Planning & Courses

Overview of planning and courses for applied mathematics (including suggested schedules)

Since students may specialize in a range of areas, the Applied Mathematics concentration is designed to be extremely flexible.

In addition, the structure encourages students to make the most of Harvard’s resources, such as taking courses in other departments, collaborating with researchers from other fields or schools, and participating in the wealth of extracurricular activities available on campus.

Planning
Courses
Sample Schedules
Course Timing
Requirements
Senior Theses

Planning

 

Mathematics Placement

Your first mathematics course at Harvard is important for building your foundational knowledge in Applied Mathematics. For choosing that course, advising is available from Applied Mathematics Concentration Advisors, starting with advising fairs before the first week of the Fall and Spring semesters.  Questions regarding placement should be directed to Concentration Advisors during office hours or advising fairs.  In addition, an advising appointment can be arranged by writing to AM advising, with advising available to any Harvard College student.  Mathematics placement is granted based on an appropriate Advanced Placement Examination, the Harvard Mathematics Placement Test, or equivalent college-level coursework taken elsewhere.  Bypassing foundation courses must be validated by successful completion (honor grades) of more advanced courses.  Students seeking placement based on college-level work done elsewhere will be asked to submit a petition, supplemented by suitable supporting materials. 

Bachelor of Arts

The A.B. degree requires 14-15 courses.

Courses

SEAS offers undergraduate and graduate courses in Applied Mathematics. SEAS faculty also offer several courses in the section entitled Freshman Seminars, Extra-Departmental Courses, and House Seminars. Many additional courses of interest to applied mathematicians can be found in the Computer Science, Engineering Sciences, Mathematics, Economics, and Statistics sections of the catalog.

Sample Schedules

Below are some suggested paths for the freshman and sophomore years in the Applied Mathematics concentration. There are many possible pathways through the degree. Interested students should consult with the Directors, Associate Director, or Assistant Director of Undergraduate Studies for guidance.

Note: Also see suggested courses based on area of application.

Schedule M - Student has no advanced placement for mathematics.

Year Fall Courses Spring Courses
Freshman Year
  • Math Ma
  • Course in possible area of application
  • Math Mb
  • Course in possible area of application
Sophomore Year
  • Math 1b
  • Compsci 50 (or AM 111 later on)
  • Math 21a
  • Applied Math 50 (not required)

Sample plans of study from students who started in Ma are available from concentration advisors

Schedule 1 - Student has no advanced placement for mathematics.

Year Fall Courses Spring Courses
Freshman Year
  • Math 1a
  • Course in possible area of application
  • Math 1b
  • Applied Math 50 (not required)
Sophomore Year
  • Applied Math 21a OR Math 21a
  • Compsci 50 (or CS50/AM 111 later on)
  • Applied Math 21b OR Math 21b
  • Course in possible area of application

Sample plans of study from students who started in 1a are available from concentration advisors

Schedule 21a - Student places out of Math 1a and Math 1b.

Year Fall Courses Spring Courses
Freshman Year
  • Math 21a
  • Course in possible area of application
  • Compsci 50 (or AM 111 later on)
  • Math 21b
  • Applied Math 50 (not required)
Sophomore Year
  • Stat 110
  • Course in possible area of application
  • Math OR Applied Math course
  • Course in possible area of application

Schedule 21b - Student documents having taken AM 21a/Math 21a.

Year Fall Courses Spring Courses
Freshman Year
  • Math 21b/23a/25a/55a
  • Course in possible area of application
  • Math 23b/25b/55b or Math 112
  • Course in possible area of application
Sophomore Year
  • Stat 110
  • Course in possible area of application
  • Additional Math OR Applied Math course
  • Course in possible area of application

Course Timing

Almost all AM students take Stat 110 to satisfy the Probability requirement.  It is commonly taken in the Fall semester after a student completes the Multivariable prerequisite.

About 85% of AM students choose CS 50 to satisfy their Computation requirement while about 15% choose AM 111.  For students who choose CS 50, most take it within their first three years at Harvard.  About 30% take it at the same time as Stat 110.

AM 50 is designed to be taken earlier in a student's AM career.  Almost half of AM students take this course and they tend to take it freshman or sophomore year.

AM 115 is designed for junior or senior year, after a student has taken Stat 110, CS 50 or AM 111, and a differential equations course such as AM 105, along with some application area courses.  Some semesters AM 105 is a required prerequisite for AM 115.

Over 80% of AM students choose to take either AM 105 or AM 121, and almost half of students take both.  AM 121 has a Stat 110 prerequisite, so is usually taken in junior or senior year, while AM 105 can be taken immediately once the foundation sequence is completed.

Requirements

Students in the Classes of 2019 or later must follow the new requirements below.  Students in the Classes of 2017 or 2018 may continue to follow the old requirements, or petition to switch to the new requirements by submitting a new plan of study.

a.  Foundation

Two to five courses in calculus and linear algebra (see Notes, part d):

        i.  Mathematics Ma/Mb, Mathematics 1a

       ii.  Mathematics 1b

      iii.  Applied Mathematics 21a, Mathematics 19a, 21a, 23b, 25b, or 55b.  

      iv.  Applied Mathematics 21b, Mathematics 19b, 21b, 23a, 25a, or 55a

b.  Breadth

Five to seven courses (see item 1.d.i, below) from the following categories. Students must take courses from at least 5 of the 8 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.

a.  Computation: First course: Applied Mathematics 111 and/or Computer Science 50.

Additional courses: Applied Mathematics 205, 207; Computer Science 51, 61, 109a, 109b, 181, 182, 205

b.  Probability and Statistics: First course: either Statistics 110 or Mathematics 154, but not both.

Additional courses: Statistics 111, 121a, 121b, 139; Mathematics 117; Applied Mathematics 126

c.  Differential Equations: Applied Mathematics 105, 108, 202; Mathematics 110

d.  Analysis: Applied Mathematics 104, 201, 202; Mathematics 112, 113, 114, 115, 118r

e.  Algebra: Linear Algebra: Applied Mathematics 120, Mathematics 121

Abstract Algebra: Applied Mathematics 106/206; Mathematics 122, 123, 124

f.  Optimization:  Applied Mathematics 121; Mathematics 116

g.  Discrete Mathematics: Applied Mathematics 107; Mathematics 152, 155r; Computer Science 121, 124, 125

h.  Modeling: Applied Mathematics 50, 91r, 115; Economics 985; or an approved advanced technical elective from outside of the student’s application area

 

Remarks: For AM/Ec students, we usually recommend real analysis (Math 112) and either AM121 or Math 116. The latter two classes cover optimization with different perspectives.

c.  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.  See Areas of Application for sample five-course plans.

 

d.  Notes

        i.  Five Foundation courses and five Breadth courses are required for students starting in Mathematics M.

       ii.  Four Foundation and six Breadth courses are required for students starting in Mathematics 1a.

      iii.  Three foundation and six Breadth courses are required for students starting in Mathematics 1b.

      iv.  Two Foundation and seven Breadth courses are required for students starting in Applied Mathematics 21b, Mathematics 19b, 21b, 23b, 25b, or 55b.  Students starting in 21a may take Mathematics 101 as a third Foundation course (if taken in the freshman or sophomore year); these students are then required to take only six courses in the Breadth category.

English Honors

Recommendations for honors are based on the grade point average of the final program of study, the rigor of the overall record, and the satisfaction of the modeling requirement.  The Committee on Undergraduate Studies in Applied Mathematics votes the level of English honors to be recommended (Honors, High Honors, Highest Honors). To be eligible for English honors, all students must satisfy a modeling requirement, in which a paper is written where mathematical analysis is used to understand some aspect of the world around us. The modeling requirement can be satisfied in three ways:

  • Writing a senior thesis, and turning it in, automatically satisfies the modeling requirement for English honors. However, it does not automatically satisfy the Breadth modeling section (v) of the plan of study. Most students who write senior theses register for one semester of Applied Mathematics 91r, which satisfies Breadth section (v) of the plan of study.  Applied Mathematics 99r cannot be used for Breadth section (v) of the plan of study, since this course is not letter-graded.  
  • Taking the modeling course, Applied Mathematics 115. The last third of this class is spent working on an independent project. A grade of B- or above automatically satisfies the modeling requirement.
  • A 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, via AM advising, for evaluation.

Recommendations for honors are based primarily on the grades and rigor of the courses in a student's final program of study, and the evaluation of the thesis.  

Students who satisfy the modeling requirement without a thesis are eligible only for Honors.  The GPA cutoff for Honors is 3.5.  

A thesis is required for High and Highest honors.  There are no set GPA cutoffs for High and Highest honors.  In any given year, final GPA cutoffs will depend on the rigor of the eligible students' programs of study.  A student's level of honors will be determined by these cutoffs and the quality of their thesis.

A.B./S.M. Option

Students with sufficient advanced placement credit to qualify for advanced standing may graduate with a bachelor’s and master’s degree in four years. Any student considering this option should discuss requirements with our Office of Academic Programs (Pierce 110) or with the Concentration Advisors.

Secondary Field

The secondary field in Mathematical Sciences is jointly sponsored by the the Mathematics Department and the Applied Mathematics concentration. Students are required to take four courses in either Mathematics, Applied Mathematics, or Statistics of which at most two can be in Statistics. The Mathematics and Applied Mathematics courses must be numbered 104 or higher; Statistics courses must be numbered 110 or higher.

Senior Theses

A thesis is a more ambitious undertaking than a project. A project that meets the honors modeling requirement (either through Applied Mathematics 115, 91r, or through other means) can be extended to a thesis with about one semester of work.  See the above section on English Honors for details on how to satisfy the honors modeling requirement.  Obviously the more time that is spent on the thesis the more substantial the outcome, but students are encouraged to write a thesis in whatever time they have. It is an invaluable academic experience.

The thesis should make substantive use of mathematical, statistical or computational modeling, though the level of sophistication will vary as appropriate to the particular problem context. It is expected that conscientious attention will be paid to the explanatory power of mathematical modeling of the phenomena under study, going beyond data analysis to elucidate questions of mechanism and causation rather than mere correlation. Models should be designed to yield both understanding and testable predictions. A thesis with a suitable modeling component will automatically satisfy the English honors modeling requirement; however a thesis won't satisfy modeling Breadth section (v) unless the student also takes Applied Mathematics 91r.

To write a thesis, student's typically enroll in Applied Mathematics 91r or 99r or both during their senior year.  Applied Mathematics 99r is graded on a satisfactory/unsatisfactory basis.  Usually concentrators will have completed their programs of study before beginning a thesis; but in those situations where this is necessary, students may take Applied Mathematics 91r for letter-graded credit, for inclusion in Breadth section (v) of the plan of study, with the thesis serving as the substantial paper on which the letter grade is based.

Students specializing in mathematical economics may substitute one of the Economics 985 thesis seminars for Applied Mathematics 99r. These seminars are full courses for letter-graded credit which involve additional activities beyond preparation of a thesis. They are open to Applied Mathematics concentrators with suitable background and interests.

Students wishing to enroll in Applied Mathematics 99r or 91r should bring the application available from the Office of Academic Programs (Pierce 110), signed by the thesis supervisor, to the Directors, Associate Director, or Assistant Director of Undergraduate Studies, who will sign the student’s study card.

Thesis Timeline

Early September:

Students often find a thesis supervisor at this time, and work with their supervisor to identify a thesis problem.  Students may enroll in Econ 985 (strongly recommended when relevant), AM 91r, or AM 99r to block out space in their schedule for the thesis.

Early December:

All fourth year concentrators are contacted by the Office of Academic Programs.  Those planning to submit a senior thesis are requested to supply certain information.  This is the first formal interaction with the concentration about the thesis.

Mid-January

A tentative thesis title approved by the thesis supervisor is required by the concentration.

Early February

The student should provide the name and contact information for a recommended second reader, together with assurance that this individual has agreed to serve. Thesis readers are expected to be teaching faculty members of the Faculty of Arts and Sciences or SEAS.  Exceptions to this requirement must be first approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  For students writing theses in mathematical economics, the second reader will be chosen by the Economics Department.

March 31

Thesis due at 4 pm. Electronic copies in PDF format should be delivered by the student to the two readers and to am-submit@seas.harvard.edu (which will forward to the Directors of Undergraduate Studies, Associate and Assistant Director of Undergraduate Studies) on or before that date. An electronic copy should also be submitted via the SEAS online submission tool on or before that date. SEAS will keep this electronic copy as a non-circulating backup and will use it to print a physical copy of the thesis to be deposited in the Harvard University Archives. During this online submission process, the student will also have the option to make the electronic copy publicly available via DASH, Harvard’s open-access repository for scholarly work. More information can be found on the SEAS Senior Thesis Submission page.

Contemporaneously, the two readers will receive a rating sheet to be returned to the Office of Academic Programs before the beginning of the Reading Period, together with their copy of the thesis and any remarks to be transmitted to the student.

Late May 

The student may pick up the readers' comments from the Office of Academic Programs in late May, after the degree meeting to decide honors recommendations.

Thesis Readers

The thesis is evaluated by two readers, whose roles are further delineated below.  The first reader is the thesis adviser.  The second and reader is recommended by the student and adviser, who should secure the agreement of the individual concerned to serve in this capacity.  The reader must be approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  The second reader is normally are teaching members of the Faculty of Arts and Sciences, but other faculty members or comparable professionals will usually be approved, after being apprised of the responsibilities they are assuming.   For theses in mathematical economics, the choice of the second reader is made in cooperation with the Economics department.  The student and thesis advsier will be notified of the designated second reader by mid-March.

The roles of the thesis adviser and of the outside reader are somewhat different.  Ideally, the adviser is a collaborator and the outside reader is an informed critics.  It is customary for the adviser's report to comment not only on the document itself but also on the background and context of the entire effort, elucidating the overall accomplishments of the student.  The supervisor may choose to comment on a draft of the thesis before the final document is submitted, time permitting.  The outside reader is being asked to evaluate the thesis actually produced, as a prospective scientific contribution — both as to content and presentation.  The reader may choose to discuss their evaluation with the student, after the fact, should that prove to be mutually convenient.

Format

The thesis should contain an informative abstract separate from the body of the thesis.  At the degree meeting, the Committee on Undergraduate Studies in Applied Mathematics will review the thesis, the reports from the two readers and the student’s academic record.  The readers (and student) are told to assume that the Committee consists of technical professionals who are not necessarily conversant with the subject matter of the thesis so their reports should reflect this audience.

The length of the thesis should be as long as it needs to be to make the arguments made, but no longer!

Thesis Examples (for download)

The most recent thesis examples across all of SEAS can be found on the Harvard DASH (Digital Access to Scholarship at Harvard) repository here

Gilcrhist_Thesis
Xu_thesis
Chin-Lee_Thesis
Lei_Thesis
ZhangC_Thesis
Buckley Thesis

 

Note: Additional samples of old theses can be found in McKay Library.  Theses awarded Hoopes' Prizes can be found in Lamont Library.