First Exam Learning Goals
First Exam Assessment, 2011
Ph.D. Program in Mathematics
1. Matriculation is a precondition for taking the First Exam. In order to complete the First Exam, the Program requires that a student pass three written examinations, selected, depending on a student’s interest, from a collection of examinations, offered in the spring and fall, which cover foundational material from a broad variety of mainstream areas leading to current mathematical research. At present, examinations are offered in Algebra, Complex Variables, Differential Geometry, Logic, Real Variables, and Topology. A matriculated student who has not already successfully completed three qualifying exams, and who is not seeking to retake an already successfully completed individual exam, is eligible to take one or more qualifying examinations.
2. The purpose of the First Exam is to verify that before progressing to a greater degree of specialization, the student has achieved a requisite degree of mastery, at the first year graduate level, of a significant cross-section of contemporary mathematics. The general wish of the Program is that a student will have concluded the entire First Exam series by the first September after the conclusion of their first year of study. This cannot be achieved in all cases, for various individual reasons, but this goal of the Program is communicated to all students upon their arrival in the Program. As noted above, the particular examinations a student takes will reflect the tastes of the student, and their selection generally provides an accurate prelude to the area in which the student will ultimately work. Preparation for the examinations consists, importantly, of the corresponding first year courses, which are always offered in the subjects covered by the examinations. This is frequently supplemented by conversations with the instructors of these courses, as well as with other faculty, and by the very common practice of students organizing systematic collaborations among themselves for purposes of common study of the examination subjects. In connection with such collaborations, it is a frequent practice for students to request and receive clarifications of subtle or difficult points from members of the faculty. A substantial collection of former examinations is maintained by the Assistant Program Officer, and made freely available to students who request them, ordinarily for the purpose of preparing for the examinations.
3. Corresponding to each of the examinations, which are given in May and September of each year, a committee of three faculty members is appointed in the spring semester, comprised of experts in the relevant
subject, and ordinarily chaired by the person teaching the associated course during the academic year which concludes that spring. This committee prepares the spring and fall exams, and is responsible for evaluating and grading them. There are two possible grades – pass or fail – and after grading the exams, the committees transmit their determinations to the Program’s Executive Officer, together with remarks, if any, about those individual cases for which a committee may have observations that seem particularly pertinent to obtaining a sense of a student’s general level of progress. The results of the examinations are communicated to the students who have taken them, generally within a few days after the examination, and ordinarily by email. In the event that a student fails an examination, the student may confer with the Program’s Executive Officer or members of the examination committee to discuss the examination. In particular cases, the Executive Officer may wish to confer with a student to discuss the examination, or more generally, the students overall level of progress in the Program. The general policy regarding repetitions of a particular subject examination is that two are permitted. Any exception to this requires the explicit permission of the Executive Officer, which is only granted in the most exceptional circumstances. As previously described, the purpose of the First Exam is to ascertain whether or not a student has achieved a suitable level of competence to permit meaningful passage to a more advanced level of study and specialization, as defined by the Program. It is our definite sense that the First Exam has succeeded in identifying both those students who should advance to that category, as well as those who require further preparation to do so.