Professor Elie Tamer, Co-Director of Undergraduate Studies
Professor Steven Gortler, Co-Director of Undergraduate Studies
Dr. Margo Levine, Associate Director of Undergraduate Studies
Dr. Sarah Iams, Assistant Director of Undergraduate Studies
Mathematical modeling is ubiquitous throughout the physical, biological, social, engineering, and management sciences. Mathematical scientists who identify themselves primarily as applied mathematicians develop, implement, and study mathematical, statistical, and computational techniques broadly applicable in various fields. In addition, they bring mathematical modeling skills to bear on particular scientific problems, using judicious approximations to obtain insights and predictions when the underlying phenomena are thought to be relatively simple and well understood, or creating conceptual frameworks for quantitative reasoning and measurement when the underlying phenomena are complicated and less well understood. In their methodological role, they may function temporarily as mathematicians, statisticians, or computer scientists; in their phenomenological role, they may function temporarily as physicists, chemists, biologists, economists, engineers, and the like. In both roles, they must possess relevant knowledge, technical mastery, and educated taste; clearly this necessitates specialization. Avowed practitioners of mathematically-oriented segments of other disciplines equally may function temporarily as applied mathematicians.
The range of activities carried on under the aegis of the principal professional organization in the field, the Society for Industrial and Applied Mathematics (SIAM), can serve as an operational definition of the scope of the discipline. Various SIAM publications are readily accessible to Harvard students and student memberships are available. Ideally, over time, an applied mathematician demonstrates substantive involvement with both the mathematical and scientific aspects of their dual roles. In the long run, their contributions must be evaluated based on both methodological and phenomenological impact. Inside academia, their activities are usually carried out in collaboration with students or colleagues; outside academia, they often serve as part of a multidisciplinary team tackling complex problems under time and resource constraints. In either context, a premium is placed on having an outstanding ability to communicate with fellow technical professionals. Applied mathematics is inherently interdisciplinary, in motivation and in operation. This vision informs the design of the concentration.
The Applied Mathematics concentration consists of a broad undergraduate education in the mathematical sciences, especially in those subjects that have proved vital to an understanding of problems arising in other disciplines, and in some specific area where mathematical methods have been substantively applied. For concentrators, a core learning objective is building and demonstrating foundational knowledge in computation, probability, discrete, and continuous mathematics through the successful completion of the foundation and breadth courses. In addition, through their coursework, concentrators should gain facility and comfort in using approximation to simplify problems and gain insight. They should learn to communicate effectively with fellow technical professionals, and should be prepared, by their senior year, to tackle mathematical modeling problems in their area of application, at the level of a senior thesis. Additionally, students can expect to be able to attain employment or, with appropriate planning, gain admission to graduate study in applied mathematics.
The concentration requirements are flexible, but structured and demanding. Individual programs should be arranged in consultation with an advisor, and are approved by the advisor and by the Co-Director, Associate Director, or Assistant Director of Undergraduate Studies. The concentration is overseen by an interdepartmental Committee on Undergraduate Studies in Applied Mathematics, and administered by the School of Engineering and Applied Sciences (SEAS).
Students select the concentration because they like mathematics, and especially the use of mathematics to solve real-world problems. Some want a deeper involvement with an area of application than may be provided within a mathematics, statistics, or computer science concentration. Others want a more mathematically-oriented approach to an area of application than that normally provided within the corresponding concentration; mathematical economics is a prime example. Yet others want a special program not otherwise available, usually involving an area of application in which mathematical modeling is less common. Applied mathematics programs will typically involve a broader range of study within the mathematical sciences and a narrower range of study within the area of application than alternate programs offered by neighboring concentrations. With a little forethought, it is ordinarily straightforward to change the chosen area of application or to transfer between this concentration and neighboring ones until the end of the sophomore year, and often beyond.
Some concentrators go on to graduate work or to employment in their area of application, or in applied mathematics. Others go on to professional schools in law, medicine, or business. Students interested in entering a PhD program should plan to take more technical electives than the minimum required for concentration, and should plan their program carefully with the Co-Director, Associate Director, or Assistant Director of Undergraduate Studies.
14-15 courses (56-60 credits)
Prospective concentrators are encouraged to make early contact with concentration representatives. Students wishing to enter the concentration should review the concentration requirements, meet with the Assistant, Associate, or Co-Director of Undergraduate Studies to discuss their proposed program, and then submit a program of study at studyplan.seas.harvard.edu/. Students should be aware that interdisciplinary and interdepartmental programs will usually be more demanding than conventional programs in an established discipline. Prerequisite or corequisite courses not included in the program of study may be needed to provide background or perspective.
In addition to the courses listed specifically below, more advanced courses may be approved by petition in the context of a particular program of study. A petition must propound in writing a coherent and persuasive argument for the intellectual merit of the proposal in question. In certain areas of application, undergraduates routinely take courses designated as primarily for graduate students. Recommendations or restrictions on course selection may flow from the choice of a particular area of application.
Total course requirements may be reduced from fifteen to no less than twelve, and the balance of foundation and breadth courses are dependent on placement in Math courses as listed below in item 1a. Such placement is granted based on an appropriate Advanced Placement examination, the Harvard Mathematics Placement Test, or an equivalent college-level course taken elsewhere, provided this bypass is validated by successful completion (honor grades) of more advanced courses. Students seeking placement based on college-level work done elsewhere must submit a petition to the Co-Director, Associate Director, or Assistant Director of Undergraduate Studies, supplemented by suitable supporting materials. Transfer students from other colleges will have their programs considered on a case-by-case basis in response to a petition documenting their previous preparation.
Foundation: Two to five courses (see note 1.d.i) in calculus and linear algebra.
- Mathematics Ma and Mb or Mathematics 1a
- Mathematics 1b
- Applied Mathematics 21a, Mathematics 21a, 23b, 25b, or 55b
- Applied Mathematics 21b, Mathematics 21b, 23a, 25a, or 55a
Breadth: Seven to five courses (see item 1.d.i, below) from the following categories. Students must take courses from at least five out of the eight categories listed below. Of those, students must take at least one course in Computation and one course in Probability and Statistics. In addition, students must take a course drawn from at least one “continuous” category (Differential Equations or Analysis) and one drawn from at least one “discrete” category (Algebra, Optimization, or Discrete Mathematics). Students must show evidence of satisfying prerequisites for a course to count towards the concentration.
- Computation: First course: Applied Mathematics 111 and/or Computer Science 50. Additional courses: Applied Mathematics 205, 207; Computer Science 51, 61, 181, 182, 205; Statistics 121
- Probability and Statistics: First course: either Statistics 110 or Mathematics 154, but not both. Additional courses: Statistics 111, 121, 139; Mathematics 117; Applied Mathematics 126
- Differential Equations: Applied Mathematics 105, 108, 202; Mathematics 110
- Analysis: Applied Mathematics 104, 201, 202; Mathematics 112, 113, 114, 115, 118r
- Linear Algebra: Applied Mathematics 120, Mathematics 121
- Abstract Algebra: Applied Mathematics 106/206; Mathematics 122, 123, 124
- Optimization: Applied Mathematics 121; Mathematics 116
- Discrete Mathematics: Applied Mathematics 107; Mathematics 152, 155r; Computer Science 121, 124, 125
- Modeling and Approved Electives: Applied Mathematics 50, 91r, 115; Economics 985; or an approved advanced technical elective from outside of the student’s application area
- Application: Five courses from an area of application in which mathematics has been substantively applied, selected to provide a coherent and cumulative introduction to mathematically-oriented aspects of the field.
The number of required courses depends on the starting Math course (see Requirements above).
Students starting in Math Ma or 1a: 15 courses
- Math Ma (5 Foundation, 5 Breadth, 5 Application)
- Math 1a (4 Foundation, 6 Breadth, 5 Application)
Students starting in Math 1b or higher: 14 courses
- Math 1b (3 Foundation, 6 Breadth, 5 Application)
- Math 21a or higher (2 Foundation, 7 Breadth, 5 Application)
- Note: Students starting in 21a may take Mathematics 101 in their freshman or sophomore year as a third Foundation course; these students are then required to take only six courses in the Breadth category. Students may count Applied Mathematics 50 only if it is taken before AM115.
- Students starting in Math Ma or 1a: 15 courses
- Honors: Recommendations for honors are based on the grade point average of the final plan of study, the rigor of the overall record, and the satisfaction of the Honors requirement. The Honors requirement is automatically satisfied with a B- or higher grade in Applied Mathematics 115 and satisfactory grades in the 115 prerequisites. The second option is a modeling project, undertaken in AM 91r, in which a mathematical analysis of a problem is undertaken. Papers describing the project must be turned in to the concentration for evaluation by the end of the semester in which the AM 91r is completed.
- Recommendations for High or Highest Honors depend on the grade average in the courses included in the final plan of study, the rigor of the overall record, and the completion and evaluation of a senior thesis.
- The number of required courses depends on the starting Math course (see Requirements above).
- Foundation: Two to five courses (see note 1.d.i) in calculus and linear algebra.
- Thesis: Optional (see item 1.d.iii).
- General Examination: None.
- Pass/Fail: All courses counted for concentration credit must be letter-graded.
- Program of Study: Students entering the concentration must file an Applied Mathematics program of study. The program must be reviewed with the student’s adviser and updated as necessary each term thereafter before the study card will be signed. Programs of study are initially approved by the adviser, and are subsequently approved by the Co-Director, Associate Director, or Assistant Director of Undergraduate Studies.
- Joint Concentration: Applied Mathematics may not be combined with any other field of concentration because of its intrinsically interdisciplinary nature; study of an area of application is already an essential part of the program.
The Directors, Professor Elie Tamer, email@example.com, (617) 496-1526; Professor Steven Gortler, firstname.lastname@example.org, (617) 496-3751; Dr. Margo Levine, email@example.com, (617) 496-8129; and Dr. Sarah Iams, firstname.lastname@example.org, (617) 495-5935—serve as interim advisers to all students entering the concentration. Subsequently, an adviser is assigned. Special arrangements are made for students whose area of application is mathematical economics, in cooperation with the Economics Department. If an adviser becomes unavailable, the student is reassigned to a new adviser. Students may seek further advice from the Co-Directors, Associate Director, or Assistant Director of Undergraduate Studies at any time.
For up-to-date information on advising in Applied Mathematics, please see the Advising Programs Office website.
Number of Concentrators as of December
*Applied Mathematics does not participate in joint concentrations.