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Comprehensive Examination

Comprehensive Examination for the Ph.D. Degree

The next Part I comprehensive exams are scheduled as follows:

Algebra - May 12, 2020POSTPONED; DETAILS TBA

Analysis - May 5, 2020POSTPONED; DETAILS TBA

All Mathematics Ph.D. students are required to pass a qualifying/comprehensive examination. This examination is in two parts. Part I consists of two written exams: (1) Algebra and (2) Analysis, with the syllabus for each being based on undergraduate and MSc-level material (see below). The aim of Part I is to ascertain that the candidate has a good overall understanding and working knowledge of the mathematics that will form a basis for further study in the PhD program.

The exams are offered in September/October and May each year.  The exams should be attempted the first or second time they are offered and must be successfully completed by the third time they are offered.  At most two attempts at each exam are permitted. This chart explains the timing:

Enter program   First attempt by   Pass by
Sep 2019 Spring 2020 Fall 2020
Jan or May 2020 Fall 2020 May 2021
Sep 2020 May 2021 Oct 2021

Part II consists of completion of a written paper and oral presentation assigned by the candidate's advisory committee. The objective here is to ascertain that the candidate has the potential to undertake research and to write down results. This is to test his/her familiarity with the background of the intended field of study. This project is assigned within two months of completing Part I and is to be completed within six months of being assigned. The project is judged on a pass/fail basis by a three-person examining committee. See Mathematics 9993 below for more details.

Syllabus for Part I

Mathematics 9991: Algebra

  1. Linear Algebra: Linear equations and matrices, rank, vector spaces, linear transformations, determinants, characteristic and minimal polynomials, eigenvalues, canonical forms, bilinear forms, duality, orthogonal bases, spectral theorems. 

    Suggested references:
    • Linear Algebra, by K. Hoffman and R. Kunze
    • Algebra, by S. Lang (2nd ed., 1984) 
  2. Groups: Subgroups, normal subgroups and quotient groups, homomorphisms, group actions, Sylow theorems, abelian groups. 

    Rings and modules: Homomorphisms, ideals and quotient rings, integral domains and fields of quotients, unique factorization domains, principal ideal domains, Euclidean rings, polynomials, fundamental theorem of modules over a PID. 

    Fields: Algebraic extensions, algebraic closure, separability, finite fields, Galois extensions, roots of unity, norm and trace. 

    Suggested references:
    • Abstract Algebra, by Dummit and Foote
    • Algebra, by Larry C. Grove

Sample past exams: Fall 2019Spring 2019, Fall 2018, Spring 2018, Fall 2017Spring 2017, Fall 2016Spring 2016, Fall 2015Spring 2015, Fall 2014Spring 2014, Fall 2013Spring 2013, Fall 2012Spring 2012, Fall 2011Spring 2011, Fall 2010Spring 2010

An even larger collection of past exams has been typed up and cross referenced to help students prepare. [pdf]

Mathematics 9992: Analysis

  1. Real Analysis: Real and complex number systems, Euclidean spaces, basic topology of metric spaces (including compactness, connectedness, completeness, separability), sequences and series of complex numbers, continuity, uniform continuity, differential of a real valued function of a real variable, mean value theorems and Taylor's theorem, Riemann-Stieltjes integral, functions of bounded variation, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem. 

    Suggested references:
    • Principles of Mathematical Analysis, by W. Rudin 
  2. Complex Analysis: Algebra of complex numbers, conjugation, absolute value, the extended plane, holomorphic functions, Cauchy-Riemann equations, elementary functions, logarithms, argument and roots, integration on paths, power series, Cauchy's theorem, Cauchy's integral formulae, Cauchy's estimates, Morera's theorem, Liouville's theorem, Fundamental theorem of Algebra, Identity theorem, Maximum Modulus theorem, Taylor and Laurent Series, classification of singularities, residue theory, Mobius transformations, Open Mapping theorem, Schwarz's lemma, argument principle and Rouché's theorem. 

    Suggested references:
    • Functions of One Complex Variable, by John B. Conway
    • Complex Analysis, by T. W. Gamelin
  3. Advanced Calculus: Differential calculus of functions of several variables; implicit and inverse function theorem; multiple integrals; line integrals; independence of path; Grad, div and curl; Green's theorem; Taylor's theorem with remainder, ordinary differential equations. 

    Suggested references:
    • Advanced Calculus, by W. Kaplan
    • Advanced Calculus, by G. B. Folland

Sample past exams: Fall 2019Spring 2019, Fall 2018, Spring 2018, Fall 2017Spring 2017, Fall 2016Spring 2016, Fall 2015Spring 2015, Fall 2014Spring 2014, Fall 2013Spring 2013, Fall 2012Spring 2012, Fall 2011Spring 2011, Fall 2010Spring 2010

An even larger collection of past exams has been typed up and cross referenced to help students prepare. [pdf]

Part II

Mathematics 9993: Presentation

After a Ph.D. candidate has successfully completed Part I of the Comprehensive Examination, he/she shall be required to prepare a review paper describing background material for the intended research topic and to defend it orally. This project may later become a part of the student's thesis. This stage is intended to test the student's potential to undertake mathematical research and to write down results. The submitted paper shall typically be between 10 and 15 pages in length and compile results from several different sources together with proofs. The presentation of the material shall be coherent and sufficiently detailed so that the members of the examining committee can evaluate its correctness without consulting special literature.

The examining committee will contain three faculty members appointed by the Graduate Affairs Committee and usually consists of the advisory committee. Within two months of the completion of Part I of the Comprehensive Examination, the examining committee shall give signed approval of a topic and a list of suggested sources. The oral presentation of the project will take place within six months of being assigned, and a final version of the paper will be submitted to the examining committee at least two weeks before the presentation.

After the presentation and audience questions, the audience is asked to leave and the examining committee meets privately with the candidate to ask additional questions. Then the examining committee meets without the candidate, decides separately whether the paper and the presentation have been satisfactorily completed, and reports its decision to the candidate and the Graduate Affairs Committee. In the event that one or both of the paper and presentation is not deemed satisfactory by a majority of the committee, the candidate may attempt the failed portion(s) a second time, within two months of the first attempt. If the candidate fails again, he/she is required to withdraw from the program.