# PhD

**Coursework Requirements**

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
- M501/502 Algebra
- M507/508 Lie Algebras and Lie Groups

- Analysis
- M511/512 Real Analysis
- M513/514 Complex Analysis

- Topology and Geometry
- M521/522/M529 Topology
- M531/M533/534 Differential Geometry

- Differential Equations
- M540/541/542 Partial Differential Equations
- M544/545 Ordinary Differential Equations

- Dynamical Systems / Probability
- M557/558 Dynamical Systems
- M560/M563/564 Probability

- Numerical Methods
- M571/572 Analysis of Numerical Methods

- Logic and Set Theory
- 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.

**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.

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.- Algebra
The Math Department "qualifying examinations" are 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.

The qualifying examinations consist of three tiers:

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.

The student is required to demonstrate proficiency in reading Mathematics in either French, German, or Russian. The Graduate Policy Committee will consider petitions to substitute other languages provided the grounds of the petition concern the performance of mathematical research.

Foreign language proficiency can be established in three different ways:

- One may take and pass with a grade of at least B (3.0) in the 492 reading course (e.g. F492 for French, G492 for German, and R492 for Russian). Students may register for the first course in the sequence _491, to prepare for the _492. Please note that the _491 and _492 courses are not counted as graduate credit, therefore these hours do not count toward your 90 hour requirement.
- One may take and pass the Graduate Student Foreign Language Test administered on a regular basis by the language departments at Indiana University. There is a charge for these exams.
- The examinee can translate a portion of an approved mathematics text. To qualify under this method, one contacts a mathematics faculty member designated by the Director of Graduate Studies and arranges with that faculty member to take the exam by translating a portion of the agreed upon text. The exam lasts about one hour and a dictionary is allowed. After consultation with the language department, a decision will be made based on the translation. Results will be communicated by the Graduate Office. One may attempt this exam in a given language at most once per semester.

The defense of a dissertation is the final requirement for the PhD degree in Mathematics. This process has several steps:

### Advancement to Candidacy

After both completing the coursework requirements and passing all qualifying exams, the student should immediately request the Director of Graduate Studies to nominate the student to PhD candidacy.

### G901 Status

Once nominated to candidacy, the student may register for the six hour course `Dissertation Research' G901. If the student registers for no other classes, then the student will pay minimal fees.

### Nomination of Research Committee

The candidate must form a `research committee' that meets the requirements of the Graduate School to be found in the Bulletin. The University Graduate School must approve of a candidate's research committee

**six months**before the candidate defends the dissertation. The candidate should visit the Mathematics Graduate Office to complete a `Nomination of Research Committtee' form well before this deadline.### Dissertation defense

The candidate must notify the Mathematics Graduate Office of the candidate's defense date at least

**thirty days**in advance. The dissertation and the dissertation defense should conform to the policies of the Graduate School to be found in the Bulletin. The candidate should pay special attention to the Graduate School's rules for Preparing Theses and Dissertations .The Mathematics Department asks that each candidate provide to the Department a black bound copy of her or his dissertation bound with the title lettered in gold ink on both the spine and front cover. This copy will be held by the Department as a record of student's achievement.