The QE – University of Colorado at Denver, Mathematics Department

Each student must pass two preliminary exams: one in Applied Analysis and one in Applied Linear Algebra. These are four-hour written exams that cover material roughly at the level of first-year graduate study. The exams are offered twice a year, from 10:00 am to 2:00 pm. The first set of exams are offered in January. The Applied Analysis preliminary exam will be held the Monday prior to the start of classes, and the Applied Linear Algebra exam is offered the Friday before the start of classes. The second exam offerings follow the spring semester. The Applied Linear Algebra exam is held the Friday before the start of the summer semester, and the Applied Analysis exam is offered the first Friday of the summer semester.

A student wishing to take a preliminary exam must sign up with the graduate program assistant at least one month prior to the administration of the exam. Students who cannot take the exam as planned must give notice of withdrawal at least one week prior to the exam.

Students have a maximum of three attempts to pass each exam. The time limit clock starts the first semester a student is admitted into the program.

• Students entering the Ph.D. program without a master’s degree in mathematics have three years to pass both preliminary exams.
• Students entering the Ph.D. program with a master’s degree in mathematics have two years to pass both preliminary exams.

No appeals will be considered if a student fails the respective exam a third time.

The QE – Stanford University, Mathematics Department

To qualify for the Ph.D. in the Stanford University Mathematics Department, students must pass two examinations: one in real analysis and one in algebra. The real analysis and algebra exams each consist of two parts. Students are given three hours for each part.

Students must pass both exams by the fall of their second year. Ordinarily first-year students take courses in real analysis and algebra throughout the year to prepare them for the qualifying exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in the following fall.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the fall. If they pass any of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

In addition to the qualifying exams, during the first year students must satisfactorily complete one of a choice of three courses sequences. The choices are:

1. Complex analysis, geometry, and topology (Math 215A,B,C)
2. Partial differential equations of applied mathematics (Math 220A,B,C)
3. Theory of Probability (Math 230A,B,C)