### Comprehensive Examination

The Comprehensive Examination is taken by all PhD graduate students, and by all non-Thesis MS graduate students, in mathematics. It consists of three sub-tests, each of three hours duration. These sub-tests cover the material commonly found in the following one-semester graduate courses in Real Analysis (7311), Algebra I (7210), and Topology I (7510). Passing these three sub-tests at the MS-Qualifying Level constitutes the major portion of passing the Final Exam for the non-Thesis MS. Passing each of the three sub-tests at the PhD Qualifying level completes the examination component of PhD Qualifying. A PhD student is encouraged to take the Comprehensive Exam as early as feasible, but must pass it no later than the beginning of the fourth regular semester of study, except by permission of the Graduate Committee. Meeting this requirement is part of the definition of satisfactory progress---a requirement for holding a graduate assistantship or fellowship. The Comprehensive Exams are scheduled twice per year: normally the week before the beginning of the fall semester, and the week before the beginning of the spring semester.

There are syllabi and sample problems available for Comprehensive Examinations. You can download them in the formats listed below. All three Comprehensive Exam sub-tests will be written according to the following policy.

### Comprehensive Exam Composition

At least 50% of the credit on each test will come from the test problem banks shown below. The sub-tests last for 3 hours. There will normally be approximately 6 to 8 problems offered, and students will typically need to turn in approximately 5 of these.

1999 Algebra Syllabus and Problems: tex, dvi, ps, and pdf.

The Algebra Syllabus for both Core-1 and Core-2 (Math 7210 and 7211 respectively) has been revised in May 2007. The new syllabus does not yet have a revised problem bank for exams, but the revised list of topics is here: Algebra Syllabus 2007.

Analysis Syllabus and Problems, revised August 12, 2011.

Topology Syllabus and Problems, revised Jan. 2010: pdf

### Past Comprehensive Exams.

## MS Final Examination

The MS Final Examination for the non-thesis MS is offered once each regular semester, at a time and date to be announced by the Department. MS Candidates must appear for the Final Examination for the MS. At this examination the Examining Sub-Committee of the Graduate Committee will review the results of the written Comprehensive Exams with the student, who will be questioned and advised. For the thesis-based MS, the Final Examination is conducted by the student's Advisory Committee and consists mainly of a thesis defense.

## General Examination

The General Examination, an oral exam for PhD students, tests a broad knowledge of mathematics plus sufficient knowledge in a specific field to begin dissertation research. The Graduate School generally assumes that those courses numbered lower than 7999 on the Program of Study form must be completed prior to the General Examination. It may be necessary for you to submit a Change of Program of Study form in order to take the General Examination. The specific content of the General Examination is established by the individual student's Advisory Committee. At the General Examination, Advisory Committees ascertain whether students have deficiencies in their broad mathematical training. The committee may recommend that these deficiencies be corrected. The General Exam should be completed before dissertation research is started; it is normally taken no later than the spring semester of the third year of study.

The Graduate School's policies state that the General Examination should be regarded as the culmination of a student's program in course work. In most cases, the remaining time spent in obtaining the degree is to be devoted to concentrated work on the dissertation and preparation for the Final Examination.

The Advisory Committee will consist of five mathematics graduate faculty members, with the chair of the committee being the dissertation advisor. The General Exam Committee membership must represent at least three of the following four areas of departmental interest: algebra, analysis, applied mathematics & combinatorics, and topology & geometry. A faculty member may serve as a representative of only one of the four listed areas on a single examination. In addition, the Dean of the Graduate School appoints a member of the graduate faculty to serve on doctoral general and final examination committees. This individual represents the Dean and the entire graduate faculty. The Dean's Representative is a full voting members of the committee, with all the rights and responsibilities of the other committee members.

## Foreign Language Examination

The Department of Mathematics requires that PhD candidates have the ability to read mathematical texts in one of these three languages: French, German, or Russian.

This knowledge can be demonstrated in any of the following ways:

- passing an examination administered by the Department of Mathematics (The departmental foreign language exam is a one-hour written exam in which the student is asked to translate, with the aid of a dictionary, a few pages of mathematical literature in one of the above languages.);
- passing either French 1020 or German 4005 (These courses are designed to develop a reading knowledge of technical literature. The Department requires only a grade of 'Pass' if one of these two courses is used to satisfy the requirement, so please be sure to sign up within the Graduate School's posted calendar limitations for "Pass/Fail" grading!
- taking at least 13 semester hours in the language, with a grade of at least B and with the last course being taken within the last five years.

Students whose native language is French, German, or Russian satisfy the requirement in that language automatically. Every student must satisfy the foreign language requirement before taking the General Exam.

## Final Examination for the PhD

The oral Final Examination for PhD candidates is primarily a doctoral dissertation defense. The student delivers a lecture presenting the original results from the dissertation for the Advisory Committee's approval. The committee members ask questions about this work and related matters. In the case of final examinations, it is the responsibility of the student to ensure that the dean's representatives receive copies of dissertations as soon as possible, and no later than two weeks before the date of the examination.