# Concentration Information

### The Applied Math Concentration Requirements: 14-15 courses.

### 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. Mathematics 22b, 21a

iv. Applied Mathematics 22a, Mathematics 22a, 21b

### 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. All courses must be taken letter graded in order to count for the concentration.

a. Computation: First course: Applied Mathematics 10 or Computer Science 32 or Computer Science 50.

Additional courses: Applied Mathematics 111, 205, 207; Computer Science 51, 61, 107, 109a, 109b, 181, 182, 205

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

Additional courses: Statistics 111, 139, 171, other courses above 110; Mathematics 117; Applied Mathematics 126

c. Differential Equations: Applied Mathematics 105, 108, 109, 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: Mathematics 122, 123

f. Optimization: Applied Mathematics 121; Mathematics 116; Computer Science 128

g. Discrete Mathematics: Applied Mathematics 107; Mathematics 152, 155r; Computer Science 120, 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:* A number of breadth courses recommend background in mathematical proof (including Math 112, Math 122, Math 154, CS 124, among others). Students considering a course that recommends prior experience with proofs should plan in advance to build a that background via Math 22a/b, Math 101, Math 121, Math 152, AM 107, or CS 20.

*Remarks:* For AM/Econ students considering pursuing an Economics PhD (and for any AM student considering a PhD in applied mathematics), we recommend real analysis (Math 112 - see the remark above about gaining proof experience before Math 112). In addition, optimization (AM 121 or Math 116) may also be of particular interest to AM/Econ students.

### 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

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 AM 21a, 22b, or Math 21a may take Mathematics 101 in their first 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 AM 50 only if it is taken before AM115.

- Students may take Math 22ab, 23ab, 23ac, 25ab, 55ab in place of AM 21ab, AM 22ab, Math 21ab. In terms of preparing for future AM coursework, these courses are appropriate for students who have previously taken multivariable calculus and linear algebra at the level of AM 21ab, AM 22ab, Math 21ab.

### Petition Form

For any course petitions related to concentration requirements, the Student Course Petition Form should be submitted to the DUS team.

### English Honors

Recommendations for English honors are based on the grade point average of the final plan of study and the satisfaction of the Honors requirement.

- The Honors requirement is automatically satisfied with a A- or higher grade in Applied Mathematics 115 and satisfactory grades in the 115 prerequisites.
- The Honors requirement can alternatively be satisfied via a modeling project, undertaken in AM 91r, in which a mathematical analysis of a problem is undertaken. A paper 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.
- Writing a satisfactory senior thesis, and turning it in, satisfies the Honors requirement for English honors. However, it does not automatically satisfy the Breadth modeling section (h) of the plan of study. Most students who write senior theses register for one semester of Applied Mathematics 91r, which satisfies Breadth section (h) of the plan of study.
**Applied Mathematics 99r cannot be used for Breadth section (h) of the plan of study, since this course is not letter-graded.**

Students who satisfy the Honors requirement without a thesis are eligible for Honors if their coursework meets the concentration GPA cutoff on their final plan of study for the concentration. The concentration GPA cutoff is 3.65 for the Class of 2024. It is 3.72 for the Class of 2025

The Committee on Undergraduate Studies in Applied Mathematics votes the level of English honors to be recommended (No Honors, Honors, High Honors, Highest Honors). A thesis is required for High or Highest honors, or for Honors with a GPA below the concentration GPA cutoff for honors without a thesis. For thesis writers, recommendations for the level of 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 the senior thesis.

### A.B./S.M. Option

The concurrent masters program makes it possible for students to graduate with a bachelor’s and master’s degree in four years. Pursuing a concurrent masters in Applied Math typically requires advance preparation. Students considering this pathway should choose Math 22a/b for their foundation sequence, will need to incorporate complex analysis and differential equations into their coursework, and may need to complete CS 51, CS 181, and real analysis, or other coursework to have appropriate background for AM 200-level courses. Any student considering this option should discuss requirements with our Office of Academic Programs.

### 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.