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Graduate Examinations

Comprehensive Examination

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The Comprehensive Examination is taken by all non-Thesis MS and PhD graduate students in mathematics. It consists of four sub-tests. The first three sub-tests cover the material commonly found in the following one-semester graduate courses in Analysis (7311), Algebra (7200), and Topology (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. The fourth sub-test of the Comprehensive Exam is based on the student's choice of one of the following six courses: Algebra I (7210), Measure and Integration (7312), Differential Equations (7320), Graph Theory (7400), Topology II (7512), or Differential Geometry (7550). Passing this fourth sub-test together with passing the first three sub-tests at the PhD Qualifying Level constitutes passing the PhD Qualifying Examination. 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. Beginning in August 2005, all three Core-1 Comprehensive Exam sub-tests will be written according to the following policy.

Core-1 Exam Composition

  • At least 50% of the credit on each test will come from the test problem bank. The Core-1 sub-tests last for 2 and 1/2 hours. There will normally be approximately 6 or 7 problems offered, and students will typically need to turn in 4 or 5 of these.


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

Analysis Syllabus and Problems, revised July 15, 2005: pdf.

Topology Syllabus and Problems, revised June 2005: tex, dvi, and pdf.

Past Comprehensive Exams.


Ordinary Differential Equations Syllabus and Problems: tex, dvi, ps, and 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.

MNS Final Examination

The final examination for the MNS degree will be of the approximate length of the comprehensive examination, but it will be individually prepared to reflect the actual courses taken by an MNS student, including questions prepared by the minor department, plus questions from the 6 hours of 7000-level mathematics courses.

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 specific content of the General Examination is established by the individual student's Examining Committee, which is appointed by the Department of Mathematics and the Graduate School. At the General Examination,

Examining 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.

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:
(1) 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.);
(2) 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! );
(3) 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 Examining Committee's approval. The committee members ask questions about this work and related matters.

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