**Doctoral Program**

Doctoral students are expected to complete their course of study in two-to-seven academic years.

Course regulations for Doctoral Program

I. In order to qualify for graduation, doctoral students are expected to earn at least 18 credit hours during their course of study.

II. Course Requirements are as follows.

(A) Basic courses include: Real Analysis, Algebra, Differential Geometry, Differential Equations, Theory of Statistical Inference or Probability Theory, Graph Theory and Numerical Analysis. All of these are one-year courses. In principle, the basic courses are offered at least once every two years, while Real Analysis and Algebra are offered every year.

(B) Advanced courses include: Functional Analysis, Algebraic Geometry or Representation Theory or Number Theory, Algebraic Topology, Partial Differential Equations, Discrete Mathematics, Theory of Stochastic Processes and Finite Element Methods.

III.Doctoral students must complete at least 18 credits in Basic and Advanced courses. Among these credits, there must be at least 12 credits from Basic courses. For those who have completed certain courses in the Master program may apply for course credit waiver.

Qualifying Exams:

I. The time for Qualifying Exams in principle is two weeks before the semester. Doctoral students can attend the Qualifying Exam given at the beginning of their first semester before their registration.

II. Doctoral students must choose two courses out of Basic courses as the subjects of Qualifying examination. The total credit for each subject is 100 points. A passing grade for the Qualifying exam for each subject is 60 points or more. A student can try the same subject for at most three times.

III. Doctoral students must pass 2 Qualifying exams in two years (before the beginning of their third year). Part-time Doctoral students must pass the Qualifying exams in two subjects in three years (before the beginning of their fourth year). Doctoral students have to quit the program if they fail to pass the Qualifying exams in time.

IV. For those who passed certain Qualifying Exams for the Doctoral program, if they quit the program and re-enter within three years, their passed Qualifying Exams will still be considered as valid.