The purpose of the Qualifying Examinations is to help the student and the Department determine whether or not the student should continue in the Ph.D. program at Yale. These exams are meant to answer whether the student has mastered the basic material in the various fields to qualify as a member of the profession, and whether the student have the ability and preliminary knowledge to carry out a research.
Passing the Qualifying Exams is one of the basic Ph.D. requirements of the math department, and among the first that students encounter. Each one is a 4-hour written exam. The exams are offered at the end of each semester (though they may be offered at other times by request) and the student is required to pass, at high enough score, all three by the end of the second year. So normally there are 4 chances to take the exams. As there is no limit to the number of times one can take the exam, students are strongly encouraged to take the exam as soon as they feel that they can pass it.
The exams cover three topics: Algebra, Analysis and Algebraic Topology. The exams in Algebra and Analysis cover many subtopics that are learned in different courses (as can be seen below in the syllabi of the exams), whereas the exam in Algebraic Topology is given in the fall semester as the final exam of the related course, Math 544 (as such, the most updated syllabus of the exam is always the syllabus of the course, find it here and here). Thus, the students are encouraged to take the course in their first year and take the exam as the final exam. Past exams are available upon request from the Department Registrar.
Though the exams in Algebra and Analysis are not linked to a specific course, there are several courses that cover some of the advanced topics in the qualifying exams. As such, even though that these courses are not mandatory, they are highly recommended for students that have not taken similar courses. The courses suggestion in Algebra are: Modern Algebra I (Math 380\500a – Fall semester) and Modern Algebra II (Math 381/501b – Spring semester). The courses suggestion in Analysis are: Measure Theory and Integration (Math 320/520a – Fall semester), Intermediate Complex Analysis (Math 315/515b – Spring semester) and Introduction to Functional Analysis (Math 325/525b – Spring semester).