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 




The choice of the exams is made by the student in consultation with their supervisor.
The firststage 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 (4.14.4)
 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 (8.18.8)
 Interlacing (9.1, 9.2, 9.3)
 Strongly Regular Graphs (10.110.5)
 Introduction to Graph Theory by D.B. West, Prentice Hall, Upper Saddle River, NJ 1996.
 Connectivity (p.136138, 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
Combinatorial enumeration exams
 Combinatorial enumeration exam 2020 (PDF)
 Combinatorial enumeration exam 2019 (PDF)
 Combinatorial enumeration exam 2018 (PDF)
 Combinatorial enumeration exam 2017 (PDF)
 Combinatorial enumeration exam 2016 (PDF)
 Combinatorial enumeration exam 2012 (PDF)
 Combinatorial enumeration exam 2009 (PDF)
 Combinatorial enumeration exam 2008 (PDF)
 Combinatorial enumeration exam 2006 (PDF)
 Combinatorial enumeration exam 2005 (PDF)
 Combinatorial enumeration exam 2004 (PDF)
 Combinatorial enumeration exam 2003 (PDF)
 Combinatorial enumeration exam 2002 (PDF)
 Combinatorial enumeration exam 2000 (PDF)
Graph theory exams
 Graph theory exam 2020 (PDF)
 Graph theory exam 2019 (PDF)
 Graph theory exam 2018 (PDF)
 Graph theory exam 2017 (PDF)
 Graph theory exam 2016 (PDF)
 Graph theory exam 2015 (PDF)
 Graph theory exam 2014 (PDF)
 Graph theory exam 2013 (PDF)
 Graph theory exam 2012 (PDF)
 Graph theory exam 2011 (PDF)
 Graph theory exam 2009 (PDF)
 Graph theory exam 2008 (PDF)
 Graph theory exam 2007 (PDF)
 Graph theory exam 2006 (PDF)
 Graph theory exam 2005 (PDF)
 Graph theory exam 2004 (PDF)
 Graph theory exam 2003 (PDF)
 Graph theory exam 2001 (PDF)
Continuous optimization exams
 Continuous optimization exam 2019 (PDF)
 Continuous optimization 2018 (PDF)
 Continuous optimization exam 2017 (PDF)
 Continuous optimization exam 2016 (PDF)
 Continuous optimization exam 2014 (PDF)
 Continuous optimization exam 2012 (PDF)
 Continuous optimization exam 2009 (PDF)
 Continuous optimization exam 2008 (PDF)
 Continuous optimization exam 2006 (PDF)
 Continuous optimization exam 2003 (PDF)
 Continuous optimization exam 2002 (PDF)
 Continuous Optimization exam 2001
 Continuous optimization exam 2000 (PDF)
Discrete optimization exams
 Discrete optimization exam 2018 (PDF)
 Discrete optimization exam 2017 (PDF)
 Discrete optimization exam 2016 (PDF)
 Discrete optimization exam 2015 (PDF)
 Discrete optimization exam 2013 (PDF)
 Discrete optimization exam 2012 (PDF)
 Discrete optimization exam 2011 (PDF)
 Discrete optimization exam 2009 (PDF)
 Discrete optimization exam 2008 (PDF)
 Discrete optimization exam 2006 (PDF)
 Discrete optimization exam 2005 (PDF)
 Discrete optimization exam 2004 (PDF)
 Discrete optimization exam 2002 (PDF)
 Discrete optimization exam 2001 (PDF)
Crytography exams
 Crytography exam 2020 (PDF)
 Crytography exam 2019 (PDF)
 Crytography exam 2018 (PDF)
 Crytography exam 2017 (PDF)
 Crytography exam 2016 (PDF)
 Crytography exam 2013 (PDF)
 Crytography exam 2011 (PDF)
 Crytography exam 2007 (PDF)
 Crytography exam 2004 (PDF)
 Crytography exam 2003 (PDF)
 Crytography exam 2002 (PDF)
Quantum computing exams
 Quantum computing exam 2020 (PDF)
 Quantum computing exam 2018 (PDF)
 Quantum computing exam 2017 (PDF)
 Quantum computing exam 2015 (PDF)
 Quantum computing exam 2012 (PDF)
 Quantum computing exam 2009 (PDF)
 Quantum computing exam 2007 (PDF)
 Quantum computing exam 2006 (PDF)
 Quantum computing exam 2005 (PDF)
 Quantum computing exam 2003 (PDF)