• A. Within the first two years, the student is expected to demonstrate proficiency in certain basic areas of mathematics by passing written examinations and doing course work. This requirement must be completed before 10 May of the student's second academic year of studies.
• B. Next, a student is expected, via the Ph.D. preliminary oral examination, to demonstrate proficiency in a major area as well as a minor area (which is either a supporting program in the School of Mathematics, or a minor in another field). This examination, along with the usual course work, completes the requirements for a Plan B Master's degree. This requirement must be completed before 10 May of the student's fourth academic year of studies.
• C. Reading proficiency in technical literature in two languages (French, German, Italian, Russian - or an approved substitute) is required. Proficiency in one language must be demonstrated prior to step B above.
• D. The student is required to write a Ph.D. thesis under the direction of an advisor. This is followed by the final oral examination (which constitutes the defense of the thesis). Proficiency in the second language must be demonstrated before this examination may be taken. The final oral exam should be passed by 10 May of the student's sixth year. Plase note that 7th year assistantship funding is not available, except in extreme circumstances.

The Ph.D. Preliminary Written Examinations are given twice each year, once before the start of fall semester, and again in the late spring; each individual examination lasts four hours. A student who wishes to become a candidate for the Ph.D. degree must pass these written examinations and related course work requirements by 10 May of the end of the second year of graduate study. No penalty is attached to unsuccessful attempts to pass these examinations. Any schedule of part-time study (which would reduce the rate at which course work and examination requirements would be met) must be approved in advance by the Director of Graduate Studies (DGS).

• i.(a) Real Analysis (Mathematics 8601-8602)
• i.(b) Complex Analysis (Mathematics 8701-8702)

• ii.(a) General Algebra (Mathematics 8201-8202)
• ii.(b) Manifolds and Topology (Mathematics 8301-8302)

The student must show proficiency in all four areas. In at least two areas, of which one must be from group (i), and one from group (ii), the written examinations must be passed. For any remaining areas, the student may choose whether to pass the written examination or to pass the appropriate 8-xxx level year-long graduate course with average grade of at least B.

The student must pass two written examinations, one chosen from the group:

• i.(a) Real Analysis (Mathematics 8601-8602)
• i.(b) Complex Analysis (Mathematics 8701-8702)

and the second chosen from the group:

• ii.(a) General Algebra (Mathematics 8201-8202)
• ii.(b) Manifolds and Topology (Math 8301-8302)
• ii.(c) Riemannian Geometry (Mathematics 8365-8366)

The student must also complete the following two year-long courses, each with average grade of at least B: Numerical Analysis and Scientific Computing (Mathematics 8441-8442), and Mathematical Modeling and Methods of Applied Mathematics (Mathematics 8401-8402).

In addition, the students in the Applied and Industrial Program must complete a year-long course that carries graduate credit in another department, subject to the approval of the DGS, by the end of the fourth year of graduate study.

To remain in good standing, international teaching assistants must receive an English Language Proficiency score of "1" by 21 May of their first year.

A student must demonstrate a reading proficiency in technical literature in two of the following languages: French, German, Italian and Russian. These requirements are handled internally by the School of Mathematics. See the Graduate Program secretary in Vincent Hall 127 for the internal procedures. The requirement in one language must be completed before taking the Ph.D. Oral Preliminary Examination. Substitutions of languages may, on occasion, be granted by the advisor and DGS. Both language requirements must be completed before the Final Oral Examination.

After having selected the major area in Mathematics where the Ph.D. thesis will likely be written, students also choose one of the following:

(i) Internal Supporting Program : Two 8-xxx year long course sequences in mathematics, outside the major area must be selected. These may not include any of the core courses (Math 8201, 8202, 8301, 8302, 8601, 8602, 8701, 8702, 8401, 8402, 8441, 8442) of the General Program. Approval from the DGS and advisor is needed. The oral examination only covers one of these supporting sequences.

(ii) Outside Minor: The DGS in the field should be consulted, and a written plan should be submitted to the DGS in Mathematics. This plan must include: the list of outside courses, a description of the content of each and how it fits into the proposed thesis work of the student, as well as the place of this material in the outside department's offerings. The plan must satisfy the minor requirements of the outside department.

(iii) Partial Internal Supporting Program: One 8-xxx sequence as in i) above, and a sequence with graduate credit outside Mathematics should be chosen. The DGS in the outside field should be consulted and prior approval from the DGS in Mathematics should be obtained by the written proposal process described in (ii).

The supporting program must total at least 12 graduate credits.

(i) The student should choose a major area advisor as soon as possible, and certainly before starting the fourth year of study. The advisor will recommend course work in the basic material, as well as expository and research papers for individual study.

(ii) This study, including the recommended research papers, will be summarized in an expository paper (approx. 10 pages), with a substantial bibliography. This will demonstrate a knowledge of the definitions and results in the area, and indicate open problems which may form the basis for the Ph.D. thesis.

(iii) At least three months before the Oral Examination, the student must select the remaining three members of the examination committee (and obtain their consent, as well as that of the major advisor, and approval of the DGS). The examination committee will normally have at least two members in the major area (including the major advisor) and at least one in the minor/supporting program. The student must obtain the signatures of the members on an Oral Examination Form, documenting the committee's composition and the members' assent. This form can be obtained from the Graduate Program secretary in Vincent Hall 127, to whom the completed form should be returned.

(iv) The student, by this time, should be writing the expository paper, and ought to be soliciting advice from the major area committee members.

(v) One month before the examination, final copies of the expository paper should be submitted to the major area advisor and committee members. The student must obtain the signatures of the major area committee members, attesting to the general acceptability of the expository paper, on another Oral Examination Form. (This form is available from, and is to be returned to, the Graduate Program secretary, Vincent Hall 127).

(vi) The examination is scheduled at the convenience of the student and the examiners. The two forms needed to schedule the examination can be obtained from the Graduate Program secretary (VinH 127) at least two weeks prior to the examination. The Examiners Report Form for the Ph.D. Oral Examination will be sent directly to the advisor a few days prior to the examination.

(vii) On the day of the examination, the student must bring four more copies of the expository paper (typed, doubled-spaced, in final form), for the examiners. After the examination, the Examiners Report Form should be returned to the Graduate Program secretary in Vincent Hall 127.

(viii) The examination itself consists of:
a) Presentation of the expository paper (about 30 minutes), combined with questions in the major area.
(Approx. 60 minutes total)
b) Examination in the minor area, or in one of the year long sequences, constituting the supporting program.
(Approx. 30 minutes)

(ix) In conjunction with passing the PhD Preliminary Oral examination (which also counts as the MS final oral exam), the student is expected to complete, and to submit to the Graduate School, all paperwork needed to receive a Plan B MS in Mathematics. Before the Ph.D. Preliminary Oral exam, please complete the Plan B MS Degree Program form, as well as the PhD Degree Program form. Please have both approved, and also pick up the MS Graduation Packet, which can be reqested by clicking here. In the MS Graduation Packet is the "Examination Report Master's Degree and Certificate of Specialist in Education". This MS "Examination Report" form should be brought to the prelim oral exam, to get the required signatures from committee members. (NOTE: The Prelim Oral "Examiners Report Form" mentioned in Items (vi) and (vii) above must also be brought and signed.) Also in the Graduation Packet, is the "UNIVERSITY OF MINNESOTA/GRADUATE SCHOOL APPLICATION FOR DEGREE" which must be submitted. The "APPLICATION FOR DEGREE" is a form that the student can fill out; it does not require committee signatures. The MS should be granted by 31 July following the fourth year.

For suggestions on the conduct of a preliminary oral exam, please click here.

Students: In contacting a potential committee member, please do remember that many faculty are very busy, and do not suggest, by your tone, that the faculty member is required to be on your committee. We recommend that you offer to meet in person in advance to discuss your work.

The expository paper and bibliography will indicate what the student has studied. The examination may also cover foundational material and literature in the general area. The nature of the expository paper is much more than a paper surveying a year long course sequence. It is not a research paper, but may include preliminary research results and it might suggest a direction for possible future research. It should be developed in consultation with the advisor and should have the general approval of the major area examiners.

For an internal supporting program, the student will select which subject is to be covered in the examination, (from the two 8-xxx sequences, at least one of which should have been completed). For an outside minor, the rules are those of the outside department.

If a student chooses a partial internal supporting program, at least one examiner must be from the other department, in which a course sequence was taken, (if the student chooses to be examined on such a sequence).

If a student wishes to change advisors or major area, the new advisor and DGS will decide whether a new examination will be required.

The oral examination may be passed, or failed with or without the committee's consent for a retake. We generally discourage a "pass with reservations". It is hoped that the system of the expository paper and associated consultation will result in students being well prepared before they attempt the examination. The oral preliminary examination should be passed by the end the fourth year of study. No more than two attempts are permitted by the Graduate School.

Advice on completing Degree Program forms: For the MS Degree Program form, do not list courses for which you have not already registered. For the PhD Degree Program form, you must list two languages; for the MS Degree Program form, you should not list any languages. Except for these two points, and a few other obvious differences, the MS degree program form can be very similar to the PhD Degree Program form, and it will save time to prepare the two degree program forms together.

Advice for paperwork at the PhD Preliminary Oral exam: Bring both the MS final oral Examiner's Report form and the PhD prelim oral Examiner's Report form to the exam, and avoid the need to seek out the examiners later to obtain signatures.

A thesis advisor (i.e. major area advisor) should be found before the Preliminary Oral Examination. The graduate school requires 24 thesis credits for the Ph.D. degree (Math 8888); students can only register for these credits after the Ph.D. Oral Examination is passed.

This examination is a defense of the thesis. The regulations of the Graduate School require that three of the four examiners be reviewers/readers and at least one reviewer be from outside the School of Mathematics. The chairperson must not be the thesis advisor, but the advisor may be a reviewer/reader. The examination consists of a presentation of the results of the thesis(which may be public) followed by questions by the examiners (which are not public).

Specific questions on the requirements for the Ph.D. programs should be e-mailed to: the Director of Graduate Studies.

Director of Graduate Studies in Mathematics (612) 625-1306     gradprog@math.umn.edu 127 Vincent Hall 206 Church St. S.E. Minneapolis, MN 55455 USA

www@math.umn.edu URL http://www.math.umn.edu/grad/phdprogrequire.html The University of Minnesota is an equal opportunity educator and employer. © 2008, The Regents of the University of Minnesota

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