Applied Mathematics

Director of undergraduate studies: John Wettlaufer, Rm. 109 KGL, 432-0892

Mathematical models are widely used throughout natural science, social science, and engineering in fields as diverse as physics, bioinformatics, robotics, image processing, and economics. Despite the broad range of mathematical settings and applications, there exists a core of essential concepts and techniques used in addressing most problems. The Applied Mathematics major provides a foundation in these mathematical techniques and prepares the student to use them 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 Data Science, and Engineering and Applied Science. Courses applying mathematics may be drawn from participating programs in Applied Physics; Astronomy; the biological sciences, including Ecology and Evolutionary Biology, Molecular Biophysics and Biochemistry, and Molecular, Cellular, and Developmental Biology; Chemistry; Economics; the various programs in engineering, including Biomedical Engineering, Chemical Engineering, Electrical Engineering, Environmental Engineering, and Mechanical Engineering; Earth and Planetary Sciences; Physics; and even Linguistics and Political Science. The Applied Mathematics degree program requires a three-course concentration in a field in which mathematics is used.

Students in the major are often sought after by graduate programs in either Applied Mathematics or in the disciplines in which they choose their concentration, as well as by industries and startup companies in which their breadth of quantitative skills are essential and often unique.

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.

Frequently Asked QuestionsStudents are encouraged to consult the Applied Mathematics FAQ for more detail about courses and policies in the major.

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 120 or ENAS 151, and MATH 222 or 225 or 226. It may also be satisfied by MATH 230, 231. Computer programming skills are also required and may be acquired by taking ENAS 130, CPSC 100, or 112. Details of individual programs must be worked out in consultation with the director of undergraduate studies (DUS), whose signed permission is required.

Requirements of the Major

The B.A. degree programThe program requires eleven term courses beyond the prerequisites, including the senior project, comprising a coherent program:

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

2. A course in probability (S&DS 241 or S&DS 238)

3. A course in data analysis (S&DS 361 or S&DS 230)

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

5. Courses in at least three of the following areas* including, but not limited to:

(a) optimization: AMTH 437, EENG 400

(b) probability and statistics: S&DS 242, 312, 351, 364, 400, 410, 411, 425, ECON 136, APHY 470

(c) partial differential equations and analysis: MATH 247, 250, 255, 256, 260, 300, 301, 302, 305, 310, AMTH 428

(d) algorithms and numerical methods: CPSC 365, 366, 424, 440, 465, 467, 468, 469, ENAS 440, 441

(e) graph theory: AMTH 562, ENAS 962

(f) mathematical economics: ECON 125, 126, 350, 351, 417, 433, 460, 471

(g) electrical engineering: EENG 397, 436, 455, AMTH 342, S&DS 364

(h) data mining and machine learning: S&DS 262, 365, 669, 671, CPSC 445, 453, 470, 474, 477, 745, AMTH 552

(i) biological modeling and computation: CPSC 453, 475, 476, BENG 352, 445, 458, ENAS 559

(j) physical sciences: ASTR 320, 420, CHEM 333**, EPS 322, 323, 421, 428, 456, PHYS 342, 343, 344, 401, 402, 410, 420, 430, 440, 442, 460, APHY 439, 448

(k) engineering: MENG 280, 285, 361, 365, 383, 463, 469, CENG 301, 315

(l) linguistics: LING 224, 227, 380

* Because departmental curricula from which the program draws regularly change, the DUS maintains a more exhaustive list of courses and areas satisfying this particular requirement. Additionally, due to rapid advances in many areas, these categories are often fluid, and their union can evolve. In order to accommodate this fluidity, students are strongly encouraged to revisit their program of study each term and share their checklist with the DUS. Students can independently and systematically plan multiple routes towards completion of the major by using the checklist and the master list of courses.

** Chemistry courses numbered 410 and above may count as a breadth requirement (either 1 full-term 1 credit course or 2 half-credit courses) with permission of the DUS.

6. At least three advanced courses in a field of concentration involving the application of mathematics to that field. Programs in science, engineering, computer science, statistics, and economics 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 those listed in point 5 above or for the B.S. below. Details of a student's program to satisfy the concentration requirement must be worked out in consultation with, and approved by, the DUS.

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

1. Topics in analysis (MATH 300) or introduction to analysis (MATH 301), Vector analysis (MATH 302), or Analysis 2 (MATH 305); the course selected may not be counted toward the requirements for the major under item 5 above. (MATH 350 and MATH 440 can in specific cases be considered in consultation with the DUS.)

2. An additional course selected from item 5 above.

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

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.

Credit/D/FailA maximum of one course credit taken Credit/D/Fail may be counted toward the requirements of the major.

Senior Requirement

Both the B.A. and B.S. degree programs require a senior thesis research project (AMTH 491).

REQUIREMENTS OF THE MAJOR

PrerequisitesMATH 120 or ENAS 151, and MATH 222 or 225 or 226, or equivalents; ENAS 130, CPSC 100, or

• $80 application fee or fee waiver
• Recommendations from two teachers and one counselor
• School Report with transcript
• Standardized test results (ACT or SAT) - not required for fall 2023 admission. (See below)
• English proficiency test results - required for non-native English-speakers (See below)
• Mid-Year Report & Final Report

See additional details about these required application components below.

Please note that if your application materials include any documents that are not in English, you must provide an official English translation in addition to the original documents.

$80 application fee or fee waiver

Applicants should pay the $80 application fee via the Common Application or Coalition Application website.

Applicants using the Common Application or Coalition Application may request that the application fee be waived. Learn more on the fee waiver page. The fee waiver scale is based on family size and income in US dollars but is applicable to all international students.

Recommendations from two teachers and one counselor

Request recommendations from two teachers who have taught you in core academic subjects (e.g. English, Foreign Language, Mathematics, Science, Social Studies) who know you well, and who have seen you at your best. It is preferable, but not required, that recommendations come from teachers who have taught you during your final or next-to-final year of secondary school.

Also request a recommendation from your school’s college counselor. If your counselor does not know you well, the recommendation may still provide helpful contextual information about your school and its academic programs. If your school does not have someone serving as a college counselor, please request a recommendation from a school administrator such as a house master, principal, or academic advisor. 

School Report with transcript

Your counselor or other school official should submit a School Report with an official transcript that includes all your secondary school courses. If a question on the School Report form is not applicable to your secondary school, leave it blank. 

Recommendations and transcripts should be submitted electronically through the Common Application or Coalition Application website. If necessary, teachers and counselors may also submit their recommendations and transcripts via mail or digital upload. Recommendations may be sent before or after you submit your application; materials that arrive before your application will be kept on file. The admissions office will make reasonable allowances for late school documents in situations where it is not possible for teachers and counselors to meet the application deadlines.

Standardized test results

In response to the extraordinary circumstances associated with the pandemic, Yale has temporarily suspended its requirement that first-year and transfer applicants submit results from the ACT or SAT. The change will be in effect during the 2022-23 admissions cycles for applicants who intend to matriculate in fall 2023. 

More information on Yale’s test policies is available on the standardized testing page. Standardized tests are just one component of a student’s application and are viewed within the context of the student’s entire file. There is no minimum score required for admission, nor is there a score that will guarantee admission.

English proficiency exams

Yale requires that non-native English-speakers who have not taken at least two years of secondary education where English is the medium of instruction submit the results from any of the proficiency tests listed below.

The TOEFL requires pre-registration for available testing dates. Yale’s most competitive applicants have scores of at least 100 on the internet-based TOEFL.

The IELTS offers proficiency tests in locations around the world. Pre-registration is required. Yale’s most competitive applicants have IELTS scores of 7 or higher.

Cambridge English exams are available at testing locations around the world. Pre-registration is required. Yale’s most competitive applicants have Cambridge English scores of 185 or higher on the C1 Advanced, C2 Proficiency, or B2 First exams.

Applicants may submit the Duolingo English Test (DET), which combines an English proficiency test with a brief video interview. Duolingo’s technology and format allows applicants to complete the test at any time or place with internet access. Yale’s most competitive applicants have DET scores of at least 120.

InitialView provides live, unscripted video interviews that candidates may submit to colleges for consideration with other application materials. Interview times must be reserved in advance. There is no scoring associated with these interviews.

Mid-Year Report & Final Report

Students who do not receive new secondary school grades by February 1, including students who have already completed secondary school, are not required to submit a Mid-Year Report. This includes many international students. Applicants who receive new grades by February 1 should submit a Mid-Year Report.

Only admitted students are required to submit the Final Report, which provides final secondary school grades and examination marks. Admitted students submit the Final Report in the summer before fall matriculation.

If your secondary school provides predicted results for external exams such as A-levels, the International Baccalaureate, and other international or national testing organizations, they should be submitted by your school alongside your transcript or Mid-Year Report.

All applicants for first-year admission must submit one of the following:

• The Coalition Application with Yale-Specific Questions
• The Common Application with Yale-Specific Questions
• The QuestBridge National College Match Application

Yale will accept any one of these applications, without preference for one over another. Students should submit one—and only one—application per admissions cycle.