### Programs by Campus

#### Bloomington

##### Mathematics

###### College of Arts and Sciences

**Departmental E-mail**: mathdept [at] indiana [dot] edu (mathdept@indiana.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. *Requirements may or may not be reflected identically in departmental URL’s.)

###### 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.

**Area Options**

In order to describe the various concentration requirements, the Department of Mathematics has classified its courses into three areas: pure mathematics, applied mathematics-numerical analysis, and probability-statistics. Each of these areas is further subdivided into fields. Pure mathematics is subdivided into analysis (real and complex), algebra, topology and geometry, and logic and foundations. Applied mathematics-numerical analysis is subdivided into applied mathematics, mechanics, and numerical analysis. Probability-statistics is subdivided into probability and statistics.

Students with a strong interest in Physics might consider the Ph.D. program in Mathematical Physics.

###### Master of Arts Degree

**Course Requirements **

A total of 30 credit hours, of which 18 credit hours must be mathematics courses at the 500 to 700 level, excluding M553, M555, M556, M595, M596, and M599. The total coursework submitted must include courses satisfying one of the following options and must be approved by the director of graduate studies.

- Applied Mathematics Option. Courses including M511, M513, and at least two of the following: M540, M541, M544, M545, M571, M572. In addition, the student must take at least one 3 credit hour graduate course outside the Department of Mathematics.
- Pure Mathematics Option. At least 6 credit hours in each of three of the following groups: algebra, analysis, applied mathematics and numerical analysis, logic and foundations, probability and statistics, topology and geometry, outside area (biology, psychology, computer science, economics, chemistry, physics, and others).
- Statistics Option. M511, M563, M566, and at least two of the following: M564, M567, M568, Economics E671, or any other approved course outside the Department of Mathematics.

###### Master of Arts for Teachers Degree

**Course Requirements**

A total of 36 credit hours, with at least one 3 credit hour course in each of the following groups: algebra, analysis, probability and statistics, topology and geometry, applied mathematics and numerical analysis.

**Restrictions**

Only Department of Mathematics graduate courses numbered 400 or higher count toward the 20 credit hours required in the major; up to 6 credit hours of courses below 400 accepted, with consent of the director of graduate studies, in partial fulfillment of the remaining 16 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 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. The total coursework submitted for the degree must satisfy one of the options below. Reading courses may not be used to satisfy the requirements of these options unless they are specifically approved by the graduate director. A dissertation is required.

1. Applied Mathematics Option. Students must complete M511-M512, M513, and M540-M541. Students must also complete either 6 credit hours of graduate credit outside the Department of Mathematics in an area conducive to mathematical treatment and approved by the student’s advisor, or 12 credit hours of graduate credit that is cross-listed with the Department of Physics. In addition, students must complete 6 credit hours of graduate credit in each of two of the following groups:

- Algebra, and logic and foundations
- Topology and geometry
- Numerical analysis, probability, and statistics

2. Pure Mathematics Option. Students must complete 6 credit hours of graduate credit in five of the following groups, including 6 credit hours of 500-level courses in two of groups 1-4, and 6 credit hours of 500-level courses in another of groups 1-6.

- Analysis
- Algebra
- Topology and geometry
- Logic and foundations
- Probability and statistics
- Applied mathematics and numerical analysis
- Outside and miscellaneous courses (cryptography, quantum computing, financial mathematics, computer science, economics, and physics are commonly used, but others may also be appropriate). Any courses in this category must be approved by your advisor and the director of graduate studies.

3. Statistics Option. Each student’s program should be arranged to include work in mathematics, mathematical statistics, probability, and the application of statistics to some particular field. Students must complete 6 hours of graduate credit in other departments in courses approved by their advisor, as well as M511-M512, M563-M564, M566-M567, and 6 hours of graduate credit in each of two of the following groups:

- Applied mathematics and numerical analysis
- Algebra
- Complex analysis
- Topology and geometry
- Computer science

**Minor**

You must complete a minor in mathematics, or in some other department. If you choose to minor in another department, you 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 you have done so. To complete a minor in mathematics itself, there are two possible options:

- Nine credit hour minor. This requires 9 credit hours of coursework at the 500-700 level in an area (Pure Mathematics, Applied Mathematics, or Probability and Statistics) different from that of the dissertation. This area then becomes the “area for minor” while forming the Research Committee (see below).
- Twelve credit hour minor. This requires 6 credit hours of coursework at the 500-700 level in each of two fields (Analysis, Algebra, Topology and Geometry, Logic and Foundations, Applied Mathematics [Differential Equations], Mathematical Physics, Numerical Analysis, Probability, Statistics) other than that of the dissertation. One of the fields chosen must be in an area (Pure Mathematics, Applied Mathematics, Probability and Statistics) other than that of the dissertation. Any of these two fields can be chosen as the “minor field” while forming the Research Committee (see below). In addition, reading courses (e.g., M800) and courses taken at the other universities will not satisfy the course requirements for the Ph.D. minor.

**Foreign Language Requirement**

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 graduate-level mathematics, exhibit the necessary abilities and self-discipline, and have prepared themselves to pursue the independent research necessary for the Ph.D. within a two- to three-year time frame.

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

Ph.D. students will take a two-part written exam on 400-level algebra and analysis. The exams will be given during the week before classes begin in the fall and in the spring. 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 page.

**Tier 2 (Committee Review)**

Each May, 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 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 report from the student’s academic advisor (see below).
- Written personal statement by student.
- Student’s performance of assistantship duties.

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 do not get 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 the Tier 2 review, students must arrange and pass a Qualifying Oral Exam before October of their fourth year. The student will seek the direction of a faculty member as a “tentative Ph.D. 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. The student will also propose a minor area, to be approved by the director of graduate studies. 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. 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.

Two aspects of the graduate program that directly support this three-tiered system bear mention:

- A faculty advisor will be assigned to every entering student. The advisor will be expected to follow the student’s progress in both coursework and instructional duties, and to write a short report for the Tier 2 review committee at the appropriate time.
- 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.

All students must maintain at least a B average in their coursework, in accordance with currently published departmental and university guidelines. Tier 1 exams are administered the week before classes begin in August and January, and will be allotted at least four hours (the intention is that time should not be a serious constraint).