## Programs by Campus

### Bloomington

#### Mathematics

###### College of Arts and Sciences

**Departmental E-mail**: mathdept [at] indiana [dot] edu

**Departmental URL**: http://www.math.indiana.edu/

(Please note that when conferring University Graduate School degrees, minors, certificates, and sub-plans, The University Graduate School’s staff use those requirements contained only in *The University Graduate School Bulletin.*)

###### Curriculum

###### Degrees Offered

Master of Arts, Master of Arts for Teachers, and Doctor of Philosophy

###### Special Departmental Requirements

(See also general University Graduate School requirements.)

**Admission Requirements**

Undergraduate mathematics major or its equivalent.

**Definitions**

The Department of Mathematics offers core courses to give our students a broad education in mathematics and to prepare them for more advanced studies in the respective subjects. These core courses are divided into topics as follows:

• Algebra

o M501/502 Algebra

o M507/508 Lie Algebras and Lie Groups

• Analysis

o M511/512 Real Analysis

o M513/514 Complex Analysis

o M518 Fourier Analysis (offered sporadically)

• Topology and Geometry

o M521/522/M529 Topology

o M531/M533/534 Differential Geometry

• Differential Equations

o M540/541/542 Partial Differential Equations

o M544/545 Ordinary Differential Equations

• Dynamical Systems / Probability

o M557/558 Dynamical Systems

o M560/M563/564 Probability

• Numerical Methods

o M571/572 Analysis of Numerical Methods

• Logic and Set Theory

o M583/584 Set Theory/Recursion Theory

• Outside and miscellaneous courses (cryptography, quantum computing, financial mathematics, computer science, economics, and physics are commonly used, but others may also be appropriate). Course choices in this category must be approved by the student’s advisor and the director of graduate studies.

These topics serve both to satisfy our breadth requirements as well as to define the possible research areas for a major and minor. Courses other than the core courses may be assigned to these topics with approval of the instructor and the director of graduate studies. Students with a strong interest in Physics are encouraged to consider the Ph.D. program in Mathematical Physics.

**Master of Arts Degree**

**General Course Requirements**

Students must complete a total of 30 credit hours, of which 18 credit hours must be from the core courses, taken from at least three different topics. With the permission of the director of graduate studies, core courses can be substituted by more advanced courses from the same topic.

**Master of Arts for Teachers Degree**

**Course Requirements**

Students must complete a total of 36 graduate credit hours, with at least one 3-credit hour course in five of the eight topics. At least 21 of the 36 credit hours must be mathematics graduate courses. This includes 400-level courses that carry graduate credit. Those courses are assigned to topics by the director of graduate studies. At most 6 credit hours of other undergraduate mathematics courses at the 300-level or above may count towards the 36 credit hours, but they require consent of the director of graduate studies.

In addition to these 6 credit hours of undergraduate mathematics courses, M391 can also count towards the 36 credit hours.

**Doctor of Philosophy Degree**

**Course Requirements**

The following course requirements are designed to provide the broad background needed for the successful pursuit of research leading to the dissertation. Students must complete 36 credit hours in mathematics at the 500, 600, or 700 level, excluding M551, M553, M555, M556, M595-M596, and M599, and, in addition, must complete 2 credit hours in M599. Their program of study will depend upon their background and interests. Students should formulate a program in consultation with their faculty advisor.

Reading courses may not be used to satisfy the requirements of these options unless they are specifically approved by the director of graduate studies. A dissertation is required.

**Field of Research (Major Area)**

The field of research or topic of the major will be one of the topics listed above, or will be listed as Pure Mathematics or Applied Mathematics and Computation, with approval of the advisor and the director of graduate studies.

**Breadth Requirements**

Students must complete 24 credit hours from the core courses, with 6 hours in each of at least four different topics. With the permission of the director of graduate studies, core courses can be substituted by more advanced courses within the same topic.

One of the topics covered must be in the major area.

**Minor**

A Ph.D. student must complete a minor in mathematics, or in some other department. If the student chooses to minor in another department, she or he must satisfy that department’s requirements as described in the University Graduate School Bulletin and have that department notify the Department of Mathematics Graduate Office that she or he has done so.

To complete a minor in mathematics itself, the student must complete 9 credit hours of courses in one of the topics above, except the Outside topic. This topic must also be different from that of the Major (Field of Research), and the courses used to cover the Minor must be different from those used to cover the breadth requirements. The chosen topic will then be the topic of the minor. Alternatively, it can be listed as Pure Mathematics or Applied Mathematics and Computation, with approval of the advisor and the director of graduate studies.

**Foreign Language Requirement**

The student must demonstrate reading proficiency in one foreign language in which major research articles in mathematics are published. Acceptable languages are German, French, and Russian or another language deemed to be more relevant by the dissertation advisor. The Graduate Policy Committee of the Department of Mathematics will consider petitions for substituting other languages.

**Qualifying Examinations**

The Department of Mathematics qualifying exam comprises a three-tier system designed to help determine as quickly and efficiently as possible whether students have mastered basic mathematics, exhibit the necessary abilities and self-discipline, and have prepared themselves to pursue the independent research necessary to earn the Ph.D. degree.

*Tier 1 (Comprehensive 400-Level Written Exams)*

Ph.D. students will take written exams on both 400- level algebra and analysis. The exams will be given during the week before classes begin in the fall and in the spring. Each part of the exam lasts four hours.

New students may take either or both of the Tier 1 exams in August when they first arrive. A student is allowed to try each exam each time it is offered, but s/he must pass both exams prior to the end of the second year of study.

Syllabi, references, and sample problems for these exams are available on the Department of Mathematics web site.

*Tier 2 (Committee Review)*

Each spring/summer, a departmental committee will review the record of every student who has either:

• Completed two years in the program without previous review, or

• Passed the Tier 1 exams on entrance to the program and elects the review at the end of the first year.

The student will:

• Provide to the graduate office a personal statement that describes the student’s plan for further study and research, including a proposal for the area of research and a topic for a minor.

• Request an “endorsement” from his or her (interim) advisor or another faculty member. By endorsing a student, the faculty member agrees to guide the student to prepare for the Tier 3 exam.

The review committee will decide which students may continue toward Ph.D. candidacy. The committee’s considerations will include:

• Performance on the Tier 1 exams.

• Performance in 500-level coursework.

• A faculty endorsement.

• Written personal statement by student.

• Student’s performance of assistantship duties.

In support of the Tier 2 review, grades in 500-level courses will be given and evaluated according to the following guidelines:

• A grade of A means that, based on the student’s work in that course, the instructor believes the student will succeed in being admitted to Ph.D. candidacy.

• A grade of B means that the student’s work in that course is satisfactory, but the instructor has reservations (based on that work) about the student’s ability to be admitted to candidacy.

• Lower grades will indicate unsatisfactory work.

All students must maintain at least a B average in their coursework, in accordance with currently published departmental and university guidelines.

As indicated above, students can accelerate their progress in the program by passing the Tier 1 exams on entrance into the program and electing to take the Tier 2 review at the end of their first year. The review committee will treat this as favorable for a student’s case. Students who elect to accelerate their progress in this way will be expected to pass the Tier 3 (Oral Exam) by the end of the Fall semester of their third year.

Students who do not receive a recommendation to continue will be encouraged to complete the M.A. degree. If they have financial support at the time of review, they will be entitled to at least one additional semester of support in order to do so.

*Tier 3 (Oral Exam)*

After passing the Tier 2 review, a student must arrange and pass an oral examination before October of his or her fourth year. The student will seek the direction of a faculty member as a scientific advisor for this exam. The faculty member will assign a reading list consisting of texts and research-level papers; this material will comprise the major topic of the exam. If and when the scientific advisor feels the student is ready for the exam, the advisor will arrange for a three-member faculty committee to administer the exam.

The student will submit for approval a proposal for the Tier 3 exam to the director of graduate studies, consisting of topics for the major and minor area of the examination, a syllabus and a reading list for the major and minor topic, and the list of three faculty members serving as the Tier 3 committee.

These exams are projected to last approximately two hours, and one of the committee members must be qualified to examine the student in the minor area, where the student must demonstrate 500-level mastery. In order to pass the exam, the student must:

• Demonstrate a level of mathematical ability and maturity sufficient for successfully undertaking a Ph.D. dissertation (normally in the major area of the exam), and

• Identify a faculty member willing to serve as Ph.D. advisor. This will typically, but not necessarily, be the faculty member who organized the exam.

**Ph.D. Minor in Mathematics**

Doctoral students in other departments may complete a minor in mathematics by satisfying one of the following options: (1) 9 credit hours of mathematics courses at the 400 level or above, or (2) M343-M344 and 6 credit hours of mathematics courses at the 400 level or above. Reading courses (e.g., M800) and courses taken at other universities will not satisfy the course requirements for the Ph.D. minor.