Department of Mathematics

The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.

Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, pure mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.

At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.

The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.

Undergraduate Study

An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.

Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.

Many mathematics majors take 18.821 Project Laboratory in Mathematics, which fulfills the Institute's Laboratory Requirement and counts toward the Communication Requirement.

Bachelor of Science in Mathematics (Course 18)

General Mathematics Option

In addition to the General Institute Requirements, the requirements consist of Differential Equations, plus eight additional 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher) that are distributed over at least three distinct areas (at least three distinct first decimal digits). One of these eight subjects must be Linear Algebra. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) as well as students who are interested in both pure and applied mathematics. More details can be found on the degree chart.

Applied Mathematics Option

Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically enroll in 18.200 Principles of Discrete Applied Mathematics and 18.300 Principles of Continuum Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of applied mathematics and may be taken concurrently with 18.03 Differential Equations. Subject 18.300, offered in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

More detail on the Applied Mathematics option can be found on the degree chart.

Pure Mathematics Option

Pure (or "theoretical") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.

The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis (18.100A, 18.100B, 18.100P or 18.100Q) or 18.700 Linear Algebra.

For more details, see the degree chart.

Bachelor of Science in Mathematics with Computer Science (Course 18-C)

Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.

The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra (18.06, 18.C06[J], or 18.700) because it is so broadly used, and discrete mathematics (18.062[J] or 18.200) to give experience with proofs and the necessary tools for analyzing algorithms. The required subjects covering complexity (18.404 Theory of Computation or 18.400[J] Computability and Complexity Theory) and algorithms (18.410[J] Design and Analysis of Algorithms) provide an introduction to the most theoretical aspects of computer science.  We also require exposure to other areas of computer science (6.1020, 6.1800, 6.4100, or 6.3900) where mathematical issues may also arise. More details can be found on the degree chart.

Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06 Linear Algebra, and, if they already have strong theorem-proving skills, may substitute 18.211 Combinatorial Analysis or 18.212 Algebraic Combinatorics for 18.062[J] Mathematics for Computer Science or 18.200 Principles of Discrete Applied Mathematics.

Minor in Mathematics

The requirements for a Minor in Mathematics are as follows: six 12-unit subjects in mathematics, beyond the Institute's Mathematics Requirement, of essentially different content, including at least three advanced subjects (first decimal digit one or higher).

See the Undergraduate Section for a general description of the minor program.

Inquiries

For further information, see the department's website or contact Math Academic Services, 617-253-2416.

Graduate Study

The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.

The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Doctor of Philosophy or Doctor of Science

The requirements for the Doctor or Philosophy or Doctor of Science in Mathematics degree include completion of a minimum of 96 units (8 graduate subjects), an oral qualifying exam, experience in classroom teaching, a thesis proposal, and a thesis containing original research in mathematics. Additional detail about these requirements can be found on the department's website.

All students pursue research under the supervision of the faculty. With the assistance of their faculty advisor, each student follows an individualized program of study encompassing the student’s area of interest. Faculty advisors may be members of the Mathematics Department or other MIT departments. Students also are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Students typically receive their degree in five years. The first two years are spent in coursework and research, culminating in an oral examination, which must be attempted by the end of their third term and completed by the end of the second year. Following the oral examination and culmination of their research, the student forms a thesis committee. The thesis committee includes the faculty advisor and at least three other faculty members. The student defends their thesis in a public defense. The thesis must meet high professional standards and make a significant original contribution to the student’s chosen research area.

Coursework must be completed with grades of A or B, and students are expected to maintain at least a B+ average in each semester. At most, one of the eight subjects should be a reading course. Harvard math graduate subjects may occasionally be used if taken for credit. With prior approval of the Graduate Co-chairs, one relevant advanced undergraduate math subject and relevant graduate subjects from other departments may be used. Note that subjects taken under the graduate P/D/F option cannot be used to fulfill this requirement. Students in Applied Mathematics must satisfy an additional breadth requirement as part of their plan of study.

Teaching is an important part of the academic profession and provides excellent experience in public presentation skills. All graduate students are required to complete at least one semester of classroom teaching as part of their graduate training and are encouraged to do more.

Interdisciplinary Programs

Computational Science and Engineering

Students with primary interest in computational science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see the CSE website.

Mathematics and Statistics

The Interdisciplinary Doctoral Program in Statistics provides training in statistics, including classical statistics and probability as well as computation and data analysis, to students who wish to integrate these valuable skills into their primary academic program. The program is administered jointly by the departments of Aeronautics and Astronautics, Economics, Mathematics, Mechanical Engineering, Physics, and Political Science, and the Statistics and Data Science Center within the Institute for Data, Systems, and Society. It is open to current doctoral students in participating departments. For more information, including department-specific requirements, see the full program description under Interdisciplinary Graduate Programs.

Financial Support

Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.

Inquiries

For further information, see the department's website or contact Math Academic Services, 617-253-2416.