Memorial University Department of Mathematics and Statistics. PhD COMPREHENSIVE EXAMINATION QUALIFYING REVIEW MATHEMATICS SYLLABUS

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1 Memorial University Department of Mathematics and Statistics PhD COMPREHENSIVE EXAMINATION QUALIFYING REVIEW MATHEMATICS SYLLABUS 1 ALGEBRA The examination will be based on the following topics: 1. Linear algebra: vector spaces, subspaces, quotient spaces, dual spaces, linear transformations, matrices, change of bases, rank and nullity, determinants, eigenvalues and eigenvectors, rational and Jordan forms of a matrix, inner product spaces, diagonalization of self-adjoint transformations and Hermitian forms. 2. Groups: subgroups, Lagrange s theorem, homomorphisms, normal subgroups, quotient groups, isomorphism theorems for groups, direct products, fundamental theorem on finitely generated abelian groups, symmetric groups and alternating groups, group actions, Sylow theorems, automorphisms, composition series, Jordan-Holder-Schreier theorem, nilpotent groups, solvable groups. 3. Fields: subfields, isomorphisms, algebraic and transcendental extensions, separable and inseparable extensions, splitting fields, fundamental theorem of Galois theory, finite fields, algebraically closed fields. 4. Rings: subrings, homomorphisms, ideals, quotient rings, direct products, matrix rings, polynomial rings, Jacobson radical, Artinian and Noetherian rings, Wedderburn-Artin theorem, rings of quotients of an integral domain, prime ideals, maximal ideals, Euclidean domains, principal ideal domains, unique factorization domains. 5. Modules: submodules, quotient modules, isomorphism theorems for modules, free modules, projective and injective modules, simple and semisimple modules, structure of finitely generated modules over principal ideal domains. 6. Introduction to representations of finite groups: irreducible and completely reducible representations, Maschke s theorem, regular representation, characters. 1

2 References: J. A. Beachy, Introductory lectures on rings and modules, London Math. Soc. Student Texts 47. Cambridge University Press, Cambridge, I. N. Herstein, Topics in Algebra, John Wiley, New York, K. Hoffman and R. Kunze, Linear algebra, second edition, Prentice- Hall, Englewood Cliffs, N.J., W. K. Nicholson, Introduction to abstract algebra, third edition, Wiley- Interscience, New Jersey Additional References: Y. A. Bahturin, Basic structures of modern algebra, Mathematics and its Applications 265, Kluwer Academic Publishers Group, Dordrecht, S. Lang, Algebra, Revised third ed. Graduate Texts in Mathematics 211, Springer-Verlag, New York, W. K. Nicholson, Linear Algebra with Applications, Fifth ed. McGraw- Hill Publishing, (especially Chapters 6-10) 2

3 2 ANALYSIS 1. Real Analysis: Properties of real numbers. Completeness (various approaches). Sequences, monotone sequences, convergence, limit superior and limit inferior. Functions and their limits, continuity, properties of continuous functions defined on a compact set. Uniform continuity. Infinite series, tests for convergence, power series, Taylor series, Taylor remainder. Differential calculus, mean value, theorems. Riemann integral, uniform convergence of sequences and series, Weierstrass approximation theorem. Transcendental functions. Elementary ideas in Fourier series. Functions of several variables; limits, continuity, partial derivatives and differentiability. Inverse function theorem. Implicit function theorem. Metric spaces: Cauchy sequences, completeness, completion of a metric space, uniform continuity, subspaces, Compactness. 2. Lebesgue Theory: Set functions, σ-fields, construction of Lebesgue measure, measurable functions, Lebesgue integral, comparison with Riemann integral, convergence theorems, Fubini s theorem for R Complex Analysis: Elementary analytic functions, characterization of analyticity, harmonic functions, Cauchy s theorem for contour integrals, power series, Laurent series, isolated singularities, the residue theorem; calculus of residues. 4. Functional Analysis: Elementary theory of normed vector spaces, Banach spaces and Hilbert spaces; linear operators on Banach and Hilbert spaces; compact operators. References: H. L. Royden. Real Analysis (Chapters 3 to 5). R. V. Churchill and J. W. Brown. Complex Variables and Applications (Chapters 1 to 7). W. Rudin. Principles of Mathematical Analysis. G. De Barra. Measure Theory and Integration (Chapters 1 to 5). M. H. Protter and C. B. Morrey (1991). A First Course in Real Analysis, Second edition. Springer. 3

4 L. Debnoth and P. Mikusinski (1990). Introduction to Hilbert Spaces with Applications. Academic Press. 4

5 3 TOPOLOGY 1. Equivalence relations, partial and linear orderings, Zorn s lemma, the axiom of choice and the well ordering principle. 2. Open and closed sets, neighborhoods, closure, interior, accumulation and interior points. Bases, sub-bases and axioms of countability. 3. Continuity and homeomorphisms. Subspaces, product spaces and quotient spaces. Function spaces with the compact-open topology. 4. Connected and path connected spaces. Compactness, the Heine-Borel theorem, the Bolzano-Weierstrass theorem, the Tychonoff theorem and one-point compactification. Metric spaces, completeness and the Baire category theorem. Urysohn s metrization theorem. References: There are many books covering most of this material in a satisfactory fashion. Two examples are: J. Dugundji. Topology. S. Willard. General Topology. 5

6 4 COMBINATORICS 1. Graph theory: basic properties and definitions of graphs and subgraphs, Eulerian and Hamiltonian graphs, matchings, coverings, independent sets, trees, connectivity, shortest paths, graph colouring including Vizing and Brooks Theorems, Ramsey s theorem, random graphs 2. Design theory: basic properties of balanced incomplete block designs (BIBDs), Fisher s inequality, basic properties of Steiner and Kirkman triple systems, covering and packing designs, mutually orthogonal Latin squares (MOLS), transversal designs (TDs) and their relation to finite planes, group divisible designs (GDDs), pairwise balanced designs (PBDs), Bose and Skolem constructions for BIBDs, Wilson s fundamental construction, some recursive constructions for triple systems, difference sets and difference methods for constructing designs, basic properties of Room squares 3. Enumeration: generating functions, principle of inclusion and exclusion, partitions, recurrence relations, distributions, Stirling numbers, Polya counting theorem, systems of distinct representatives References: D. West, Introduction to Graph Theory (2nd ed), Prentice Hall, 2001 J. Bondy and U. Murty, Graph Theory with Applications, Macmillan Press, 1977 R. Diestel, Graph Theory, Springer, 1997 I. Anderson, Combinatorial Designs and Tournaments, Oxford Science Publications, 1993 C. Lindner and C. Rodger, Design Theory, CRC Press, 1997 T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, 1993 P. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press,

7 Additional References: Ryser, Combinatorial Mathematics, MAA, 1963 Wallis, Combinatorial Designs, Dekker,

8 5 APPLIED MATHEMATICS 1. Ordinary differential equations: First and higher order linear equations. Existence and uniqueness of solutions. The Laplace transform and its application to IVPs including impulse functions and the convolution integral. Series solutions of second order linear equations. Systems of first order linear equations. Introductory phase plane techniques and critical points for nonlinear DEs. 2. Partial differential equations: Linear, quasilinear and nonlinear firstorder PDEs. The method of characteristics. Initial and two-point boundary value problems involving second-order linear PDEs: the heat equation, the Poisson equation and the wave equation. Maximum principle. Separation of variables. Cauchy problem, Sturm-Liouville problem. Fourier series. 3. Numerical analysis and methods: Interpolation and polynomial approximation. Numerical differentiation and integration. Direct methods for solving linear systems, iterative solution of sparse matrix systems. Euler s method and Runge-Kutta methods for ODEs. Finite difference methods for parabolic, hyperbolic and elliptic PDEs. Finite element methods for elliptic PDEs. Consistency, stability and convergency of approximate solutions. References W.E. Boyce and R.C.DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., (Eighth Edition), R. Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhouse Publishing Comp., K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, R. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, A. Iserles, Numerical Analysis of Differential Equations, Cambridge University Press,

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section