Qualifying Exam Syllabi

The purpose of the written qualifying exams, as endorsed by the Policy Committee in Spring 1990, is to indicate that the student has the basic knowledge and mathematical ability to begin advanced study. 

• algebra,
• analysis,
• logic,
• numerical analysis,
• differential equations:
• ordinary differential equations,
• partial differential equations,
• probability,
• statistics,
• applied statistics:
  • (PhD level),
  • (MA level),
  and
• topology.

In some subjects, there will be a separate examination for students seeking the M.A., that will be given at the same time as the Ph.D. examination. The M.A. level examination may cover less material, but some of the questions may be the same as those on the Ph.D. level examination. M.A. students may exercise the option of taking the Ph.D. examination and only being required to receive an "M.A. pass".

Each examination will last four hours and no two will be scheduled on consecutive days.

Each MATH student will be required to take three examinations. The only restrictions are: (1) not both ODE and PDE can be taken; (2) the three exams cannot be analysis, numerical analysis, and one of the differential equations exams.

Each STAT student must take probability, statistics, and one other part. Students taking the M.A. or Ph.D. in applied statistics may take the following three exams: applied statistics, mathematical statistics, and probability.

Each AMSC student takes at least three examinations (generally only three), chosen in consultation with the study advisory committee but only one or two are chosen from the list above; the others are in areas of application.

In accordance with previously announced policy, the Geometry/Topology exam will be offered only for Math 730 and MATH 734 beginning in January, 1997. There will be 3 questions for each course.

The attached syllabi are current and reflect the latest thinking of the respective field committees. In some cases there is a separate syllabus for the M.A. exam but generally the only difference between the two exams is the difficulty.

MATH 600

GROUPS

Review of elementary group theory including Lagrange's theorem: symmetric groups and Cayley's theorem: normal subgroups, quotient groups and the homomorphism and isomorphism theorems; abelian and cyclic groups (one week).

Groups with operators, normal series, Jordan-Hölder theorem, solvable groups; unsolvability of S_n for n > 4.

Group actions; class formula; Sylow's theorems; solvability of p-groups.

Language of categories: objects, maps and functors; Hom; universal mapping properties used to define quotients, products, coproducts (direct sums) and free objects in categories of groups and abelian groups. Constructive existence proofs of the above objects, especially generators and relations in category of groups. Internal direct sums in abelian groups; primary decomposition of abelian torsion groups.

RINGS

Definitions and examples; left, right and two-sided ideals; quotients, homomorphism and isomorphism theorems; products; examples should include matrix rings, group rings, and real quaternions. Simple rings: proof that a matrix ring over a division ring is simple. Definition and a few words about Artinian and Noetherian rings. Statement (no proof) of Wedderburn's theorem for simple Artinian rings. Hilbert Basis theorem.

Integral domains and fields; prime and maximal ideals-Zorn's lemma; operations on ideals; Chinese remainder theorems.

Localization: multiplicative sets; rings of quotients and quotient fields; local rings.

Factorization: P.I.D.'s UFD's; Euclidean rings; polynomial rings; Gauss's lemma. Proof that polynomial rings over UFD's are UFD's.

MODULES

Definitions and example; exact sequences; exactness properties of HOM; quotients, products, direct sums (internal and external), examples of modules over matrix rings and group rings.

Free modules; invariance of rank over a commutative ring; non-invariance in general.

Finitely generated modules over P.I.D.; applications to canonical forms of matrices and to abelian groups.

MATH 601

(NOTE: Math 601 is not required on the M.A. exam.)

MODULES

Tensor product; localization; algebras and base change; exactness properties of tensor products.

Exterior algebra.

Projective and injective modules. Homology including the snake lemma. Statements (not proofs) of facts on derived functors including Tor and Ext.

FIELD THEORY

Extensions, algebraic and transcendental; characteristic; finite fields; algebraic closure; transcendence basis.

Galois theory: The Galois correspondence; Galois groups of polynomials as permutation groups; cyclic extensions; roots of unity; ruler and compass constructions; solvability by radicals; norms and traces; computations of Galois groups.

Introduction to representations of finite groups over the complex numbers (as in Serre Part 1 Chapters 1,2,3, and the part of 5 dealing with finite groups).

MATH 405

The contents of Linear Algebra by K. Hoffman and R. Kunze, probably the most comprehensive and readable book on the subject.

References

As references for MATH 600-601, we recommend the following general treatises on algebra:
• Algebra by S. Lang
• Algebra by van der Waerden (2 volumes)
• Algebra by J.K. Goldhaber and G. Ehrlich
• Commutative Algebra by M. Atiyah and I.G. MacDonald
• Basic Algebra by N. Jacobson (2 volumes)
• Topics in Algebra by I. Herstein
• Algebra by T. Hungerford
• Linear Representations of Finite Groups by J-P. Serre

ANALYSIS

The written examination in Analysis consists of six questions roughly based on the material of MATH 630 and MATH 660 (three questions from each course).

MA students may request a special MA level exam or can opt to take the PhD exam seeking an MA level pass.

Students are responsible for all the material on the exam syllabus, even if it was not covered in a particular semester's course. Much of the material on the exam syllabus (especially for the MA level exam) is often covered in undergraduate analysis courses.

Syllabus for both the MA and PhD Written Exams in Analysis:

Review from Advanced Calculus
1. Real and complex numbers. Continuity, sequences and series, compactness on the real line. Vector spaces (over R and C), linear maps. Green's Theorem in the plane. ([BG], Ch. 1, 1-5)
Real Analysis
1. Lebesgue measure and integration on the real line, differentiation of monotone functions, absolute continuity, functions of bounded variation. ([Be], Ch. 2,3,4; [Ro], Ch. 3,4,5; [WZ], Ch. 3,4,5,7)
2. L^p spaces on the real line, including the Hölder and Minkowksi inequalities, the Riesz-Fischer Theorem (on the real line), L²-theory of Fourier series, the bounded linear functionals on L^p(R), p from 1 to infinity. ([Ro], Ch. 6;[WZ], Ch. 8)
3. Abstract measure and integration theory, including the Dominated Convergence Theorem, Fubini's Theorem, L^p spaces. ([Ro], Ch. 12)
Complex Analysis Syllabus for Complex Exam with suggestions for study in a pdf file. References

[A]

L. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979

[Be]

J. Benedetto, Real Variable and Integration, Teubner, 1976

[C]

J. Conway, Functions of One Complex Variable, 2nd edition, 1978

[Ha]

P. Halmos, Measure Theory, Graduate Texts in Mathematics, no. 18, Springer-Verlag, 1981

[HS]

E. Hewitt and K. Stromberg, Real and Abstract Analysis , Springer-Verlag, 1969

[Ro]

H. Royden, Real Analysis, 3rd ed., MacMillan, 1988

[Ru]

W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987

[WZ]

R. Wheeden and A. Zygmund, Measure and Integral: An introduction to Real Analysis, M. Dekker, 1977

LOGIC

The student is responsible for the material in the following outline. Most of this material will be adequately covered in MATH 712-713, with Dr. Kueker's Notes as primary reference. The other references cover certain topics more fully and are recommended supplementary reading.

ELEMENTARY LOGIC:

formal languages, truth assignments and tautologies, first-order languages and structures, formal deductions, the completeness theorem, theories and their models.
• Enderton: 1.1-1.5, 1.7, 2.1-2.7
• Kueker: Chapters 1, 2 and 3
• Shoenfield: Chapters 2 and 3, 4.1 and 4.2

MODEL THEORY:

the Henkin method for constructing models, compactness, the Lowenheim-Skolem theorems, realizing and omitting types, elementary extensions, prime and saturated models, categoricity, interpolations, definability and preservation theorems.
• Chang, Keisler: 2.1-2.3, 3.1, 3.2, 3.4
• Kueker: Chapters 4 and 5, 6.1 and 6.4
• Shoenfield: Chapter 5

INCOMPLETENESS AND UNDECIDABILITY:

recursive functions, representability, diagonalization, Gödel's first incompleteness theorem, the theorems of Church and Tarski, undecidability of arithmetic and other theories, Gödel's second incompleteness theorem.
• Kueker: Chapters 7,8 and 9
• Shoenfield: Chapter 6 and 8.1, 8.2

RECURSION THEORY:

recursively enumerable relations, weak representability, universal r. e. relations, the recursion theorem, partial recursive functions, Turing reducibility, the arithmetical hierarchy, Post's theorem.
• Kueker: Chapters 10 and 11
• Soare: Chapter I, II.1-II.3, III.1, IV.1, IV.2

References

• C.C. Chang and H. J. Keisler, Model Theory. North-Holland Pub. Co., 1973
• H.B. Enderton, A Mathematical Introduction to Logic. Academic Press, 1972
• D.W. Kueker, Notes on Mathematical Logic.
• J.R. Shoenfield, Mathematical Logic. Addison-Wesley, 1967
• R.I. Soare, Recursively Enumerable Sets and Degrees. Springer-Verlag, 1987

NUMERICAL ANALYSIS

(Last revised Fall, 2010.)

The numerical analysis graduate courses, AMSC 666 (Numerical Analysis I) and AMSC 667 (Numerical Analysis II), are survey courses to give students an overview of the subject and to prepare some students for more advanced course work in the area. Students interested in numerical analysis but having no previous experience should start with 466 and proceed to 666-667. More experienced students should begin with 666.

The CMSC/AMSC 666 NUMERICAL ANALYSIS EXAM

• INTERPOLATION AND APPROXIMATION
• Least squares fitting
• Chebyshev polynomials and min-max approximations
• Polynomial interpolation
• Spline interpolation
• Error analysis
• Weierstrass' Theorem

References: [SB] Chapter 2, [Atkinson] Chapters 3-4.

• QUADRATURE
• Newton-Cotes methods, Peano kernel theorem
• Euler-McClaurin Formula, Romberg integration
• Orthogonal polynomials and Gauss quadrature
• Adaptive quadrature

References: [SB] Chapter 3, [Atkinson] Chapter 5.

• NONLINEAR SYSTEMS
• Fixed point methods
• Newton's method
• Modified Newton's methods, Broyden's method
• Rates of convergence

References: [SB] Chapter 5, [Kelley-Iter] Chapters 4-8.

• OPTIMIZATION
• Nonlinear least squares methods
• Steepest descent and conjugate gradient methods
• Newton's method and quasi-Newton methods (DFP, BFGS)
• Rates of convergence
• Gauss-Newton and Levenberg-Marquardt methods

References: [Kelley-Opt] Chapters 1-4.

References

• [Atkinson] K. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley and sons, 1989.
• [Kelley-Iter] C. T. Kelley, Iterative methods for Linear and Nonlinear Equations, SIAM, 1995.
• [Kelley-Opt] C. T. Kelley, Iterative methods for Optimization, SIAM, 1999.
• [SB] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd edition, Springer-Verlag, 2002.

The CMSC/AMSC 667 NUMERICAL ANALYSIS EXAM

• NUMERICAL SOLUTION OF INITIAL VALUE PROBLEMS
• Runge-Kutta methods
• Multistep methods
• Consistency, stability and convergence analysis
• Error estimation and stepsize control
• Methods for stiff systems

References: [SB] Sections 7.1-7.2, [Atkinson] Chapter 6.

• NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS
• Two-point boundary value problems
• Finite difference methods
• Variational formulation and the finite element method
• Introduction to error analysis
• Discretization methods for multidimensional problems

References: [SB] Sections 7.4-7.7, [LT] Chapters 4-5.

• ITERATIVE METHODS
• The conjugate gradient method for symmetric positive definite systems
• Preconditioning techniques and relation to stationary iterative methods (such as Jacobi, Gauss-Seidel, SOR)
• Application to boundary value problems
• Convergence analysis
• Multigrid
• Krylov subspace methods for indefinite and nonsymmetric systems

References: [SB] Chapter 8, [Saad] Chapters 4, 6, 9, 10, 13, [Kelley-Iter] Chapters 1-3.

• EIGENVALUE METHODS
• Linear algebra concepts (similarity transformations, singular value decomposition, Rayleigh quotients)
• Power and inverse power methods
• QR algorithm
• Lanczos and Arnoldi methods

References: [SB] Chapter 6, [Stewart] Chapters 1-2, 5.

References

• [Atkinson] K. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley and sons, 1989.
• [Kelley-Iter] C. T. Kelley, Iterative methods for Linear and Nonlinear Equations, SIAM, 1995.
• [LT] S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods, Springer-Verlag, 2005.
• [Saad] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM 2003.
• [Stewart] G. W. Stewart, Matrix Algorithms, Volume II: Eigensystems, SIAM, 2001.
• [SB] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd edition, Springer-Verlag, 2002.

DIFFERENTIAL EQUATIONS

Students choosing to be examined in Differential Equations must select an examination covering ONE of the following full-year sequences:
• MATH 670-671 (Ordinary Differential Equations)
• MATH 673-674 (Classical Partial Differential Equations)
Students are urged to indicate the course they wish to be examined on well in advance o the examination date, and to discuss this choice with the Chairman of the appropriate Field Committee (PDE or ODE).The field committees have prepared detailed syllabi for these courses.

Ordinary Differential Equations

(This was last revised Spring 1992.)

In addition to the topics listed below, it is assumed that the student has facility with the necessary background material. This includes (at least ) linear algebra through the Jordan canonical form, advanced calculus through the implicit function theorem and vector calculus, topological concepts such as compactness in metric spaces, equicontinuity, and other preparatory topics of a similar level.

Existence, uniqueness, and continuation of solutions. Continuous and differentiable dependence on parameters and on initial data. This include Gronwall's inequality, Peano's existence theorem, and the uniqueness theorem for systems that satisfy a Lipschitz condition.

Autonomous and non-autonomous systems, the time-t map, phase-space and flow, first integrals, phase volume and the divergence of the vector field. Poincaré sections.

Poincaré-Bendixon theory.

Linear systems. Constant coefficients, including generalized eigenvectors. Variable coefficients, Wronskian, the fundamental matrix, Floquet theory, monodromy.

Stability (including uniform, Lyapunov, and asymptotic) for linear and non-linear autonomous and non-autonomous systems, Lyapunov functions.

Topological methods in ODE's including index theory for planar systems and the Brouwer Fixed Point Theorem.

Hamiltonian systems with one degree of freedom.

Sturm-Liouville theory: Sturm's comparison theorem and Sturm's oscillation theorem. Eigenvalues and eigenfunctions.

Horseshoes, related results of topological and symbolic dynamics.

Stable and unstable manifolds of fixed (=equilibrium) points.

Bifurcation theory: Eigenvalues of the linearization at a stationary point crossing the imaginary axis. The Hopf bifurcation.

References:

• J. Hale: "Ordinary Differential Equations"
• V. Arnold: "Ordinary Differential Equations", MIT Press, 1978
• P.H. Hartman: "Ordinary Differential Equations
• J. Guckenheimer, J. Moser, S. Newhouse: "Dynamical Systems"
• D. Ruelle: "Theory of Ordinary Differential Equations"
• L. Perko: "Differential Equations and Dynamical Systems"
• N. Markley: "Principles of Differential Equations", Wiley, 2004

PARTIAL DIFFERENTIAL EQUATIONS

MATH-AMSC 673-674

LAPLACE'S EQUATION

Dirichlet and Neumann boundary value problems. Fundamental solution. Mean value property. Maximum principle, uniqueness, subharmonic functions. Harnack's inequality. Regularity of harmonic functions. Green's function, Poisson kernel. Energy methods: Uniqueness, Dirichlet's principle. Boundary integral method for the Dirichlet problem.

HEAT EQUATION

Initial boundary value problems. Fundamental solution, Duhamel's principle. Weak maximum principle. Energy methods, uniqueness.

WAVE EQUATION

D'Alembert's formula. Solution by spherical means (n=3), method of descent (n=2). Huygen's phenomenon, finite propagation speed. Duhamel's principle. Energy methods: Uniqueness, domain of dependence.

FIRST ORDER NONLINEAR PDE

1. Characteristic curves. Local existence of solutions.
2. Conservation laws (scalar laws and the p-system) Solution by characteristics, development of singularities, Riemann invariants. Weak solutions, Rankine-Hugoniot jump conditions, nonuniqueness. Shock waves, Lax shock conditions.

CLASSICAL METHODS FOR CLASSES OF PDES

1. Maximum principles for elliptic and parabolic PDE. Weak and strong maximum principles. Hopf's principle.
2. Distributions - elementary properties. Localization, convergence. Derivatives, integrals, convolution. Fundamental solutions for ODE's, basic PDE's.
3. Fourier transform. Convolution, solution of PDE's.

HILBERT SPACES OF FUNCTIONS

1. Fundamental notions. Definition of Banach space. L² as a space of distributions.
2. Orthogonal projection, orthonormal bases, Fourier series.
3. Duality in Hilbert space. Riesz representation theorem. Uniform boundedness principle. Weak convergence. Weak compactness of the unit ball in separable Hilbert space.
4. Sobolev spaces based on L² : H^k(R^m). Sobolev embedding theorem, compact embeddings.

BOUNDED OPERATORS ON HILBERT SPACES OF FUNCTIONS

1. Domain, range, inverse, norm of operators.
2. Spectrum and resolvent.
3. Compact operators. Finite rank, Hilbert-Schmidt operators. The Fredholm alternative.
4. Spectrum of compact symmetric operator. The spectral theorem ( Hilbert-Schmidt theorem).

LINEAR ELLIPTIC EQUATIONS

1. Ellipticity. Classical, strong, weak, and distributional solutions.
2. Lax-Milgram lemma. Existence of weak solutions.
3. Eigenfunction expansions.
4. Sturm-Liouville boundary value problems.

References:

• L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
• M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Texts in Appl. Math. 13, Springer, 1993

Other References:

• F. John, Partial Differential Equations (4th ed), Springer, 1982
• D. Gilbarg and N. Trudinger, Elliptic PDE of Second Order (2nd ed), Springer, 1983
• M.H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, 1967
• R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, vol I (1953, vol II (1962).
• G. B. Folland, Introduction to PDE, Princeton, 1976.
• I. Gohberg and S. Goldberg, Basic operator theory, Birkhauser, 1981.
• S. G. Mikhlin, Mathematical Physics; an advanced course, North-Holland, 1971 (In Russian).

PROBABILITY

Master's Level Requirements (STAT 410, 650)

Foundations: Probability spaces, axioms, conditional probability and independence, Bayes' Theorem.

Discrete random variables: combinatorial probability, discrete densities, Bernoulli trials, expectations and moments, Poisson Limit Theorem.

General Random Variables and Vectors: definitions, distribution functions, densities, moments, change-of-variable formulas, joint distributions, conditional distributions, mixed distributions, moment generating functions, and characteristic functions.

Limit Theorems of Probability: convolutions, concepts of convergence, laws of large numbers, Central Limit theorem.

Discrete-time discrete-state Markov chains: definitions, transition probabilities, classification of states, periodicity, ergodicity, limiting and stationary behavior, absorption probabilities, recurrence.

Continuous-time Markov chains: definitions, birth-and death processes, Kolmogorov forward and backward equations, compound Poisson process.

Branching processes; extinction probabilities.

References

• Hoel, Port, Stone; Introduction to Probability Theory, Houghton-Mifflin (1971). All chapters.
• Karlin & Taylor: A First Course in Stochastic Processes, Academic Press (1975) (2nd edition) ch. 1-5, 8-9.
• Ross: A First Course in Probability, (3rd edition) Macmillan (1988). All chapters, except chapter 10.

Doctoral Level Requirements (STAT 600, 601) Revised June, 2010

PhD exam requirements:

1. Probability space, distribution functions and densities for random variables and vectors; particular distribution functions, Poisson limit theorem, de Moivre-Laplace theorem.
2. Measure-theoretic definition of expectation, Borel sigma-algebra, measure induced by a random variable, classification of measures on the real line, different types of convergence of random variables and their properties, Radon-Nikodym theorem (without proof), L^p spaces.
3. Conditional probabilities, independence of events, sigma-algebras and random variables, Bayes' theorem, pi-systems and Dynkin systems.
4. Markov chains with discrete phase space, law of large numbers, ergodic Markov chains, recurrence and transience, random walks, gambler's ruin problem.
5. Poisson Process, definition of a Markov chain on a general phase space.
6. Borel-Cantelli lemmas, Kolmogorov inequality, three series theorem, laws of large numbers; equivalent sequences and truncation.
7. Weak convergence of measures, Prokhorov theorem (in Euclidean space), Characteristic functions, Gaussian random variables, CLT with the Lindeberg condition.
8. Definition of a random process, Kolmogorov consistency theorem. Conditional expectations and martingales (proofs in the discrete time only), Doob's inequality, Optional stopping theorem, Convergence of Martingales. Definition and basic properties of Brownian motion.

References

• P. Billingsley: Probability and Measure (second edition) J. Wiley (1986). Chapters 1-6 (except starred sections).
• K.L. Chung: A Course in Probability Theory, Academic Press (1974) (2nd Edition). All chapters.
• S. Karlin, H. M. Taylor: A First Course in Stochastic Processes, Academic Press, 1975 (2nd Edition). All chapters.
• L. Koralov, Y. Sinai: Theory of Probability and Random Processes, Springer, 2007.

MATHEMATICAL STATISTICS

STAT 700-701 (Revised Nov. 1993)

Sampling Distributions: functions of samples of normal variables, sample moments, order statistics and quantiles.

Classes of distributions and statistics: Exponential family sufficiency, completeness

Point Estimation: unbiasedness, consistency, methods of estimation including method of moments, maximum likelihood, least squares, UMVU estimators, Rao-Blackwell and Lehmann-Scheffe Theorems, efficiency, information, Cramer-Rao lower bound

Testing and Interval Estimation: Neyman-Pearson lemma, monotone likelihood ratio, UMP tests, likelihood ratio principle and its applications, including multivariate normal, chi square tests, relation between testing and interval estimation.

Asymptotics: modes of convergence, Slutsky's theorem, multivariate central limit theorem, delta method consistency and asymptotic behavior of estimators, including maximum likelihood estimators. Asymptotic properties of likelihood ratio tests.

Linear Models and Least Squares: general model, Gauss-Markov theorem, simple regression, multiple regression, basic ANOVA models, correlation (testing and estimation).

Count Data: multinomial goodness of fit, contingency tables and chi-square tests.

Decision Theory: basic concepts, loss, risk, priors, etc. admissibility and complete classes (Ph.D. only) Minimax principle, least favorable distributions (Ph.D. only) Bayes estimators and tests

Sequential Analysis: (Ph.D. only) SPRT and optimality Wald's identity

Non-Parametric Methods: quantiles, tolerance regions, order statistics, distribution free statistics goodness, of fit tests, empirical distribution function, important tests, e.g., sign, signed rank, Mann-Whitney-Wilcoxon, etc.

References:

• P.J. Bickel and K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day, 1977.
• C.R. Rao, Linear Statistical Inference and Its Applications, Wiley,1973
• V.K. Rohatgi, An Introduction to Probability Theory and Mathematical Statistics, J. Wiley, 1976.
• G.G. Roussas, A First Course in Mathematical Statistics, Addison-Wesley, 1973, Chapters as appropriate to syllabus.

APPLIED STATISTICS

(PhD Level)

STAT 440 Sampling Theory

STAT 740-741 (Linear Models I,II).

1. Sampling Theory. Simple random sampling. Sampling for proportions. Estimation of sample size. Sampling with varying probabilities. Sampling: stratified, systematic, cluster, double, sequential, incomplete. Ratio and regression estimates. Neyman allocation.

2. General Linear Models. Matrix formulation multivariate normal distribution, geometric formulation, least squares, estimable functions. Confidence sets, tests of linear hypotheses under normality, connection with likelihood-based methods. Analysis of residuals, graphical diagnostics, assessment of model fit. Special regression models: polynomial regression and dummy variables. Generalized linear models (GLM).

3. Fixed Effect Analysis of Variance. Comparison of means and one way analysis of variance (ANOVA), full rank and reduced rank models, estimable functions and contrasts, multiple comparisons. Two way ANOVA, interaction, analysis of unbalanced layouts. Nested and crossed factors, incomplete designs.

4. Random Effects and mixed models. Definitions, ANOVA estimates in balanced models, distribution theory. Unbalanced random effect designs, maximum likelihood (ML) and restricted maximum likelihood (REML) estimates. Goodness of fit.

References:

• Cochran, W.G. Sampling Techniques, (1977, 3rd ed.) New York: J. Wiley.
• Cody, R.P. and Smith, J.K. Applied Statistics and the SAS Programming Language, (1997) Upper Saddle River, NJ: Prentice-Hall.
• Draper, N.R. and Smith, H. Applied Regression Analysis. (1998, 3rd ed.) New York: J. Wiley.
• Hocking, R. Methods and Applications of Linear Models. (1996) New York: J. Wiley.
• Lohr, S.L. Sampling: Design and Analysis. (1999) Pacific Grove, CA: Duxbury.
• McCullagh, P. and Nelder, J.A. Generalized Linear Models . (1989, 2nd ed.) New York: Chapman and Hall.
• Milliken, G. and Johnson, D. Analysis of Messy Data, Vol. I: Designed Experiments (1984) New York: Van Nostrand Reinhold.
• Rao, C.R. Linear Statistical Inference and Its Applications (1973, 2nd ed.) New York: J. Wiley.
• Rao, P.S.R.S. Variance Components Estimation (1997) New York: Chapman and Hall.
• Rencher, A.C. Linear Models in Statistics (1999) New York: J. Wiley.
• Sarndal, C.E., Swensson, B., and Wretman, J. Model Assisted Survey Sampling (1992) New York: Springer-Verlag.
• Scheffe, H. The Analysis of Variance (1958) New York: J. Wiley.
• Searle, S.R., Casella, G., and McCulloch, C.E. Variance Components (1992) New York: J. Wiley.
• Stapleton, J.H. Linear Statistical Models (1995) New York: J. Wiley.

APPLIED STATISTICS (MA Level)

STAT 440, 450, 740

1. Sampling Theory. Simple random sampling, stratification, ratio and regression estimates, systematic sampling, cluster sampling, Horvitz-Thompson estimator, two stage sampling, double sampling.
2. Linear Statistical Models. Method of least squares, estimability, Gauss-Markov theorem, hypothesis testing and confidence ellipsoids under normality.
3. Regression and Correlation. Simple and multiple regression models,distribution of correlation coefficient, inference on coefficients (t and F tests), multiple and partial correlation, weighted least squares, effects of model misspecification, analysis of residuals, multicollinearity, alternatives to least squares.
4. Analysis of Variance. One way classification, multiple comparison, balanced two-way classification, fixed vs. Random effects, ANOVA in regression context, incomplete designs, factorial designs, analysis of covariance, effect of non-normality, heteroscedasticity and dependence of errors.

References

• Cochran, W.G. Sampling Techniques, (1977, 3rd.) J. Wiley (Chapters 1-5, 6-9, 9A.1-9A.3, 9A.7, 10.1-10.4, 12.1-12.9).
• Draper, N. and Smith, H. Applied Regression Analysis. (1981, 2nd ed.)J. Wiley (Chapters 1-6, 9).
• Scheffe, H. The Analysis of Variance (1959) J. Wiley (Chapters 1,2.1-2.9, 3, 4.1-4.3, 5-8, 10).
• Rao, C.R. Linear Statistical Inference and Its Applications (1973, 2nd ed.) J. Wiley Chapter 4)

TOPOLOGY

MATH 730 - 734


Students prepare for the topology exam by taking the first year topology sequence MATH 730, MATH 734. Any of the following topics, as well as any of the material in the listed sections of Bredon's "Topology and Geometry", may be tested on the Qualifying Examination.


• Survey of basic point set topology
• Identification spaces
• Simplicial complexes, cell complexes
• Continuous maps
• Metric spaces
• Fundamental Group
• Covering Spaces
• Group presentations
• Van Kampen's theorem
• Definition of higher homotopy Groups
• Smooth manifolds
• Statement of the Classification of surfaces
• Implicit function theorem, Inverse function theorem
• Critical points, differentiable maps, transversality
• Definition of Lie group, topological group
• Homology and cohomology
• Singular theory
• Cellular theory
• Hurewicz Theorem
• Euler characteristic
• Cup products, cap products, Poincaré duality
• Universal Coefficient Theorems
• Elementary homological algebra
• Cofibrations

References:

• Text: Bredon, "Topology and Geometry"
• Chapter I (except 6, 16, 17)
• Chapter II (1-7)
• Chapter III (1-6,8,9)
• Chapter IV (1-6, 8-10, 12-19, 21-23)
• Chapter V (6-8)
• Chapter VI (1-5, 7-9)
• Chapter VII (1)
• Armstrong, "Basic Topology"
• Hatcher, "Algebraic Topology"
• Kinsey, "Topology of Surfaces"
• Singer and Thorpe, "Lecture notes on Elementary Topology and Geometry"
• Massey, "Algebraic Topology"
• Greenberg and Harper, "Algebraic Topology: A First Course," Chapters 7-29
• Massey, "Singular homology Theory"
• Vick, "Homology Theory," Chapters 1-5
• Spanier, "Algebraic Topology"
Last updated 12/02/02 by jmr.