# Comprehensive Examination

### Upcoming offerings

The next Comprehensive Exams in Mathematics are scheduled as follows:

• Algebra - May 9, 2023 (Tuesday) at 9 AM in MC 107.
• Analysis - May 16, 2023 (Tuesday) at 9 AM in MC 107.

The next Comprehensive Exams in Applied Mathematics are scheduled for:

• Part 1 - May 22, 2023 (Monday) at 9 AM in MC 204.
• Part 2 - May 25, 2022 (Thursday) at 9 AM in MC 204.

The next round of Departmental Presentations will take place on Thursday at 3:30.

### 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:

 Enter program First attempt by Pass by Sep 2022 May 2023 Oct 2023 Jan or May 2023 Oct 2023 May 2024 Sep 2023 May 2024 Oct 2024

### 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. Part 2 consists of a single 3-hour written exam covering three areas: Partial Differential Equations, Numerical Analysis, and the candidate's research area.

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.

### Syllbus 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 2502A/B: Differential calculus of functions of several variables: level curves and surfaces; limits; continuity; partial derivatives; total differentials; Jacobian matrix; chain rule; implicit functions; inverse functions; curvilinear coordinates; derivatives; the Laplacian; Taylor Series; extrema; Lagrange multipliers; vector and scalar fields; divergence and curl.
• Calculus 2503A/B: Integral calculus of functions of several variables: multiple integrals; Leibnitz' rule; arc length; surface area; Green's theorem; independence of path; simply connected and multiply connected domains; three dimensional theory and applications; divergence theorem; Stokes' theorem.

#### Linear Algebra

• Math 1600A/B: Properties and applications of vectors; matrix algebra; solving systems of linear equations; determinants; vector spaces; orthogonality; eigenvalues and eigenvectors.
• AM 2811A/B: Vector space examples. Inner products, orthogonal sets including Legendre polynomials, trigonometric functions, wavelets. Projections, least squares, normal equations, Fourier approximations. Eigenvalue problems, diagonalization, defective matrices. Coupled difference and differential equations; applications such as predator-prey, business competition, coupled oscillators. Singular value decomposition, image approximations. Linear transformations, graphics.

#### 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 2814F/G: Introduction to numerical analysis; polynomial interpolation, numerical integration, matrix computations, linear systems, nonlinear equations and optimization, the initial value problem. Assignments using a computer and the software package, Matlab, are an important component of this course.

### Part 2

Part 2 consists of three sections.

The first section treats the material covered in the core PhD course, AM9505A/B, a graduate course in Partial Differential Equations.  Topics include: transform methods for PDEs; kinematic waves; effects of nonlinearity; asymptotic analysis; domain decomposition methods. The first section is mandatory for all students.

The second section treats material covered in the core PhD course, AM9561A/B, Numerical Analysis.  Topics include: IEEE floating point standard 754; forward and backward error analysis; numerical linear algebra; complexity of numerical algorithms; numerical solution of ordinary and partial differential equations; spectral methods. The second section is mandatory for all students.

The third section consists of questions from specific research areas that reflect the students writing the examination in a particular year.  The material covered in each of these sections is closely based on graduate course(s) offered in that research area in the preceding year.  Students will have some degree of choice in which questions to answer in this section. For example, the 2021 Applied Math comprehensive examination included two questions each from Dynamical Systems, Materials Sciences, Mathematical Biology and Neuroscience.  Students were required to answer two of eight questions in this section.