Mathematics and Minor (Ma)

The principal aim of the graduate program is to develop the student’s ability to do original research in mathematics. Independent and critical thinking is fostered by direct contact with faculty members. (Read more about the current research interests of the faculty.) Faculty advisers help students plan their programs of study leading to a Ph.D. in mathematics. Entering students are advised by the graduate option representative, who assists them in selecting appropriate courses, depending upon their previous studies.

Entering graduate students are admitted directly to the Ph.D. program, since the Institute does not offer a regular program in mathematics leading to the master’s degree. A master’s degree may be awarded in exceptional circumstances either as a terminal degree or preliminary to the Ph.D. Sufficiently advanced undergraduates may be admitted to graduate standing to pursue a master’s degree simultaneously with the bachelor’s program.

The recipient of a master’s degree will be expected to take 135 units in advanced mathematics (numbered 110 and higher). Unless the student has placed out of some of them, these must include at least two among Ma 110, 120, 121, 140, and 151 (the basic courses in analysis, algebra, combinatorics, probability, and topology) and at least two terms of discrete mathematics. Reading and/or research courses may only be included in this 135 units if approved by the executive officer for mathematics.

The general Institute requirements specify that the recipient of a master’s degree must have taken at least 135 units of graduate work as a graduate student at the Institute, including at least 81 units of advanced graduate work in mathematics. This advanced work is interpreted as work with a course number greater than 109 and may include a master’s thesis.

The graduate courses are listed in section five. Two among the five core course series—Ma 110 abc in analysis, Ma 120 abc in algebra, Ma 121 abc in combinatorics, Ma/ACM/IDS 140 abc in probability and Ma 151 abc in geometry and topology—are required of all graduate students. Students are expected to complete two core course series of their choice during the first year of study, unless they complete the corresponding qualifying exam upon entrance to the program (see section on qualifying exams). In addition, students must complete three courses in the core areas not associated with the two chosen qualifying exams; these are to be completed by the end of second year of study.

Students are required to complete nine additional advanced mathematics courses Ma 110 and above. Unless these courses are given pass/fail only, they must be taken for grades. Reading and research do not normally qualify to meet these requirements. Under special circumstances (e.g., finishing the degree in three years), exceptions to these requirements may be considered by the graduate option representative. Beginning no later than the second year, students will be expected to begin independent research work and will be strongly encouraged to participate in seminars.

Qualifying examinations in the five core areas—analysis, algebra, combinatorics, probability, and geometry/topology—emphasize mastery of the basic concepts and theorems and the ability to apply them to specific cases. Students are required to take and pass two of the five examinations, and must earn a B or better in all three additional courses in the remaining core areas. The exams are offered at the beginning of the Fall term and end of the spring term each academic year, with exceptional offerings made at other times of the year if needed and approved by the graduate option representative. The examinations must be attempted by the end of the spring term during the first year of study, and completed by the end of the first academic year. Entering students are permitted to take one or more qualifying examinations at the beginning of the fall term of their first academic year. If completed successfully, the student may drop the corresponding core course and will be excused from the remainder of the corresponding core requirement. Students must complete the three additional quarters of course courses by the end of their second year of study. Special exceptions or extensions to the qualifying examination timeline may be considered by the graduate option representative with support from the primary research adviser.

Teaching is an integral part of the mathematics graduate education. It is expected that students serve as teaching assistants in each term courses are offered unless alternative arrangements have been made with the option and approved by the executive officer.

Although there are no courses given in the summer, graduate students are expected to carry out studies and research in their chosen area of mathematics. In the summer after the first year, they will work under the guidance of a faculty member to investigate a possible area for their thesis research.

It is expected that by the fall quarter of the second year, students will find a member of the faculty who agrees to serve as their thesis adviser. The formal change of adviser process should be complete by the end of the second academic year.

The progress of all continuing students is assessed by the faculty each fall, and students will consult with their advisers about their progress and planning of their studies and research. Students receive help and advice not only from their thesis adviser and other faculty mentors, but also whenever needed from the graduate option representative and the executive officer. (See also the section with guidelines for graduate student advising.) In addition to the primary thesis adviser, each student will have a candidacy committee, Thesis Advisory Committee (TAC), and final defense committee. In most cases, the members of all of these committees are the same, but the student has the right to change the members. The student in consultation with their adviser will arrange the formation of the committee, which will have at least four members and meet the requirements listed in the subsection Graduate Policies and Procedure entitled ‘Degree of Doctor of Philosophy.’ At least two of the committee members should be members of the mathematics professorial faculty and the chair of the committees must be a professorial faculty who is not the primary research adviser.

Before the end of their third year, students are expected to finish the process of applying for admission to candidacy for the Ph.D. degree. This formal step requires completion of the qualifying examinations, core courses, and advanced mathematics courses, as well as a satisfactory oral presentation to a committee of faculty members. The presentation will describe both the general area of the student’s proposed thesis research and the specific problem or problems to be addressed. The candidacy meeting should be complete by the beginning of the fourth year of study. It normally takes place during the spring term of the third year. In some cases, during the summer term of the third year, but scheduling must be done by the end of the spring term. Special exceptions or extensions to the admission to candidacy timeline may be considered by the graduate option representative with support from the primary research adviser.

The Thesis Advisory Committee (TAC) will track the student’s progress through annual meetings between the candidacy exam and final defense. It is the responsibility of the student to convene the TAC during the fourth and fifth year of study. The fourth-year meeting should take place no later than the end of the winter term of the fourth year. The fifth-year meeting should take place no later than the end of the fall term of the fifth year.

At least two weeks prior to the final thesis defense, candidates for the degree of Doctor of Philosophy must deliver copies of their theses to their advisers, to the Graduate Office, and to the members of the committee that will conduct the final oral examination on the thesis. The examination must be held at least three weeks before the date on which the degree will be conferred and at least two weeks after the delivery of the copies of the thesis.

Students majoring in other fields may take a subject minor in mathematics. Minor programs must include 54 units of advanced work approved by a representative of the mathematics department, who will ensure that the work represents a concentrated study in one or more of the main fields of mathematics. A special oral examination in the subject minor will be given soon after completion of the minor program.