Oral Comprehensive Exam

Every doctoral student in mathematics shall pass their oral comprehensive exam within 3 years of first enrolling in the doctoral program.

Purpose: The Oral Comprehensive Exam is designed to allow a student to show they are ready to work on their PhD thesis and receive input from a committee of experts on the direction of their work. Some key components of the exam are:

  • The students will demonstrate that they have mastered enough material in a specialized area of mathematics to begin working on their thesis.
  • The students will demonstrate that they can synthesize and critically evaluate research literature in their area of specialization.
  • The students will demonstrate that they can formulate and discuss interesting open problems and/or areas in need of development in their area of specialization.
  • The students will demonstrate their ability to communicate mathematics ideas orally. This will include providing background, context and motivation for the problems considered as well as being able to express mathematical arguments or ideas in a cogent manner.
  • The student will demonstrate their ability to communicate mathematical ideas in a written form through the oral exam proposal.
  • The students will receive feedback from the committee members about potential directions their proposed work can take and connections with other parts of mathematics.

Format: The oral exam involves a 40 to 45 minute talk by the students on their area of research making sure to address the above issues. This will be followed by 10 to 15 minutes of questions and comments from the audience that will hopefully include constructive ideas about the work and/or their presentation. The exam committee will consist of at least 4 faculty members: the dissertation advisor, two more members chosen by the student and his or her advisor, and an additional member appointed by the graduate director or the graduate committee. The general public is allowed in oral exams during the talk portion and can ask questions, but there should be some time for the committee to ask questions and make comments to the student in private. Lastly the committee will meet privately for a few minutes to discuss the performance, vote on whether the student passes, and fill out the appropriate forms.

Exam proposal: The oral exam proposal must be completed at least one week before the exam, and submitted to all committee members. The proposal should be between one and three pages, preferably typeset with TeX or LaTeX, consisting of the following elements:

  • List of the committee members for the exam.
  • A brief, but careful and precise, discussion of area and direction of the student's research.
  • A list or discussion of the literature read to prepare for the exam.

This material will help the committee provide better feedback to the student, allow the student to practice mathematical exposition, and provide a record of the student's direction of research at one point in time (but this is not meant to limit the direction a student's research ultimately takes).

Required Forms: All committee members should sign the Oral Comp Form at the conclusion of the exam. It is the student's responsibility to print this form, take it to the exam, gather all the required signatures, and then submit the form to the graduate director. In addition, the student should print and take to the exam a copy of the Oral Comp Survey form for each committee member. The student's advisor will then collect these forms and hand them to the graduate director. The student should provide the committee members with a link to this website and the survey form prior to the exam, so that they are fully aware of the parameters of the exam. The survey forms may also be filled out online and emailed to the graduate director, in case some committee members were participating in the exam via remote video conferencing.