Comprehensive exams
Refer to the Graduate Calendar for Comprehensive Exam requirements.
Syllabus
The general part of the examination consists of 5 topics, standard in any undergraduate degree in Mathematics:
- Single Variable Real Analysis
Properties of the real numbers, infimum and supremum of sets. Numerical sequences and series. Limits of functions, continuous functions, intermediate value theorem, uniform continuity. Differentiation, mean value theorem, L’Hospital’s rule. Riemann integral, fundamental theorem of calculus. Sequences and series of functions, power series, uniform convergence, Taylor expansion.
References: M. Spivak, Calculus, 3rd edition; W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.
- Linear Algebra
Matrices and systems of linear equations, Gauss reduction, elementary matrix operations. Vector spaces, subspaces, basis, dimension. Determinants. Inner product, orthogonality, orthonormal bases. Linear mappings, change of basis, kernel and image. Eigenvalues and eigenvectors, diagonalization, Jordan normal forms. Bilinear, quadratic and hermitian forms. Spectral theorem, Jordan canonical form, minimal polynomial.
References: S. Lipschutz, Linear Algebra, Schaum’s Outline Series, McGraw-Hill; S.H. Friedberg, A.J. Insel, L.E. Spence, Linear Algebra, 2nd edition, Prentice-Hall.
- Complex Analysis
Analytic functions, Cauchy-Riemann equations. Power series representation. Line integrals, Cauchy’s theorem and Cauchy’s integral formulas. Residue theorem and its application, Rouche’s theorem. Open mapping theorem. Morera theorem. Liouville’s theorem, fundamental theorem of algebra. Meromorphic functions and Laurent expansions. Fractional-linear (Mobius) transformations.
References: J.B. Conway, Functions of One Complex Variable, Springer Verlag; R.A. Silverman, Introductory Complex Analysis, Dover.
- Metric Spaces
Metric spaces, function spaces. Compactness, completeness, fixed-point theorems. Continuous functions and their properties. Ascoli-Arzela theorem. Weierstrass approximation theorem.
References: H. Royden, Real Analysis, 3rd edition, Macmillan; A. Friedman, Foundations of Modern Analysis, Dover.
- Measure Theory
Lebesgue measure and integration on real line, convergence theorems, absolute continuity, Lp spaces. General theory of measure and integration, Radon-Nikodym theorem.
References: H. Royden, Real Analysis, 3rd edition, Macmillan.
The common part of the examination consists of the following 6 topics comprising two fundamental mathematics topics, two probability topics and two statistics topics:
- Linear Algebra
Vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization.
Reference: S.H. Friedberg, A.J. Insel and L.E. Spence, Linear Algebra, 4th edition, Prentice-Hall.
- Real Analysis
The Real number system, basic topology, numerical sequences and series, continuity, differentiation, sequences and series of functions.
Reference: W. Rudin, Principles of Mathematical Analysis. Volume 3. McGraw-Hill.
- Fundamentals of Probability
Events and their probabilities, random variables and their distributions, discrete random variables, continuous random variables.
Reference: G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press.
- Statistical Inference
Some special distributions, elementary statistical inferences.
Reference: R.V. Hogg and A.T. Craig, Introduction to Mathematical Statistics, 6th or 7th Edition, Prentice Hall.
- Linear Regression
General linear regression, least Squares, statistical inference on regression outputs, diagnostic tests on residuals (homoscedasticity, normality, absence of trends, non-correlation), analysis of variance (ANOVA), weighted least squares, variable selection procedures.
Reference: N. R. Draper and H. Smith. Applied regression analysis. Volume 326. John Wiley & Sons.
The specialized part is taken in the area of study chosen by the student and the supervisor for the student's thesis work. The following is a partial list of some possible topics.
- Algebra and Number Theory
Group theory: group actions, Sylow’s theorems, finitely generated abelian groups. Ring theory: polynomial rings, Euclidean domains, unique factorization domains, principal ideal domains. Commutative algebra: localization, tensor products, rings and modules of fractions, integral extensions, Noetherian and Artinian rings, local rings, valuation rings, Dedekind domains, Hilbert basis theorem, Hilbert Nullstellensatz. Field theory and Galois theory: normal and separable extensions, solvable extensions, solvability by radicals, cyclic and cyclotomic extensions.
References: M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley; D. Dummit and R. Foote, Abstract Algebra, 2nd edition, Prentice-Hall; N. Jacobson, Basic Algebra I, Freeman; N. Jacobson, Basic Algebra II, Freeman; S. Lang, Algebra, Addison-Wesley.
- Ergodic Theory and Dynamical Systems
One-dimensional dynamics, higher dimensional dynamics, complex dynamics. Invariant measures, basics of ergodic theory. Absolutely continuous invariant measures, Frobenius-Perron operator.
References: R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley.
- Mathematical Physics
TBA
- Probability and Statistics
Random variables. Conditional probability and expectation. Markov chains. Poisson process. Continuous time Markov chains. Distributions, random vectors, random samples. Point estimation, testing of hypotheses, interval estimation. Decision theory. Non-parametric methods. Analysis of variance. Linear regression.
References: S.M. Ross, Introduction to Probability Models, Academic Press; G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press; A.M. Mood, F.A. Graybill and D.C. Boes, Introduction to the Theory of Statistics, McGraw-Hill; R.V. Hogg and A.T. Craig, Introduction to Mathematical Statistics, Macmillan; G. Casella and R.L. Berger, Statistical inference, Duxbury Press.