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Ph.D. Comprehensive Examinations

 A. Ph.D. Qualifying And Comprehensive Examinations

All Ph.D. students must pass a qualifying exam, which is a written exam covering basic undergraduate material. Students must select one exam composed of a core section plus a specialized section.

Core (linear algebra, complex analysis, real analysis)

Based on material from the Mathematics courses 2R, 3A, 3X.

Sample Topics

  • Abstract vector spaces. Linear transformations, inner product spaces. spectral theorems, orthogonal bases.
  • Sequences of real numbers; supremum, continuity, Riemann integral, differentiation, sequences and series of functions, uniform continuity and uniform convergence.
  • Analytic functions, Cauchy’s theorem, Cauchy’s integral formula, residues, zeroes of analytic functions; Laurent series, the maximum principle.

Applied Mathematics Option (ordinary and partial differential equations)

Based on material from the Mathematics courses 3F, 3FF.

Sample Topics

  • Systems of ordinary differential equations, autonomous systems in the plane, phase portraits, linear systems, stability, Lyapunov’s method, Poincare-Bendixson theorem, applications.
  • First order equations, well-posedness, characteristics, wave equation, heat equation, Laplace equation, boundary conditions, Fourier series, applications.

Sample Exams (Applied Math – Preliminary)

Pure Mathematics Option (groups and rings)

Based on material from the Mathematics courses 3GR, 4GR.

Sample Topics

  • Group theory, Sylow theorems and structure of finitely generated Abelian groups, applications of group theory.
  • Ring and module theory, principal ideal domains, unique factorization domains, Euclidean rings, field theory and Galois theory.

Sample Exams (Pure Math – Preliminary)

Probability and Statistics Option (basic probability and statistics)

Based on material from the Statistics courses 2D, 3D.

Sample Topics

  • Combinatorics, independence, conditioning; Poisson-process; discrete and continuous distributions with statistical applications; expectation, transformations moment-generating functions; introduction to statistical inference.
  • Sampling distributions, limiting distributions; maximum likelihood methods; sufficiency and its statistical inference implications; pivotal quantities; interval estimation; tests of hypotheses, optimality.

Sample Exams (Probability & Statistics – Preliminary)

Students who do not pass the qualifying exam on their first attempt are permitted a second attempt at the same exam. Students at the MSc level are allowed unlimited attempts at the exam.

B. Comprehensive Exam

All PhD students must pass an oral examination on subject matter related to their research area. Students who do not pass the oral exam on their first attempt are permitted a second attempt at the same exam.

Purpose of the Exam

This exam is intended to check that the student has sufficient knowledge of the general area of their proposed research to undertake original research and produce publishable results.

Examining Committee

The examining committee consists of three faculty members, including the supervisor, who is the chair of the examining committee. The examining committee agrees on a written description of the topics to be covered by the exam, and communicates this written description to the student at least four weeks in advance of the exam. The written description should include some suggested references (published papers and/or monographs).


The oral exam normally lasts about one hour, and in no case longer than two hours. The format is to be determined by the candidate and their examination committee; two commonly used formats are:

Advanced topics

An oral examination by the examining committee on advanced topics in the student’s general research area. The examination topics should not be limited to graduate courses the student has taken.

Research proposal

A 15-minute oral presentation by student, followed by questions from the examining committee. The student should explain what contribution their proposed research would make to the existing body of knowledge. The student should provide the committee with a 10-page written summary of their presentation, including a literature survey and a short research proposal, at least one week before the exam.