beyond prerequisites (including the senior sem); B.S. degree program—fourteen term courses beyond prerequisites (including the senior sem)

Specific courses required:  B.A. degree—ENAS 194a or b or MATH 246a or b; STAT 238a or 241a; STAT 361a or 230b; AMTH 244a or CPSC 202a; B.S. degree—same, as well as MATH 300b

Distribution of courses:  B.A. degree—at least three courses in a field of concentration concerning the application of mathematics to that field, at least two of which are advanced; three addtl courses as specified; B.S. degree—same, as well as two addtl courses as specified

Senior requirement:  Senior sem (AMTH 490b)

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