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Preliminary Exams

To attain the level of doctoral candidacy and begin research, students admitted to the PhD or DA program must first pass a doctoral preliminary examination consisting of two written exams, a minor sequence, and an oral examination (Statistics concentrators only). This process helps to determine the student's readiness to undertake thesis research.

Students who have completed 64 hours of graduate coursework and have passed the doctoral preliminary examination are formally recommended to the Graduate College as doctoral candidates. Students should consult the Graduate Handbook  for additional details and specific deadlines for completion.

Preliminary Examination FAQ

When are the written prelims offered?

Prelims are offered once each year at the end of the Spring semester, during the 2 weeks after final exams. The date and time of each exam is arranged by the examiner(s) in consultation with the students pre-registered for the exam. Students are allowed 3 hours to complete the exam.

How do I register for a prelim?

Students should register online for each exam by early April. Students who register for a prelim, then decide not to take the exam must notify the examiner at least 10 days prior to the exam. Failure to do so will result in a failing score on the exam.

Which prelims must I take?

The prelims are listed below by subject area. Students must complete two different written prelims and select a minor sequence from a third prelim area. Students should choose their prelims in consultation with their faculty mentor.

What topics are covered in the prelim?

By default, each prelim is based on two courses given in the academic year in which the prelim is taken. Normally the instructor for those courses sets the prelim, which is based on the topics covered in the course that year. Accommodations will be made, where reasonable, for students who wish to take the prelim based on other previously taught courses listed in that prelim.

How can I prepare for the exams?

Sample exams are posted on the website for review. Check the Exam Topics link to see if a prelim syllabus is available.

What is the minor sequence?

The minor sequence consists of two 500-level courses that lead to a written prelim, without overlapping an area in which a written prelim is taken. The minor sequence requirement can be satisfied either by passing a written prelim in a third area OR by completing the sequence of courses and earning grades that convert to a qualifying numerical score.

What is a qualifying score on the doctoral prelim exam?

Scores assigned for the written prelim exams are: 1 (best), 2, 3, and 5 (fail). Students may retake a written prelim. Although scores of 2 and 3 are passing scores, the sum of the scores on the two written prelims cannot exceed 4. To be eligible for doctoral candidacy, the combined scores of the two written prelims along with the numerical score earned in the minor sequence cannot exceed 6; in addition, Statistics students must pass an Oral Examination.

When do I take the Oral prelim exam (Statistics students only)?

Shortly after completing the written prelims and minor sequence, Statistics students should contact their advisor to plan for their Oral Prelim exam. Students must submit the Committee Recommendation form at least 30 days prior to their planned Oral Prelim exam date. Specific deadlines for nomination to doctoral candidacy are outlined in the Graduate Handbook.

Prelim Exams

The written prelims are based on a sequence of graduate courses identified below. The sum of the scores on the two written prelims must not exceed 4; combined with the minor sequence, the sum cannot exceed 6. Students should consult the Graduate Handbook for additional details.

Note: All statistics PhD students are required to pass the Probability and Statistics prelim. Students who select two written prelims in Mathematical Computer Science must select a minor sequence in another area (Math or Stat).

Written Prelims and their associated courses:


  • MATH 516 (Second Course in Abstract Algebra I)
  • MATH 517 (Second Course in Abstract Algebra II)


  • MATH 533 (Real Analysis I)
  • MATH 535 (Complex Analysis I)

Geometry and Topology

  • MATH 547 (Algebraic Topology I)
  • MATH 549 (Differentiable Manifolds I)


  • MATH 502 (Metamathematics I) plus one of the following
  • MATH 504 (Set Theory I)
  • MATH 506 (Model Theory I)
  • MATH 511 (Descriptive Set Theory)

Differential Equations

  • MATH 576 (Classical Methods of Partial Differential Equations)
  • MATH 585 (Ordinary Differential Equations)

Methods in Applied Analysis

  • MATH 539 (Functional Analysis I)
  • MCS 571 (Numerical Analysis of Partial Differential Equations)

Number Theory

  • Math 514 (Number Theory I)
  • MATH 515 (Number Theory II)

Combinatorics: Two of the following courses*

  • MCS 521 (Combinatorial Optimization)
  • MCS 582 (The Probabilistic Method)
  • MCS 583 (Extremal Combinatorics)
  • MCS 591 (Advanced Topics in Combinatorial Theory)

Algorithms and Complexity: Two of the following courses*

  • MCS 501 (Computer Algorithms II)
  • MCS 521 (Combinatorial Optimization)
  • MCS 541 (Computational Complexity)
  • MCS 548 (Mathematical Theory of Artificial Intelligence)
  • MCS 549 (Mathematical Foundations of Data Science)
  • MCS 590 (Advanced Topics in Computer Science)

Computational Science: Two of the following courses*

  • MCS 507 (Mathematical, Statistical and Scientific Software)
  • MCS 563 (Analytic Symbolic Computation)
  • MCS 571 (Numerical Analysis for Partial Differential Equations)
  • MCS 572 (Introduction to Supercomputing)

Probability and Statistics

  • (This prelim is required for all Statistics PhD students)
  • Stat 501 (Probability Theory I)
  • Stat 511 (Advanced Statistical Theory I)

Linear Inference, Sampling, and Design

  • Stat 521 (Linear Statistical Inference) plus one of the following:
  • Stat 522 (Multivariate Statistical Analysis)
  • Stat 531 (Sampling Theory I)
  • Stat 535 (Optimal Design Theory I)

Mathematics Education

  • Only Doctor of Arts students may take this prelim

History of Mathematics

  • Only Doctor of Arts students may take this prelim

*Note that the same course may not be counted towards two prelims.