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Comprehensive Examination

Comprehensive Exams in Mathematics

All Mathematics Ph.D. students are required to pass a qualifying/comprehensive examination. This examination consists of two written exams: (1) Algebra and (2) Analysis, with the syllabus for each being based on undergraduate and M.Sc.-level material (see below). The aim of this exam 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 Ph.D. 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:

Syllabus 

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 - Mathematics and Applied Mathematics

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

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 - Mathematics and Applied Mathematics

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

Comprehensive Exams in Applied Mathematics

The Comprehensive Exams in Applied Mathematics are in two parts.

Part 1 consists of a single 3-hour written exam covering four areas: Calculus, Linear Algebra, Ordinary Differential Equations, and Numerical Methods. Its syllabus is based on the corresponding undergraduate courses offered by the Department of Mathematics (see below). A score of 80% on each of these four areas is required to pass part 1.

Part 2 consists of a single 3-hour written exam covering recent graduate courses taken by the student. An overall score of 60% is required to pass part 2.

Comprehensive Exams are offered in May of each year, roughly one week apart. Students are required to pass both parts during their first year to remain in the program.

Syllabus for Part 1

Calculus

• Calculus 1500A/B: Basic set theory and an introduction to mathematical rigour. The precise definition of limit. Derivatives of exponential, logarithmic, rational trigonometric functions. L'Hospital's rule. The definite integral. Fundamental theorem of Calculus. Integration by substitution. Applications.
• Calculus 1501A/B: Techniques of integration; The Mean Value Theorem and its consequences; series, Taylor series with applications; parametric and polar curves with applications; first order linear and separable differential equations with applications.
• Calculus 2500A/B: A third course in the calculus series. Limits and continuity in two or three variables. Partial derivatives and multivariate integrals. Line integrals and parameterized curves. Parameterized surfaces and surface integrals. Vector fields and Green’s theorem. Stokes’s and Gauss’s theorems. This is a computational course without proofs.

Linear Algebra

• Math 1600A/B: Properties and applications of vectors; matrix algebra; solving systems of linear equations; determinants; vector spaces; orthogonality; eigenvalues and eigenvectors.
• Math 2700A/B: The Gram-Schmidt process; similarity and orthogonal diagonalization; abstract vector spaces and linear transformations over arbitrary fields; change of basis; inner product spaces; norms and distance; least squares and Fourier approximation; singular value decomposition. Applications to differential equations and other topics will be emphasized throughout the course.

Ordinary Differential Equations

• AM 2402A/B: Introduction to first order differential equations, linear second and higher order differential equations with applications, complex numbers including Euler's formula, series solutions, Bessel and Legendre equations, existence and uniqueness, introduction to systems of linear differential equations.

Numerical Methods

• AM 2814A/B: Introduction to mathematical computation. Approximate solution of linear systems, nonlinear systems, and the initial value problem for ODEs. Matrix computations and practical computational complexity. Optimization. Contemporary applications of mathematical computation. Assignments using a computer and mathematical software are an important component of this course.

Part 2