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Program Learning Goals

                                    Dissertation Learning Goals 2011

                                      Ph.D. Program in Mathematics

In our program, successful completion of the dissertation is the final step in a process consisting of three formal stages. The first of these is the successful completion of the First Exam. The second is the successful completion of the Second Exam. The last is successful completion of the dissertation. After completing the dissertation and thus the formal require ments for graduation from the program, the new graduate should be well equipped for successful productive employment, most frequently in academia at a tenure-track or post-doctoral research position, or alternatively in the private or government sector in a position requiring a Ph.D. in mathematics. In brief, this is the goal which the interrelated stages of the above process are designed to achieve.

In greater detail, the preparation for, and successful passage of, the First Exam is intended to assure that a student has acquired sufficiently broad foundational knowledge and grounding to continue on to the more specialized and advanced study required for both the Second Exam as well as for the dissertation itself. The kind of broad general knowledge required for the First Exam can be expected to importantly contribute to later versatility in both teaching and research, in any of the academic, private sector, or government settings. In particular, completion of the Second Exam and the dissertation alone would not ensure that the student has acquired the very important general component of a mathematical education corresponding to the First Exam.

After the First Exam, The Second Exam is a kind of small-scale dress rehearsal for the dissertation, but without the requirement of an original research contribution to the field. It is an oral examination, taken before a committee of three faculty members, and is ordinarily chaired by the student's advisor. In format, the examination consists of a formal presentation, generally one to two hours in length, of a part of mathematics of contemporary relevance. Additionally, the topic should generally be one whose mastery will facilitate the student's work towards an eventual Ph.D. degree. Specifically, the subject should be one in which the student has demonstrated aptitude and interest, and has secured the guidance of a member of the faculty who is both knowledgeable in the area and interested in taking on a Ph.D. level advisement role with the student. Passage of the exam requires that the student be capable of putting together and delivering a clear and coherent exposition of a high-level topic in mathematics, and be able, when necessary, to clarify various points, and to answer pertinent questions from a knowledgeable audience. The student's preparation for the exam is ordinarily formed from a combination of relevant prior course and seminar work, individual reading and study, and consultations with appropriate members of the faculty, as well as with other students.

These requirements strongly connect the Second Exam to the dissertation. By the time the Second Exam is passed, the student will ordinarily have secured an advisor, and the dissertation research will usually be in an area to which the student has become exposed in depth by preparing for the Second Exam. In this sense, the dissertation is often a deeper probe, involving original research, into areas acquired in detail by means of the preparation for the Second Exam.

The dissertation is expected to contain an original contribution to its field by the student, and should ideally not be an end or final point in itself, but should involve the student in an active area, with future research and growth potential. This is an important consideration, especially for students beginning academic careers.

Our department, like most Ph.D. programs in mathematics, does not require a formal dissertation proposal by the student, and none of us is aware of such a requirement in any of the mathematics doctoral programs with which we are familiar. Fundamentally, this is because of the nature of mathematics. Such a requirement, in which what needs to be carried out can be extensively mapped out in advance, does not carry over well to mathematics.

Mentoring of the student during, and often for a period prior to embarking on the Ph.D., is predominantly conducted by the advisor, who has usually supervised the student's Second Exam as well. Often, along the way to the Ph.D., the student also interacts with other faculty in cognate fields, as well as with other advanced students. When, in the advisor's opinion, the time has arrived for the thesis defense, a defense committee is constituted, ordinarily with major input from the advisor. This committee contains three or more faculty members, knowledgeable about the field of the student's dissertation, and who have been provided, several weeks before the defense, with copies of the dissertation, so that they may have adequate time to study, evaluate, and familiarize themselves with it.

These Dissertation Learning goals were developed by a committee apppointed by the Executive Officer of the Mathematics Program. The committee consisted of
1. Mr. Brent Cody, Student
2. Professor Richard Churchill, Deputy Executive Officer
3. Professor Jozef Dodziuk, Executive Officer
4. Ms. Jeanne Funk, student
5. Professor Clayton Petche
6. Professor Burton Randol