Curriculum and Degree Information
The Ph.D. Program in Mathematics provides students of high ability and strong preparation with an opportunity to begin study for the doctoral degree either immediately upon graduation from college or after completion of some graduate work in the colleges of the City University or at other accredited institutions.
The program is designed to give students the background they will need to pursue careers as pure or applied mathematicians.
- Algebraic Geometry
- Analysis and Harmonic Analysis
- Complex Analysis and Teichmuller Theory
- Group Theory
- Lie Theory
- Number Theory
- Riemannian Geometry and Analysis
- Quantum Information Theory
To view the faculty members that are associated with each specialization, visit the Faculty page.
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 requirements 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 appointed 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
Students are expected to pass 3 qualifying exams by the Spring of their second year. The students should be able to complete 45 credits and reach Level II by the end of the second year.
During the third year students select a mentor who will be willing to chair the Second Examination Committee and assign the topic for the Second Examination. The Second Examination should take place no later than the fourth year. By the end of the fourth year students are expected to complete the coursework and all other requirements for Advancement to Candidacy. In most cases, the Chair of the Second Examination Committee will become the dissertation adviser.
Thesis research is conducted during the fourth, fifth and sixth year and culminates in the defense sometime during this period.
In addition to the general University requirements stated earlier in this catalog, the student must meet the following requirements:
Each student must follow a plan of study, approved by a faculty adviser, which usually includes three years of course and seminar work. A minimum of 81 graduate credits of course work are required for the degree, at least 60 of which must be in mathematics and the rest may be in closely related fields. Course offered by other programs which are listed under Math as “See Also” may be taken for credit toward the Math degree. A determination regarding other courses not so designated will be made on a case-by-case basis. At least 36 of the 60 credits in mathematics must be in non-introductory courses or seminars.
Students are expected to pass 3 qualifying exams by the Spring of their second year.
The Mathematics Qualifying (or Preliminary) Examinations are given in six areas. The exams are scheduled twice a year, in September and May. Students must pass three exams (of their choosing) to be considered to have passed the Qualifying exams (also called the "First Examination;" it is listed this way on the transcript.) Students may retake a failed exam once. Exams are graded Pass/Fail. Students must sign up at least three days in advance of any exams that they wish to take so that the proctor will know how many students to expect. Each exam is scheduled for three hours.
First Exam Learning Goals
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.
There is one foreign language requirement with a stipulation that it cannot be your native language. Consult the mathematics office for details.
After passing the First Examination, the student specializes in some area of advanced mathematics. A faculty committee will be appointed to help the student arrange a program of study in an area of special interest. When this program of study is completed, the student takes an oral examination given by the faculty to determine readiness to pursue dissertation research.
Second Exam Learning Goals
In order to be eligible to take the Second Exam, a student must be at Level II, and in particular must have passed the Qualifying Examinations, and have accumulated at least 45 credits of course work.
The Second Exam 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.
The student’s performance on the exam is evaluated by the exam committee immediately following the exam, and the results communicated to the student and the Executive Officer, as soon as a determination has been made. Whether the student passes or fails (the only two possible outcomes) is based on the level of knowledge displayed by the student, as well as the quality of the presentation, and the evidence it presents for the student’s suitability to move to the level of working on the Ph.D.
As noted above, the student receives very quick notification, usually just after the exam, about whether they passed or failed. Historically, failures are quite unusual, since if a student is not going to do well, this has usually become apparent well before a Second Exam is reached, and the advisor would have ordinarily made it clear to the student that they are not, in the advisor’s opinion, prepared to pass a Second Exam in the selected field. If, under such circumstances, or after an actual failure of a Second Exam, an advisor feels unable to work successfully with a student towards a Ph.D., the student would be directed to have a conference with the Executive Officer, in order to explore various alternative options, both as regards suitable fields of study and/or advisement, or rarely, the suitability of continuation in the Program. If the student continues in the Program and successfully changes fields, they would be permitted to take another Second Exam in their new field. Under exceptional and quite rare circumstances, a student may be permitted to attempt to retake and pass a Second Exam in the previous field.
Each student must complete a dissertation embodying the results of original research in mathematics. The thesis is usually written in a field of specialization recommended by the candidate's thesis advisor and approved by the thesis committee. The completed dissertation must be approved by the thesis committee and must be defended at an oral examination.
Dissertation Learning Goals
All Ph.D. graduates of our Program are expected to have acquired several skills.
Among the most important of these are:
The attainment of mastery, to the level of publishing in respected research journals, of a significant, active field of mathematical research.
The ability to clearly present, e.g., in a colloquium or seminar context, developments in their field, either in the form of accounts of their own work or that of others. In a student’s progression toward completion of the Ph.D., these capabilities are developed by the requirements of the presentation involved in the Second Exam, by the defense of the Ph.D. thesis, and frequently, by presentations given in seminars at the Graduate Center and elsewhere, and at mathematical meetings and conferences.
The capacity to teach effectively, in a general academic or business context. Our students are particularly experienced in this requirement, since an overwhelming majority of them have teaching experience at undergraduate colleges of CUNY, and, in contrast to the practice in many other otherwise similar programs, have full responsibility for the courses they teach.
As a general point of information, the majority of our graduates enter teaching careers, although some are employed in the business world. In connection with the latter career path, experience has made clear that advanced mathematical training is often regarded, both outside as well as within academia, as valuable general training for careers in several fields.