## General Information

#### 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.*

The Department Written Examination for the Ph.D and M.A. is administered in January and August of each year during the month preceding the first week of classes and is given in the following fields:

- algebra
- analysis
- probability
- statistics
- applied statistics:
- (PhD level)
- (MA level)

- geometry/topology.

NOTE: MATH students do not take the Applied Stat exam, which is for STAT students. See the requirements for the PhD and MA.

M.A. students take the Ph.D. examination and are 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 two examinations.

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 examinations, 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.

The syllabi below are current and reflect the latest thinking of the respective field committees.

## ALGEBRA

### 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

#### 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).

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 is often covered in undergraduate analysis courses.

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

**Review from Advanced Calculus**

- Real and complex numbers. Continuity, sequences and series, compactness on the real line. Vector spaces (over
**R**and**C**), linear maps. ([F])

**Real Analysis**

- Lebesgue measure and integration on the real line, differentiation and monotone functions, absolute continuity, functions of bounded variation, the Fundamental Theorem of Calculus. ([BeCz], Ch. 2, 3, 4; [FiRo], Ch. 2, 3 ,4, 6)
*L*spaces on the real line, including the Hölder and Minkowksi inequalities, the Riesz-Fischer Theorem, bounded linear functionals on^{p}*L*(^{p}**R**). ([BeCz] Ch. 3, 5; [FiRo] Ch. 7)- Convergence theorems: Fatou's Lemma, monotone convergence theorem, dominated convergence theorem, Egoroff's theorem, Vitali's convergence theorem, convergence in measure, convergence in
*L*. ([BeCz] Ch. 3; [FiRo], Ch. 3, 4, 5, 7)^{p}

**References**[BeCz] J. Benedetto and W. Czaja, "Integration and Modern Analysis," Birkhauser, Boston, MA, 2009.

[F] P. M. Fitzpatrick, "Advanced Calculus," 2nd edition, American Math. Soc., 2009.

[FiRo] P. M. Fitzpatrick and H. Royden, "Real Analysis," 4th edition, Pearson, 2010.

**W. Rudin,**

General references

General references

*Real and Complex Analysis*, 3rd ed., McGraw-Hill, 1991

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

Syllabus for Complex ExamSyllabus for Complex Exam with suggestions for study in a pdf file.

## 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:

- Probability space, distribution functions and densities for random variables and vectors; particular distribution functions, Poisson limit theorem, de Moivre-Laplace theorem.
- 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.
- Conditional probabilities, independence of events, sigma-algebras and random variables, Bayes' theorem, pi-systems and Dynkin systems.
- Markov chains with discrete phase space, law of large numbers, ergodic Markov chains, recurrence and transience, random walks, gambler's ruin problem.
- Poisson Process, definition of a Markov chain on a general phase space.
- Borel-Cantelli lemmas, Kolmogorov inequality, three series theorem, laws of large numbers; equivalent sequences and truncation.
- Weak convergence of measures, Prokhorov theorem (in Euclidean space), Characteristic functions, Gaussian random variables, CLT with the Lindeberg condition.
- 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. 2023)**

Sampling Distributions: standard distributions, functions of samples of normal variables, order statistics, conjugate families of priors.

Parameters and statistics: identifiability of parameters, sufficiency, completeness, minimality, Basu’s Theorem, Exponential family definitions and properties

Point Estimation: unbiasedness and consistency, methods of estimation including method of moments, maximum likelihood, Estimating Equations, UMVU estimators, Rao-Blackwell and Lehmann-Scheffe Theorems, efficiency, Fisher Information, Cramer-Rao lower bound

Algorithms for Estimation, including Newton-Raphson and EM Algorithm

Testing and Interval Estimation: Neyman-Pearson lemma, monotone likelihood ratio, UMP tests, likelihood ratio test, including multivariate normal, chi square tests, duality 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 and estimating-equation estimators. Asymptotic properties of likelihood-ratio and Rao-score tests.

Linear Models and Least Squares: general model, multivariate normal distribution, Gauss-Markov theorem, multiple regression, projection and prediction

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

Decision Theory: basic concepts, loss, risk, priors, admissibility and complete classes, Minimax principle, Bayes estimators and tests.

Non-Parametrics: quantiles, ranks, empirical distribution function

**References:**

- P.J. Bickel and K.A. Doksum,
*Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I*, 2nd ed. Chapman & Hall/CRC, April 2015. - Casella, G. and Berger, R.,
*Statistical Inference*, 2nd ed. Duxbury, 2002. - Jun Shao,
*Mathematical Statistics*, 2nd ed. Springer, 2003.

### V. Rohatgi, and A.K. Saleh, *An Introduction to Probability and Statistics*, 2nd ed., Wiley 2001..

## 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

- Sampling Theory. Simple random sampling, stratification, ratio and regression estimates, systematic sampling, cluster sampling, Horvitz-Thompson estimator, two stage sampling, double sampling.
- Linear Statistical Models. Method of least squares, estimability, Gauss-Markov theorem, hypothesis testing and confidence ellipsoids under normality.
- 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. - 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)

## GEOMETRY/TOPOLOGY

Students prepare for the geometry/topology exam by taking the first year topology sequence ** MATH 730, MATH 740. **

### MATH 730

- Review of basics of relevant general topology: topological spaces, compactness, connectedness, the Hausdorff axiom, metrizability and second countability, compactly generated spaces (1 week)
- Quotient spaces, attaching cells (1 week) (Hatcher, Ch. 0)
- Homotopy extension theorem, CW complexes (rest of Hatcher, Ch. 0)
- Fundamental group (Hatcher, Section 1.1)
- Van Kampen's Theorem (Hatcher, Section 1.2)
- Path lifting, Covering spaces (Hatcher, Section 1.3)
- Classification of (topological) surfaces, with calculation of their fundamental groups
- Basics of homology theory (Hatcher, Ch. 2)

### REFERENCES for MATH 730:

**Hatcher, "Algebraic Topology"**(primary reference)- Kinsey, "Topology of Surfaces"
- Singer and Thorpe, "Lecture notes on Elementary Topology and Geometry"
- Bredon, "Topology and Geometry"
- Spanier, "Algebraic Topology"

### MATH 740

- Differentiable manifolds: transversality, inverse and implicit function theorems, immersions and submersions, submanifolds.
- Fiber bundles, basics of Lie groups, vector bundles, tangent and cotangent bundles.
- Differential forms, Lie derivative, integration on manifolds and Stokes' theorem.
- Riemannian metrics, connections, curvature, covariant differentiation.
- Minimizing properties of geodesics, Hopf-Rinow theorem, Jacobi equations.

### REFERENCES for MATH 740:

- M. DoCarmo, "Riemannian Geometry"
- F. Warner, "Foundations of Differentiable Manifolds and Lie Groups"
- J. Lee, "Introduction to Smooth Manifolds"
- B. O'Neill, "Semi-Riemannian Geometry"
- P. Petersen, "Riemannian Geometry"
- J. Milnor, "Topology from the Differentiable Viewpoint"

*Last updated 07/05/17*