The Pure Maths Version and Syllabi are omitted from this document.
The Statistics Version and Syllabi are omitted from this document.
The Department of Mathematics and Statistics requires all doctoral students to pass two preliminary examinations, first the Part A and then the Part B. The Part A examination has to be passed by the end of the first year of registration at the PhD 2 level. All graduate students thinking in terms of doctoral studies at McGill are urged to familiarise themselves with the regulations governing these crucial examinations and to plan their studies taking these examinations into account.
MATH 700 — Ph.D. Preliminary Examination Part A.
General Regulations
Introduction
The Ph.D. Part A examination is a written examination consisting of two papers, which, with the exceptions noted below, must be taken and passed together within the same examination period. There are three versions: Applied Mathematics, Pure Mathematics, and Statistics. In the Applied and Pure versions the papers are called alpha and beta with Paper alpha the same for both versions; in the Statistics version they are called theory and methodology. Each graduate student will be examined in one of these versions and the choice is made by the student in consultation with the Advisory Committee of the student. Two thirds of Paper alpha covering topics in real analysis and linear algebra at the level of our first year undergraduate honours courses is traditionally referred to as the compulsory part.
Deadline to Pass the Part A Examination
This examination has strict deadlines. Only students who have successfully passed the Part A examination will be permitted to register for the September semester of the Ph.D. 3 year.
Schedule of Examination
Each graduate student is allowed only two official attempts at the Part A examination. The examination will be administered twice a year, once in May and once in August. The parts alpha and beta, or theory and methodology, are held on separate days within one week. So under usual circumstances, a student will make the first attempt in May preceding September of the Ph.D. 3 year, so that if necessary, a second attempt can be made before the September deadline, that is, in August. However students may take the examination earlier if they feel ready for it and scheduling permits. Notices of the precise time and place of the examination are posted a few weeks beforehand. A student intending to write the Part A examination must notify the Department office in writing of theit intention by March 30 for the May examination and June 30 for the August examination. Each paper will involve a 4hour sitting. The papers will be set in such a way that a strong candidate can be expected to complete the paper within 3 hours.
Exceptions
Normally, a student who fails at the first attempt will be required to retake and pass the entire examination, although in exceptional cases the committee may allow the student to take only one part . However, this permission must be sought and given before the examination. Also, in case of exceptional circumstances such as illness, the Committee of Graduate Affairs (CGA) may relax the deadline to pass the Part A examination as seems appropriate.
Free Trials
Students are allowed one free attempt at the examination. "Free" here means that in case of failure, the student still has the 2 official attempts (see Schedule of Examination), whereas success is just as valid here as in the official attempts. This attempt, however, must be made before registration in the September semester of the Ph.D. 2 year. This gives the opportunity to Master's and Ph.D. 1 students to take the examination on a variety of occasions, but also includes the possibility that a Ph.D. student admitted at Ph.D. 2 level (which is the usual case) to take a free trial in August, before registering for the September semester of this Ph.D. 2 year. Note that a student enrolled in a Master's program, who passes a free trial examination is entitled to apply for direct promotion to Ph.D. standing without getting a M.Sc. degree. In any case, a student who passes such a free try has fulfilled the Part A requirement. For these reasons we cannot stress enough the benefits for our Master's students of making a serious free attempt at the Part A examination. in anticipation of later enrolment at the Ph.D. level.
Results
The Part A Committee will recommend to the CGA the results of the examination. The result will be reported as PASS/FAIL. While the evaluation will be based on the overall performance of the student on both papers, candidates taking the Applied or Pure examinations will be expected to perform at least satisfactorily on the compulsory part of Paper alpha. There will be no partial passes but the examiners may recommend a specific additional requirement (such as grading or taking certain courses) to be fulfilled by individual students who have displayed a conspicuous weakness. The CGA will present the results of the examination to the Department and the results may be appealed to the Department by the student or by any member of the Department. However, the student will not be permitted to register, pending an appeal of a failure. A student who writes the Part A exams in May and fails the compulsory (undergraduate) part of paper alpha, but has an overall passing performance, may, at the recommendation of the Part A Subcommittee, be given the option to sit one 4hour exam on the compulsory material of paper alpha. Regarding deadlines, this option would be treated the same way as a full attempt at the Part A exams.
Applied Mathematics Version
The compulsory part of the Paper alpha examination requires students to answer 3 out of 4 questions on Single Variable Real Analysis and 3 out of 4 questions on Linear Algebra. In addition students have to answer 3 out of 4 questions on Vector Calculus, Ordinary Differential Equations and Complex Analysis.
For Paper beta (Applied Mathematics), students are required to choose 3 modules from the following list:
Discrete Mathematics,Students, in conjunction with their advisor, will select three modules. Normally this selection will be made well in advance of the examination, but in any event must be made at least 14 days before the exam is sat. Four questions will be set on each chosen module and students will be assessed on their answers to seven questions: at most three questions may be taken from each of the three selected modules. For modules based on two 500 level courses, normally two questions will be set on each half of the material.
Numerical Analysis,
Optimization,
Partial Differential Equations,
Probability,
One of the pure mathematics modules
Syllabus for Paper Alpha — Applied and Pure
The compulsory part of this examination requires students to answer 3 out of 4 questions on Single Variable Real Analysis and 3 out of 4 questions on Linear Algebra.
In addition students have to answer 3 out of 4 questions on Vector Calculus, Ordinary Differential Equations and Complex Analysis.
Single Variable Real Analysis: The properties of the natural numbers and the real numbers, infimum and supremum of sets of real numbers. Limits of sequences, monotone sequences, the Cauchy criterion. Limits of functions. Continuous functions, the intermediate value theorem, maxima and minima. Uniform continuity, monotone functions, inverse functions. Differentiation, the mean value theorem, L'Hospital's rule, Taylor's expansion with remainder term. The Riemann integral, the fundamental theorem of calculus. Sequences of functions, pointwise and uniform convergence, continuity of the limit of a sequence of functions , interchange of limit with derivative and integral. Infinite series of numbers and functions, absolute convergence. Power series, the Taylor expansion. The rigorous introduction of the exponential, logarithmic, trigonometric and inverse trigonometric functions. Reference: R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 2nd edition, John Wiley and Sons (1992). 
Linear Algebra: Matrices and systems of linear equations. Gaussian reduction, matrix operations. Vector spaces, subspaces, linear independence, spanning sets, basis, dimension. Systems of linear equations and vector spaces. Determinants, Cramer's rule. Inner product, orthogonality, orthonormal bases. Linear mappings, matrix of a linear mapping, change of basis. Kernel and image of a linear mapping, the dimension equation. Eigenvalues and eigenvectors, diagonalization, Jordan normal forms. Bilinear, quadratic and hermitian forms. Adjoint operators, selfadjoint, orthogonal and unitary operators. Diagonalization in Euclidean and unitary spaces. The spectral theorem. Reference: S. Lipschutz, Linear Algebra, Schaum's Outline Series, McGraw Hill (1991). 
Vector Calculus (Common with Applied Mathematics): The derivative of a multivariable vectorvalued function as a linear transformation. The multivariable inverse and implicit function theorems. Multiple integrals, path and surface integrals. Change of variables theorem for integrals. Calculation of areas, volumes, arclengths, momenta, etc. The integral theorems of vector analysis: Green's, Stokes' and Gauss'. Course: Honours Advanced Calculus (MATH 248)Reference: J.E. Marsden and A. Tromba, Vector Calculus, W.H. Freeman, 4th edition (1996). 
Ordinary Differential Equations: Existence and uniqueness theorem for first order systems. Basic solution techniques for first order O.D.E.'s. Second order linear equations. Series solutions, at both regular and regularsingular points. Systems of first order linear O.D.E.'s. Course:Honours Ordinary Differential Equations (MATH 325)Reference: W.E. Boyce and R.C. diPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley and Sons (1992). 
Complex Analysis: Analytic functions, the CauchyRiemann equations. Entire functions. The exponential, trigonometric and logarithmic functions as analytic functions; Euler's formula. Line integrals, the Cauchy theorem, Cauchy's integral formula. Power series representation in a disk of analyticity. Consequences: uniqueness theorem, mean value theorem, maximum modulus theorem, open mapping theorem. Morera's theorem. Liouville's theorem and the fundamental theorem of algebra. Meromorphic functions, Laurent expansions. Residue theorem and its applications. Fractional linear transformations. Course: Honours Complex Analysis (MATH 366)References:

Syllabi for Paper beta modules — Applied Mathematics
Discrete Mathematics: Graph theory: trees and cycles; matching theory; connectivity; planar graphs: Kuratowski's theorem, crossing number; graph colouring; perfect graphs; regularity lemma; graph minors; treewidth; random graphs and the probabilistic method. Combinatorics: enumerative methods; extremal set theory; systems of distinct representatives; generating functions; de Bruijn sequences; sieve methods; partially ordered sets: order ideals, lattices, Mobius inversion; permutation statistics; compositions and partitions; Stirling numbers; Polya theory; the Matrixtree theorem; Gaussian numbers; Lagrange inversion. Courses: Graph Theory and Combinatorics (MATH 350), Combinatorics (MATH 550).References:

Numerical Analysis: Direct methods for linear systems: matrix factorization, triangular matrices, QR factorization, Gaussian elimination, LU factorization and variants, conditioning and stability. Least squares problems: normal equations, QR factorization, singular value decomposition. Eigenproblems: reduction to Hessenberg or Tridiagonal form, QR algorithm, Jacobi method. Polynomial and trigonometric interpolation and approximation, applications to quadrature. Iterative methods for linear systems: Jacobi, GaussSeidel, SOR, GMRES, Conjugate Gradients, Preconditioning. Iterative methods for nonlinear systems: fixed point iteration, NewtonRaphson and variants. Numerical methods for ordinary differential equations: Linear multistep and RungeKutta methods, consistency, stability, convergence, stiffness, adaptive integration. Hamiltonian problems, symplectic integration. Boundary value problems: shooting and RungeKutta methods. Finite difference and finite element methods for elliptic partial differential equations, including consistency and convergence. Finite difference methods for hyperbolic and parabolic partial differential equations, implicit and explicit time integration. Courses: Numerical Analysis 1 (MATH 578), Numerical Differential Equations (MATH 579)References:

Optimization: Optimization in several variables, convex sets and functions, theorems of the alternative (Farkas, Gordan, DubovitskiiMilyutin, hyperplane separation theorem. Characterizations of optimality for singleobjective and multiobjective convex programs. Convexification and stabilization of nonlinear programs. First and second order optimality conditions. Elements of nonsmooth optimization. Basic nonlinear programming methods: steepest descent. Newton's, variable metric, feasible directions and penalty function methods. Introduction to optimization with functionals. Combinatorial Optimization: algorithmic and structural approaches in combinatorial optimization: matchings; shortest paths and network flows; the ellipsoid method and applications; polyhedra and total unimodulatrity; total dual integrality; network connectivity; packings and coverings of branchings, arborescences and directed cuts. Courses: Combinatorial Optimization (MATH 552), Optimization (MATH 560).References:

Partial Differential Equations: Definition of wellposedness of problems, uniqueness and regularity of solutions. Familiarity with separation of variables techniques assumed. First order PDE: First order Cauchy problems. Linear and nonlinear firstorder PDE, their solution by the method of characteristics. Conservation laws, shocks, entropy conditions. Systems of conservation laws and Riemann invariants. HamiltonJacobi equations and Legendre transforms. Complete Integrals and envelopes. Characteristics for 2nd order linear PDE. The CauchyKovalewsi theorem. Laplace's equation: fundamental solution, mean value theorems and maximum principles, representation formulae, Green's function, energy methods, weak solutions, properties of solutions. Heat equation: fundamental solution, mean value theorems, representation formulae, energy methods, weak solutions, Duhamel's principle, properties of solutions Wave equation: spherical means, method of descent, d'Alembert's solution, nonhomogenous problems, dispersive waves, shocks and rarefactions, weak solutions, properties of solutions. Transforms methods (Fourier and Laplace), scaling methods, similarity solutions, including plane waves, solitons, travelling wave solutions, series solutions. Courses: Applied Partial Differential Equations 1 (MATH 580), Applied Partial Differential Equations 2 (MATH 581)

Probability: Measure and Integration including Lebesgue measure, Probability measures and Expectation, Random Variables and Distributions, Convergence of Random Variables (almost surely, in L1, in L2, and in probability), Product measures and Fubini's Theorem, the Laws of Large Numbers, Kolmogorov's ZeroOne Law, Poisson Process, Weak Convergence, Characteristic Functions, the Central Limit Theorem, Limit Theorems in Rk, RadonNikodym Theorem, Conditional Probability, Markov Chains, Conditional Expectation, Martingales. Courses: Advanced Probability Theory 1 (MATH 587), Advanced Probability Theory 2 (MATH 589). Reference: P. Billingsley, Probability and Measure, third edition, Wiley, 1995. [Chapters 16, unstarred sections and topics.] 
Syllabi for Paper beta modules Pure Mathematics
Analysis: Metric spaces, compactness, completeness, Baire's category theorem, AscoliArzela theorem. StoneWeierstrass theorem. Measure theory and integration on general spaces. Egoroff's theorem. Lusin's theorem. Convergence theorems. Product measures and Fubini's theorem. Regular Borel measures on compact spaces, Riesz Markov theorem (Riesz representation theorem for the space of continuous functions). Lebesgue measure on ndimensional Euclidean spaces, RadonNikodym theorem. Elementary theory of Banach and Hilbert spaces: HahnBanach, uniform boundedness, open mapping theorems. L^{p} spaces. Convergence in L^{2}. Theorems of Fejer and Parseval. Convergence of Fourier series. Courses: Honours Analysis 3 (MATH 354), Honours Analysis 4 (MATH 355) and with some material from Advanced Real Analysis 1 (MATH 564), Advanced Real Analysis 2 (MATH 565).References: The first set of topics in covered by: W. Rudin, Principles of Mathematical Analysis, McGraw Hill (1976). (The AscoliArzela theorem (not so named) is on pages 156158. The Baire category theorem is in Exercises 2.30 (p. 46) and 3.22 (p. 82); the Baire category theorem as well as the AscoliArzela theorem are given explicitly in the other Rudin book (see below)). The second and third sets of topics on real analysis can be found in any of the following books. The topics in complex analysis are covered by Rudin's book listed below:

Algebra: Groups: basic concepts, groups acting on sets, stabilizers, orbits, the counting formula. Actions of a group on itself, left and right multiplication, conjugation, the Sylow theorems. Symmetric and alternating groups: cycle decomposition, conjugacy classes. Simplicity of the alternating groups. Free groups, generators and relations. Rings and modules: basic concepts, the Chinese remainder theorem for rings, applications to modular arithmetic. Integral domains and fraction fields. Factorization in integral domains: unique factorization domains, principal ideal domains, Euclidean domains. Primitive polynomials, Gauss' lemma, Factorization of integers, of polynomials, of Gaussian integers. Irreducible polynomials, Eisenstein's criterion. Commutative rings: prime and maximal ideals, polynomial rings. Localization. Integral extensions. The Hilbert Nullstellensatz. Modules: Artinian and Noetherian modules, Hilbert basis theorem. Structure of finitely generated modules over a PID, with applications to abelian groups and linear transformations. Projective and injective modules, tensor product of modules. Fields: field extensions, algebraic and transcendental elements. Finite extensions and degree. Adjunction of roots. Finite fields. Galois theory: the main theorem of Galois theory. Symmetric functions. Primitive elements. Roots of unity, the cyclotomic equation. Cyclic extensions, solvable extensions. Solvability by radicals. The computation of the Galois group of a polynomial in simple cases. Category theory: the basic language of categories, functors and natural transformations, products, coproducts, free constructions, inductive and projective limits. Courses: Honours Algebra 3 (MATH 370), Honours Algebra 4 (MATH 371), with some material from Higher Algebra 1 (MATH 570), Higher Algebra 2 (MATH 571).
For certain topics detailed below, the following books may be consulted:
The simplicity of the alternating groups is an exercise in ARTIN (p. 233); the Chinese remainder theorem for rings is not in ARTIN; both these topics are exercises in JACOBSON I. Both topics are done in detail in the other books except JACOBSON II. For Galois theory, the little HUNGERFORD is not adequate. For cyclic and solvable extensions and the computation of Galois groups of polynomials, any one of the last three books should be used. In category theory, LANG does not have natural transformations, and the big HUNGERFORD does not have inductive and projective limits; otherwise, any one of the last three books is adequate. 
Geometry and Topology: Basic point set topology, fundamental group, covering spaces, classification of surfaces. Basic facts on differentiable manifolds, differential forms, Stokes's theorem, de Rham cohomology. Courses: Geometry and Topology 1 (MATH 576), Geometry and Topology 2 (MATH 577). Note that the syllabus includes slightly more material than is taught in these courses.

Old Examinations
Old examinations are available in the library; they go back until 1984. Some of the more recent examinations are available online. Although the syllabus and format of the examinations have changed, they still give an idea of what to expect. The more recent examinations are closer to the present syllabus. Note that starting in May 2007 there have been major changes in the applied mathematics version of the exam and minor changes in the pure mathematics version.
MATH 701 — Ph.D Preliminary Examination Part B.
 The purpose of the Part B Examination is to ensure that a student getting a Ph.D. degree from our Department be well versed in at least two areas of Mathematics and Statistics which go beyond the basic knowledge required for the Part A Examination.
 A Standing Committee on the Part B Examination shall be set up by the Graduate Affairs Committee and shall consist of a Chair and two other members. This subcommittee will be responsible for approving the choice of examination topics for each candidate, for designating the members of the student's Examining Committee and for arranging the place, date and time of the examination. The Part B Subcommittee will assume total responsibility for informing students of deadlines both impending and expired concerning the Part B Examination.
 The student, in consultation with the supervisor, will propose to the Part B Subcommittee two topics and a principal examiner for each. Each topic should be based on a one semester 600 or 700 level course, or similarly advanced material, and will usually be defined by a book, a designated part of a book or a collection of research papers. A reasonably detailed description, countersigned by the prospective principal examiner, will be submitted in writing to the Part B Subcommittee. The Part B Subcommittee will inform the student in writing of its approval of the proposals. The exam is to be held not earlier than 2 and no later than 6 months from this date. The student will inform the Part B Subcommittee at least one month in advance of the date the exam is to be taken. The Part B Subcommittee will then set up an examining committee consisting of the 2 principal examiners, two other examiners, and a member of the Part B Subcommittee who will act as Chair of the examining committee.
The exam will be open to members of the Department. The exam will be conducted mainly by the principal examiners. The Part B examination is an oral examination: notes, books, or other materials are not allowed. After the principal examiners have finished their part, the other members of the examining committee and other members of the Department present may pose further questions. The student's status at the end of the exam is to be determined by a majority vote of the examining committee.  The student must take a Part B Examination not later than five semesters (not including summers) after beginning the Ph.D. program. If the student fails in one or both topics presented, he or she may repeat the examination once, and this within the following four months, at a time to be approved by the Part B Subcommittee.
 The Part B Examination must be completed successfully before scheduling the thesis defense.
 These regulations are to take effect beginning in September 1997.