Qualifying and Breadth
Requirements

The Department of Mathematics offers 5 distinct qualifying exams in
the following subjects:
Each qualifying exam has a duration of 4 hours, and is offered
twice a year (January and August). The syllabi and schedules can be
found in the Qualifying Exam
section of the Graduate Program web page.- The Qualifying Exams are designed to assess students' competence in basic mathematical skills and knowledge in the corresponding areas, at the level of the core courses displayed in the table below.
- The Breadth Requirement is designed to ensure that students acquire a broad education and a solid background in mathematics, thus requiring that they demonstrate basic knowledge in the various areas displayed in the table below.

- Core Courses:

Set I |
Set II |
Set III |
Set IV |

Algebra Math 653/Math 654 |
Real Analysis Math 607/Math 608 |
Diff. Geometry Math 622/Math623 |
Applied Analysis Math 641/Math 642 |

Discrete Math/Number Theory Math 613/Math 630/Math 627 |
Complex Analysis Math 617/Math 618 |
Topology Math 636/Math 637 |
Numerical Analysis Math 609/Math 610 |

#### * Qualifying Exam Requirement:

- Students must pass at least 2 qualifying exams by the end of the second year of enrollment.
- These two qualifying exams must come from different sets in the above table.

#### * Breadth Requirement:

- Students must take at least 4 regular courses at a level greater or equal to the core courses and obtain a B or higher in each course.
- In
each of the 4 sets of subjects displayed in the table above,
the student must either take a corresponding
course or pass a qualifying exam. (See the list below for courses
that can be used to fulfill the breadth
requirements.)
- A typical student is expected to have fulfilled these requirements by the end of the second year of enrollment.

#### * Courses for Breadth
Requirement:

The following courses can be used to fulfill the breadth
requirements in the respective areas. Additional courses may be
used with the approval of the Graduate Committee.
Subject |
Courses |

Algebra |
Math 653, Math 654 |

Discrete Math/Number Theory |
Math 613, Math 630, Math 626, Math 627 |

Real Analysis |
Math 607, Math 608, Math 655, Math
656 |

Complex Analysis |
Math 617, Math 618, Math
650 |

Differential Geometry |
Math 622, Math 623 |

Topology |
Math 636, Math 637, Math 643, Math
644 |

Applied Analysis |
Math 611, Math 612, Math 641, Math 642, Math 658, Math
670 |

Numerical Analysis |
Math 609, Math 610, Math
661 |