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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:听Applied Mathematics,听Pure Mathematics, and听Statistics. In the Applied and Pure versions the papers are called听alpha听and听beta听with Paper alpha the same for both versions; in the Statistics version they are called听theory听and听methodology.听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听the compulsory part.

Deadline to Pass the Part A Examination

This examination has strict deadlines. Only students who have successfully completed the Part A examination by August 31 of their Ph.D. 2 year will be allowed to register for the subsequent semester. Students who entered the graduate program in January will therefore have to take and pass their examinations at the end of their first semester in PhD2. Only in exceptional circumstances will the deadline be relaxed, with a formal letter to this effect from the Graduate Program Director acting in consultation with the Chair of the Department and the Progress Tracking Committee (or at least the advisor) of the student.

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 4-hour 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 re-take 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 4-hour 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.

The compulsory part of the听Paper alpha听examination requires students to answer 3 out of 4 questions on听Single Variable Real Analysis听and 3 out of 4 questions on听Linear Algebra. In addition students have to answer 3 out of 4 questions on听Vector Calculus,听Ordinary Differential Equations听and听Complex Analysis.

For Paper beta (Applied Mathematics), students are required to choose three modules. At least two of the modules must be chosen from the following list:

Continuous Optimization***,
Discrete Mathematics,
Discrete Optimization*,听
Convex Optimization**,
Mathematical Economics*,
Numerical Analysis,
Partial Differential Equations,
Probability.

听(**Because of the overlap between the modules, students opting to take the Convex Optimization module, may not take either of the Continuous Optimization, Discrete Optimization or Mathematical Economics modules. For the same reason, students opting to take the Continuous Optimization module, may not take either of the Convex Optimization, Discrete Optimization or Mathematical Economics modules. Students may opt to sit both the Discrete Optimization and Mathematical Economics modules.)
(***Following the retirement of MATH 560, the Continuous Optimization module will be offered for the last year in 2023.)听

Students may take at most one of the Pure Mathematics modules

Algebra,
Analysis,
Geometry and Topology.

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.

Syllabus for Paper Alpha 鈥� Applied and Pure

The compulsory part of this examination requires students to answer 3 out of 4 questions on听Single Variable Real Analysis听and 3 out of 4 questions on听Linear Algebra.

In addition students have to answer 3 out of 4 questions on听Vector Calculus,听Ordinary Differential Equations听and听Complex Analysis.

Single Variable Real Analysis:听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:听R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 2nd edition, John Wiley and Sons (1992).

Linear Algebra:听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, self-adjoint, orthogonal and unitary operators. Diagonalization in Euclidean and unitary spaces. The spectral theorem.

Reference:听S. Lipschutz, Linear Algebra, Schaum's Outline Series, McGraw Hill (1991).

Vector Calculus (Common with Applied Mathematics):听The derivative of a multi-variable vector-valued function as a linear transformation. The multi-variable inverse and implicit function theorems. Multiple integrals, path and surface integrals. Change of variables theorem for integrals. Calculation of areas, volumes, arc-lengths, momenta, etc. The integral theorems of vector analysis: Green's, Stokes' and Gauss'.

Course:听Honours Advanced Calculus听(MATH 248)
Reference:听J.E. Marsden and A. Tromba, Vector Calculus, W.H. Freeman, 4th edition (1996).

Ordinary Differential Equations:听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 regular-singular points. Systems of first order linear O.D.E.'s.

Course:Honours Ordinary Differential Equations听(MATH 325)
Reference:听W.E. Boyce and R.C. diPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley and Sons (1992).

Complex Analysis:听Analytic functions, the Cauchy-Riemann 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:听Honours Complex Analysis听(MATH 366)
References:

J. Conway, Functions of One Complex Variable, Springer Verlag, 2nd edition (1978).
J.E. Marsden, Basic Complex Analysis, W.H. Freeman and Co, 2nd edition (1987).
J. Bak and D.J. Newman, Complex Analysis, Springer Verlag (1982).

Syllabi for Paper beta modules 鈥� Applied Mathematics

Continuous Optimization:

Honours Linear Optimization (MATH 517): Convexity, polyhedrality, duality theory, fundamental theorem, sensitivity analysis, simplex algorithm, interior-point methods, quadratic optimization, and applications in game theory.

Optimization (Math 560): Theoretical foundations: convexity, matrix norms, directional derivatives, optimality conditions. Line search methods for unconstrained optimization, step-size rules, gradient method, Newton, Quasi-Newton and inexact Newton method, conjugate gradient methods (linear and nonlinear), trust-region methods. Theory of general (smooth) constrained optimization: KKT conditions, constraint qualifications, Lagrange duality. Numerical methods for constrained optimization: Penalty methods, SQP methods.

Courses: Linear Optimization (MATH 517), Optimization (MATH 560).

References: Numerical Optimization, J. Nocedal, S. Wright, Springer, 2006.

Discrete Mathematics:

Graph theory: Trees and cycles: Kruskal's theorem, Cayley formula, Dirac's theorem. Matching theory: Hall's, Konig's and Tutte's theorems, Tutte-Berge formula. Connectivity: Menger's and Berge-Fulkerson theorems. Cliques and Independent sets: Ramsey's theorem, Gallai's equations. Graph coloring: Brooks' and Konig's theorems. Graph minors. Planar graphs: Kuratowski's theorem, 5-color theorem.

Combinatorics: Extremal set theory: Sperner systems, Littlewood-Offord problem, Erdos-Ko-Rado theorem, shadows, compression and Kruskal-Katona theorem, Turan-type problems, Erdos-Stone theorem, and hypergraph lagrangians. Ramsey theory: van der Waerden, Hales-Jewett and Erdos-Szekeres theorems. Discrete geometry: convexity, Helly, colorful Caratheodory and Tverberg theorems, incidence problems, Szemeredi-Trotter theorem. Algebraic methods: combinatorial Nullstellensatz, Shannon capacity, cap-set problem and the sensitivity conjecture.

Courses: Graph Theory and Combinatorics (MATH 350), Combinatorics (MATH 550).

References:

Introduction to Graph Theory, second edition, D, West, Prentice Hall, 2001.
Graph Theory, fifth edition, R. Diestel, Springer, 2015.
Combinatorics, B. Bollobas, Cambridge University Press, 1986.
Lectures on Discrete Geometry, J. Matousek, Springer, 2002.

Discrete Optimization:

Honours Convex Optimization (MATH 563): Honours level introduction to convex analysis and convex optimization: Convex sets and functions, subdifferential calculus, conjugate functions, Fenchel duality, proximal calculus. Subgradient methods, proximal-based methods. Conditional gradient method, ADMM. Applications including data classification, network- flow problems, image processing, convex feasibility problems, DC optimization, sparse optimization, and compressed sensing.

Combinatorial Optimization (Math 552): Algorithmic and geometric approaches in combinatorial optimization: matchings; shortest paths and network flows; T-joins; the Ellipsoid method and applications; polyhedra and total unimodulatrity; total dual integrality; network connectivity; packings and coverings of branchings, arborescences and directed cuts. Matroids and submodular optimization.

Courses: Combinatorial Optimization (MATH 552), Honours Convex Optimization (MATH 563).

搁别蹿别谤别苍肠别蝉:听

Combinatorial Optimization, W. Cook, W. Cunningham, W. Pulleyblank and A. Schrijver, Wiley, 1998
First-order Methods in Optimization, A. Beck, MOS-SIAM Series on Optimziation, SIAM, Philadelphia, 2017.
Convex analysis and nonlinear optimization,听J. Borwein and A.S. Lewis, 2006
Fundamentals of Convex Analysis,听J.B. Hiriart-Urruty and C. Lemar茅chal,听Springer, Berlin,听
Introductory lectures on convex optimization, Yu. Nesterov, 2003. Convex Analysis, T. Rockafellar, 1970.鈥澨� 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听 听

Convex Optimization:

Honours Linear Optimization (MATH 517): Convexity, polyhedrality, duality theory, fundamental theorem, sensitivity analysis, simplex algorithm, interior-point methods, quadratic optimization, and applications in game theory.

Honours Convex Optimization (MATH 563): Honours level introduction to convex analysis and convex optimization: Convex sets and functions, subdifferential calculus, conjugate functions, Fenchel duality, proximal calculus. Subgradient methods, proximal-based methods. Conditional gradient method, ADMM. Applications including data classification, network- flow problems, image processing, convex feasibility problems, DC optimization, sparse optimization, and compressed sensing.

Courses: Linear Optimization (MATH 517),听Honours Convex Optimization (MATH 563).听

References:

First-order Methods in Optimization, A. Beck, MOS-SIAM Series on Optimziation, SIAM, Philadelphia, 2017.
Convex analysis and nonlinear optimization, J. Borwein and A.S. Lewis, 2006.
Fundamentals of Convex Analysis, J.B. Hiriart-Urruty and C. Lemar茅chal, Springer, Berlin, Heidelberg, 2004.
Introductory lectures on convex optimization, Yu. Nesterov, 2003. Convex Analysis, T. Rockafellar, 1970.

Mathematical Economics:

Honours Linear Optimization (MATH 517): Convexity, polyhedrality, duality theory, fundamental theorem, sensitivity analysis, simplex algorithm, interior-point methods, quadratic optimization, and applications in game theory.

Algorithmic Game Theory (MATH 553): Foundations of game theory. Computation aspects of equilibria. Theory of auctions and modern auction design. General equilibrium theory and welfare economics. Algorithmic mechanism design. Dynamic games.

Courses: Honours Linear Optimization (MATH 517), Algorithmic Game Theory (MATH 553).

搁别蹿别谤别苍肠别蝉:听Twenty lectures on Algorithmic Game Theory, Tim Roughgarden, Cambridge University Press, 2016.

Numerical Analysis:

Direct methods for linear systems: QR factorization, LU factorization and variants, conditioning and stability. Least squares problems: normal equations, QR factorization, and singular value decomposition. Eigen-problems: reduction to Hessenberg form, QR algorithm, Jacobi method. Polynomial and trigonometric interpolation and approximation, applications to quadrature. Iterative methods for linear systems: Jacobi, Gauss-Seidel, SOR, GMRES, Conjugate Gradients, Preconditioning. Iterative methods for nonlinear systems: fixed point iteration, Newton and variants. Numerical methods for ordinary differential equations: Linear multistep and Runge-Kutta methods, consistency, stability, convergence, stiffness, adaptive integration, shooting method for boundary value problems. Hamiltonian problems, symplectic integration. 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:

L.N. Trefethen and D. Bau. Numerical Linear Algebra, SIAM, 1997. Lectures 1-8, 10-15, 20-30, 32-38, and 40.
A. Quateroni, R. Sacco, F. Saleri. Numerical Mathematics, Springer, 2000. Sections 7.1, 8.1, 8.3, 8.6, 9.1-9.6, 10.1-10.10.
E. Hairer, S.P. Norsett and G. Wanner. Solving Ordinary Differential Equations I, 2nd Ed, Springer 1993. Sections II.1-4, 7, 16. III 1,2,3,4.
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, 2nd Ed, Springer, 1996. Sections IV.2-3, V1.
A. Iserles. A First Course in the Numerical Analysis of Differential Equations. Chapters 7, 8, 13, 14.

Partial Differential Equations: Definition of well-posedness of problems, uniqueness and regularity of solutions. Familiarity with separation of variables techniques assumed. First order PDE: First order Cauchy problems. Linear and nonlinear first-order PDE, their solution by the method of characteristics. Conservation laws, shocks, entropy conditions. Hamilton-Jacobi equations and Legendre transforms. Complete Integrals and envelopes. Characteristics for second order linear PDE. The Cauchy-Kovalewski theorem.

Laplace's equation: fundamental solution, mean value theorems and maximum principles, representation formulae. Distributions, Green鈥檚 function, energy methods, weak solutions, and properties of solutions.

Heat equation: fundamental solution, mean value theorems, representation formulae, energy methods, weak solutions, Duhamel's principle, properties of solutions.

Wave equation: d'Alembert's solution, spherical means, method of descent, non-homogenous problems. Transforms methods (Fourier and Laplace), scaling methods, similarity solutions, including plane waves, solitons, travelling wave solutions.

Sobolev Spaces and weak solutions to second order elliptic PDEs: Lax Milgram Theorem, Fredholm Alternative.

Courses: Partial Differential Equations 1 (MATH 580), Partial Differential Equations 2 (MATH 581)

搁别蹿别谤别苍肠别蝉:听L.C. Evans, Partial Differential Equations, AMS (1998). Chapters 1-6.
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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 Zero-One Law, Poisson Process, Weak Convergence, Characteristic Functions, the Central Limit Theorem, Limit Theorems in Rk, Radon-Nikodym 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 1-6, unstarred sections and topics.]

The compulsory part of the听Paper alpha听examination requires students to answer 3 out of 4 questions on听Single Variable Real Analysis听and 3 out of 4 questions on听Linear Algebra. In addition students have to answer 3 out of 4 questions on听Vector Calculus,听Ordinary Differential Equations听and听Complex Analysis.

Paper beta (Pure Mathematics)听consists of 4 questions in each of听Algebra,听Analysis听and听Geometry and Topology. Students will be assessed on their answers to 7 questions with at least 2 from each area.

Students can choose 1 module from the following: Discrete Mathematics,听Numerical Analysis,听Optimization,听Partial Differential Equations,听Probability,

Syllabi for Paper beta modules Pure Mathematics

Analysis:听Metric spaces, compactness, completeness, Baire's category theorem, Ascoli-Arzela theorem. Stone-Weierstrass 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 n-dimensional Euclidean spaces, Radon-Nikodym theorem. Elementary theory of Banach and Hilbert spaces: Hahn-Banach, uniform boundedness, open mapping theorems.听Lp听spaces. Convergence in听L2. Theorems of Fejer and Parseval. Convergence of Fourier series.

Courses:听Honours Analysis 3听(MATH 454),听Honours Analysis 4听(MATH 455)听and with some material from听Advanced Real Analysis 1听(MATH 564),听Advanced Real Analysis 2听(MATH 565).
搁别蹿别谤别苍肠别蝉:听The first set of topics in covered by: W. Rudin, Principles of Mathematical Analysis, McGraw Hill (1976). (The Ascoli-Arzela theorem (not so named) is on pages 156-158. The Baire category theorem is in Exercises 2.30 (p. 46) and 3.22 (p. 82); the Baire category theorem as well as the Ascoli-Arzela 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:

W. Rudin, Real And Complex Analysis, McGraw Hill (1987).
H. Royden, Real Analysis, third edition, Macmillan (1988).
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley Interscience (1987).

Algebra:听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, co-products, free constructions, inductive and projective limits.

Courses:听Honours Algebra 3听(MATH 456),听Honours Algebra 4听(MATH 457),听with some material from听Higher Algebra 1听(MATH 570),听Higher Algebra 2听(MATH 571).
搁别蹿别谤别苍肠别蝉:听Most of the material can be found in any one of the following three books:

M. Artin, Algebra, Prentice-Hall (1991).
T.W. Hungerford, Abstract Algebra: An Introduction, Saunders (1990).
N. Jacobson, Basic Algebra I, Freeman and Company (1989).

For certain topics detailed below, the following books may be consulted:

T.W. Hungerford, Algebra, Springer-Verlag (1974).
S. Lang, Algebra, Addison-Wesley (1993).
N. Jacobson, Basic Algebra II, Freeman and Company (1989).

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:听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:听Geometry and Topology 1听(MATH 576),听Geometry and Topology 2听(MATH 577). Note that the syllabus includes slightly more material than is taught in these courses.
References:

Singer and Thorpe, Lectures on Elementary Topology and Geometry, Springer Verlag (1967). [Chapters 1, 2, 3, 5]
Munkres, Topology 2nd edition, Prentice Hall. Part I, chapters 2-5, 7. Part II, chapters 9, 11-13.
Boothby: An Introduction to Differentiable Manifolds and RiemannianGeometry 2nd edition, Academic Press, Chapters vi.1, vi.2, vi.3-vi.8. (Exercises in vi.8 are encouraged as preparation.)
Alternate references:
An alternate reference for the de Rham cohomology is: Bott & Tu, Differential Forms in Algebraic Topology, Springer-Verlag. Chapter 1, sections 1-5.
An alternate reference for the material related to the fundamental group and covering spaces is Chapter I of: Hatcher, Algebraic Topology, Cambridge University Press.

The Statistics Part A exam consists of a听Theory Paper听(in mathematical statistics and measure theoretic probability) and a听Methodology Paper听(in linear models and generalized linear models).

The exam format has been thought out so that a student who has training in linear regression and has taken听Mathematical Statistics 1听(MATH 556),听Advanced Probability Theory 1听(MATH 587),听Mathematical Statistics 2听(MATH 557),听Generalized Linear Models听(MATH 523)听during their first year has all the requisite notions fresh on her or his mind by May. The best time to attempt them is thus at the end of the first year.

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Syllabus for the Theory Paper 鈥� Statistics

Measure, Integration and Foundations of Probability:听Open, closed and compact sets in Rk. Basic properties of probability spaces: s algebras, axioms of probability functions and calculus of probability. Outer measure, measure extension theory and Lebesgue measures on Rk. Measurable mappings, limits and measurability of functions. Distribution functions and expectations: Chebychev's inequality, Markov's inequality and Jensen's inequality. Moment generating functions; Stieltjes integrals on Rk; Uniform integrability; Lebesgue's dominated convergence theorem, monotone convergence theorem and Fatou's lemma. Independence. Product spaces, product measure and Fubini theorem. Integration by parts. Almost sure convergence, convergence in probability and weak convergence. Law of large numbers; Kolmogorov's zero-one law. Basic properties of the Poisson process. Definitions and basic properties of conditional expectation and conditional probability.听

Course:听Advanced Probability Theory 1听(MATH 587)听
Reference:听Billingsley, P. (1986). Probability and Measure. 2nd ed. Wiley. [Unstarred sections of Chapters 1 to 4.]

Mathematical statistics:听The calculus of probabilities. Counting. Equally likely outcomes. Conditional probability and independent random variables. Distribution Functions: Density and mass functions. Expected values. Moments and moment generating functions. Common families of distributions: exponential families, location and scale families. Joint and marginal distributions. Conditional distributions and independence. Multivariate transformations. Covariance and correlation. Mixture distributions. Order statistics. Sums of random variables from a random sample. The Lindeberg-Levy Central Limit Theorem. The sampling distributions: Student's t and Snedecor's F. Sufficiency, minimal sufficiency, ancillarity, completeness, Basu's Theorem, Rao-Blackwell Theorem, Lehmann-Scheff Theorem, minimum variance unbiased estimation. Method of moments, maximum likelihood estimation, Bayes' estimation, invariant estimation, consistency. Asymptotic properties of maximum likelihood estimators, the "delta method" for functions of random variables. Hypothesis testing: concepts of significance and power, the Neyman-Pearson lemma, likelihood ratio tests, Bayesian tests. Interval estimation: methods of finding confidence intervals including inverting a test statistic and pivotal quantities. Bayesian credibility intervals.

Courses:听Mathematical Statistics 1听(MATH 556),听Mathematical Statistics 2听(MATH 557)
Reference:听Casella, G. & Berger, R. L. (2001). Statistical Inference. 2nd ed. Wadsworth [Chapters 1-10].

听

Syllabus for the Methodology Paper 鈥� Statistics

Linear models: Least squares estimators and their properties. Simple linear regression, multiple regression, Gauss-Markov theorem. Analysis of variance. Linear models with general covariance. Distribution of estimators. General linear hypothesis: F-test and t-test, prediction and confidence regions. Categorical and continuous covariates, collinearity, interactions. Multivariate normal and chi-squared distributions. Model selection, residual analysis, detecting influential observations, testing for lack of fit, transformations, weighted least squares, and variable selection techniques.

Course: Regression and Analysis of Variance (MATH 533)
Reference: Weisberg, S. (1980). Applied Linear Regression. New York: Wiley.

Generalized linear models: Exponential families. Link function, variance function. Iteratively reweighted least squares, asymptotic distribution of maximum likelihood estimators. Analysis of deviance, goodness of fit tests. Log-linear models, analysis of contingency tables, overdispersion. Logistic regression, case-control studies, multinomial regression. Gamma models. Quasi-likelihood models. Mixed models. Bayesian estimation.

Course: Generalized Linear Models (MATH 523)
Reference: McCullagh, P. & Nelder, J. (1989). Generalized Linear Models. 2nd ed. Chapman & Hall: New York.
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