In general, the Candidacy Exam, formerly known as the Advanced Topic Exam (ATE) is a presentation that all Math Ph.D. students have to do.
Purpose: To demonstrate a student's research abilities and/or their possible research direction.
Content: This should be discussed with your main advisor. The student does not need to have new research in order to schedule their ATE. For example, they may present a summary of a research paper that was read to prepare for future dissertation work. It is important to have this conversation with your advisor so that you are on the same page.
Depending on the declared area of the student, a summary report is also required.
Areas that require an additional report:
Biomathematics
Financial Mathematics
Other areas are not required to have a report there are however benefits to creating one.
The Ph.D. Committee: Before you chose a date for your Candidacy Exam, you must have your committee members selected and approved. Your committee should be made up of 4 (or 5 if you have two advisors) professors. Two in-department professors and one university representative, out-of-department.
Ideally your committee members should have an idea of what your research is about before going into the exam.
Providing them with a brief summary documenting your work is one of the best ways to fulfill that need
Have a discussion with your main advisor on which faculty would be a good fit for your committee before contacting them.
Format: Once a date has been set and time has been set, your major professor sends out an email to the department, inviting anyone who can make it to attend.
The presentation and audience questions would typically take place during a 50 minute class period.
40-minute presentation
5-10 minutes for audience questions
The general audience is then dismissed, leaving you with the committee to answer any other questions they have for you
Finally you are dismissed, leaving the committee time to deliberate your "fate".