Applied Mathematics

Professor Michael P. Brenner, Director of Undergraduate Studies

We can characterize what applied mathematicians should learn by examining what they do. Mathematical modeling is ubiquitous throughout the physical, biological, social, engineering and management sciences. Mathematical scientists who identify themselves primarily as applied mathematicians function in complementary dual roles in varying proportions. First, they develop, implement and study mathematical, statistical and computational techniques broadly applicable in various fields. Second, they bring mathematical modeling skills to bear on particular scientific problems, through judicious approximations to obtain novel insights and predictions when the underlying phenomena are thought to be relatively simple and well understood, or through the creation of 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, applied mathematicians demonstrate over time substantive involvement with both the mathematical and scientific aspects of their dual roles. In the long run, their contributions must be evaluated based on some balanced measure of 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 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 involves a broad undergraduate education in the mathematical sciences, especially in those subjects that have proved vital to an understanding of the world around us, and in some specific area where mathematical methods have been substantively applied. The goal is to acquire experience at a mature level, consistent with the nature of a Harvard undergraduate education. The requirements are flexible, but structured and demanding. Individual programs are arranged in consultation with an adviser, and are approved by the adviser and by the Director of Undergraduate Studies. The concentration is overseen by an interdepartmental Committee on Undergraduate Studies in Applied Mathematics, and administered by the Division of Engineering and Applied Sciences.

Generally, students select the concentration because they like mathematics, 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. Inevitably, there are tradeoffs and compromises to be worked out. 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 sometimes 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.

REQUIREMENTS
16 half-courses

The concentration requirements are discussed in detail in the Applied Mathematics Concentration Guidelines document available from the Academic Office, Pierce Hall 110, or on the DEAS website (www.deas.harvard.edu). The Guidelines contain an exegesis of the overall requirements and of specific areas of application. Placement information relevant to first year-students is also included. Prospective concentrators are encouraged to make early contact with concentration representatives. Students wishing to enter the concentration must obtain the Applied Mathematics program of study and related instructions from the Academic Office and review these materials before meeting with the Director of Undergraduate Studies. 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: see Guidelines.

Total course requirements may be reduced from 16 to no less than twelve half-courses by placement out of basic courses 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 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.

1. Required courses:
1. Four half-courses in calculus, linear algebra, and differential equations:
1. Mathematics 1a. Mathematics 1b.
2. Applied Mathematics 21a, Mathematics 21a, or Mathematics 19a.
3. Applied Mathematics 21b, Mathematics 21b, or Mathematics 19b.
4. Theoretically-inclined students may substitute Mathematics 23a and 23b or 25a and 25b for Mathematics 21a and 21b. Mathematics Xa and Xb may be substituted for Mathematics 1a. Consult Guidelines regarding placement issues.
2. Three half-courses from the following two categories, including at least one half-course from each category:
1. Analysis: Applied Mathematics 105a, 105b, 147; Mathematics 106, 112, 113,115, 116, 118r.
2. Algebra: Applied Mathematics 106, 107, 120; Mathematics 121, 122, 123, 152.
3. Three half-courses from the following three categories, including at least one half-course from each of the first two categories:
1. Statistics: Either Statistics 110 or Mathematics 191 or Engineering Sciences 101; Statistics 111, 139, 171.
2. Computation: Applied Mathematics 111, 205; Computer Science 50, 51.
3. Physics: Physics 11a, 11b, 15a, 15b, 15c, 16.
4. Five half-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. At most two half-courses designated as primarily for undergraduates (numbered below 100 or 1000 depending on the department involved) may be included. Programs in specific areas of application are discussed in detail in the Guidelines.
5. Applied Mathematics 115 or 91r , or an advanced technical elective.
2. Tutorial: Optional (available as Applied Mathematics 91r).
3. Thesis: Optional (see item 5c).
4. General Examination: None.
5. Other information:
1. Pass/Fail: All courses counted for concentration credit must be letter-graded.
2. 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 approved by the adviser, filed with the Academic Office, and subsequently approved by the Director of Undergraduate Studies.
3. Honors: Requirements for honors degree recommendations are complicated and in transition. Details are discussed in the Guidelines. To be eligible for honors, all students must satisfy a writing requirement. Recommendations for Honors are based primarily on the grade average in the courses included in the final program of study. Recommendations for High or Highest Honors are based primarily on this grade average and on faculty evaluations of a senior thesis; however, alternatives to a senior thesis may be permissible for High Honors (see Guidelines).
4. 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.

ADVISING

The Director of Undergraduate Studies serves as interim adviser to all students entering the concentration. Subsequently, an adviser appropriate to the student's chosen area of application is assigned. Special arrangements are made for students whose area of application is mathematical economics, in cooperation with the Economics Department. If students do not request a change in adviser, they ordinarily will have the same adviser until they graduate. However, if an adviser becomes unavailable, the student is reassigned to a new adviser. Students may seek further advice from the Director of Undergraduate Studies at any time.

RESOURCES

See the Mathematical Sciences at Harvard booklet and other materials available from the Academic Office.

HOW TO FIND OUT MORE

Further information and advice is available from Sandra Godfrey, Academic Programs Administrator, Academic Office, Pierce Hall 110, 617-495-2833, godfrey@deas.harvard.edu, or Dr. Marie Dahleh, Assistant Dean for Academic Programs, Pierce 111, 617-495-1485, mdahleh@deas.harvard.edu. Ms. Godfrey can also arrange an appointment with an appropriate faculty member, or with the Director of Undergraduate Studies.

ENROLLMENT STATISTICS

Number of Concentrators as of November

*Applied Mathematics does not participate in joint concentrations.