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## Admissions & Aid

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- PhD Mathematics

## PhD Mathematics

### STRUCTURE

**Programme Structure**

All courses taken by a student are subject to approval by the Graduate Committee and student's supervisor (if any). The Graduate Committee will interview each student at least once each year to their review progress and approve the course schedule. For PhD Mathematics programme, a student has to complete at least 90 units. At least 27 credits are to be earned out of 500-level (or above) mathematics courses. At least 40 units must be research units and a maximum of 20 units of these can be counted towards the 90 units requirement. At least 9 units must be earned from other departments but no more than 15 such units can count towards the 90 units requirement. At least 55 units must be from mathematics courses. A student may not take an undergraduate-level course which he/she has already taken for his/her previous degree.

**Course Waiver**

A student having an MPhil or MS degree or who has done advanced course work (subject to proof) may request the Graduate Committee, through an application, at the start of the first year to waive off some of the course work requirements. Such students will have to take an oral examination on the course (s) for which waiver is demanded. At most 6 units can be waived off out of 27 units of 500-level or above mathematics courses. Research units cannot be waived off. Out of 55 mathematics units, a maximum of 24 can be waived off. Out of 90 units, at most 30 can be waived off. The final decision will be made by the Graduate Committee.

**CGPA Requirement**

Graduate students are expected to maintain a minimum CGPA of 3.3. A student whose CGPA is below 3.3 will not be hired as a TA or RA (or any other position). If someone's CGPA falls below 3.0, except in the first quarter, he/she will be separated from the programme.

**Qualifying Exam**

Mathematics students have to pass a written qualifying exam from the following subjects:

- Numerical Analysis
- Differential Geometry
- Partial Differential Equations
- Functional Analysis
- Topology
- Complex Analysis
- Measure Theory
- Algebra

The qualifying exam shall be given once a year. A student will have to clear this exam in prefereably one ot at the most two attempts, within the first two years of study. Students will be assigned a grade of Fail, Pass, or High Pass.

Exit with Master degree: A student who has earned 45 units of regular courses towards the LUMS PhD programme, which include at least 21 units of 500 level regular courses, and is separated from the programme, may be awarded the MS degree.

**Quarter Load**

A student has to carry at least twelve units every quarter to be a full time student. Before passing the Qualifying Examination, at least six of these twelve credits should be from 500-level regular courses. However, undergraduate courses may count towards this requirement if allowed by the Graduate Committee.

**Teaching and Colloquium Requirement**

After passing the qualifying exam, the student is required

- To teach one undergraduate-level course per year
- To give a colloquium talk every year. Every student has to give three such colloquium talks during his/her study programme

**Candidacy Examination**

After passing the PhD qualifying examination, before starting substantial work on a dissertation, the student is required to take an oral candidacy examination. This exam will be from the proposed field of specialisation on which the dissertation is to be written.

After passing the candidacy examination, the student has to write and defend a dissertation, which must be submitted within five years from the date of admission to PhD programme. However an extension of up to two years may be granted in special cases. Supervisor's approval is required for the submission of the dissertations.

**Dissertation Evaluation**

The dissertation will be evaluated by three referees, of whom one will be the supervisor and the other two will be foreign referees. These foreign referees will be chosen by the VC from a list of five people nominated by the Head of the Department of Mathematics. The Board of Advanced Studies and Research, which comprises the Dean, HOD, supervisor and two other senior faculty members from the Department of Mathematics will examine the reports from referees and decide whether the reports are positive or not. If at least two of the three referees give positive reports, the student will go through the final defence of his/her dissertation.

**Dissertation Defence Committee**

The Dissertation Defence Committee will comprise the Head of the Department of Mathematics (convener), Dean, supervisor, one professor (from LUMS but not from the Department of Mathematics) nominated by the VC and one external examiner who will be chosen by the VC from a list of three people nominated by the Head of the Department of Mathematics.