First stage comprehensive exams

The first stage comprehensive exams are written exams that are offered once a year in the Spring term and must normally be attempted within four terms of the first registration. Students starting their PhD in the Winter term are allowed to attempt the exam within five terms of the first registration. The student must write one exam from two of the following three categories:

Category one Category two Category three
  • Combinatorial enumeration
  • Graph theory
  • Continuous optimization
  • Discrete optimization
  • Cryptography
  • Quantum computing

The choice of the exams is made by the student in consultation with their supervisor.

The first-stage comprehensive examination requirement is satisfied by passing both exams.  If a student does not pass, the student may be asked to fulfil other requirements or to retake both exams. Students retaking the exams are required to do so in the Fall term immediately following the first attempt.  Students who fail a retake will be required withdraw from the program.

The comprehensive exam is intended to assess the student's grasp of the core background of their primary research area and a breadth area, as well as the student's ability to write formal mathematics.

Students are encouraged to refer to the GSPA guidelines for details, espically if accommodations are warranted for health or other reasons.

The syllabus for the exams is listed below.

Combinatorial enumeration suggested references

Graph theory suggested references

  • Graph Theory by R. Diestel, Second Edition Graduate Texts in Mathematics 173 Springer, New York, 2000.
    • Matchings (2.1, 2.2)
    • Connectivity (3.1, 3.2, 3.3)
    • Planarity (4.1-4.4)
    • Colourings (5.1, 5.2, 5.3)
    • Flows (6.3, 6.4, 6.6)
    • Turán's Theorem (7.1)
    • Ramsey's Theorem (9.1)
    • Hamiltonicity (10.1, 10.2)
    • PLUS the problems associated with the above sections
  • Algebraic Graph Theory by C. Godsil and G. Royle, Graduate Texts in Mathematics 207, Springer, New York, 2001.
    • Transitivity (3)
    • Matrices and Graphs (8.1-8.8)
    • Interlacing (9.1, 9.2, 9.3)
    • Strongly Regular Graphs (10.1-10.5)
  • Introduction to Graph Theory by D.B. West, Prentice Hall, Upper Saddle River, NJ 1996.
    • Connectivity (p.136-138, 4.2)
    • Ramsey's Theorem (8.3)
  • Introduction to Graph Theory by R.J. Wilson, Third Edition Longman, Essex, 1985.

        Directed Graphs (7)

Continuous optimization suggested references

Discrete optimization suggested references

Quantum computing suggested references

Cryptography suggested references

Old first stage comprehensive exams