Department of Mathematics

The University of Western Ontario

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• Application Procedure
• Degree Requirements
• Comprehensive Examination
• Members of the Graduate Faculty
• Graduate Courses
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Comprehensive Examination for the Ph.D. Degree

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 the material in the third and fourth years of the undergraduate program (see below). The aim of Part I is to ascertain that the candidate has a good overall understanding and working knowledge of the core of the undergraduate curriculum.

The exams are offered in September 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:

Students who entered our program before April 2011 must successfully complete the exams within twenty-one months of the start of their Ph.D. program.

Copies of the three most recent exams are available from Janet Williams. Students are encouraged to use these to prepare for the exam before starting the Ph.D. program. Students who have received offers of admission to the Ph.D. program in Mathematics may receive copies of these sample examinations upon request.

Part II consists of completion of a project (typically a seminar 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 test his/her familiarity with the background of the intended field of study. This project is assigned within a month 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

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

Part II

Mathematics 9993: Presentation

After a Ph.D. candidate has successfully completed Part I of the qualifying exam, he/she shall be required to perform satisfactorily in a less formal qualifying procedure dealing with graduate (9000-level or higher) mathematical material. This stage is intended to test a student's potential to undertake mathematical research.

An examining committee of three faculty members shall be appointed by the Graduate Affairs Committee and usually consists of the advisory committee.  Within one month of completing Part I of the Comprehensive Examination, the examining committee shall assign a project to be accomplished by the candidate. The completed project is to be submitted to the examining committee within six months of being assigned.

The project will typically be in the form of a lecture on a topic related to the candidate's likely area of study.  For example, it may consist of a review of one or more published articles.  After the lecture and audience questions, the audience is asked to leave and the examining committee meets privately with the candidate to ask additional questions.

The examining committee shall decide whether or not the candidate's project has been satisfactorily completed, and shall report its decision to the Graduate Affairs Committee and the candidate. In the event that the project is not deemed satisfactory, the candidate shall be required to withdraw from the program.