## Qualifier Examination Information

All Ph.D. students must pass three examinations in different areas of advanced mathematics or statistics. Following is the relevant part of the Ph.D. program description:

*"Students must pass three written examinations. Two of these will be chosen from the areas Algebra, Combinatorics and Real Analysis. The third will be chosen from the areas Mathematical Modeling, Applied Statistics and Probability & Mathematical Statistics. Normally, these will be taken within a year of completion of the core coursework. These examinations need not be taken together. Usually, at most two attempts at passing this examination will be permitted. Students who wish to make a third attempt must petition the Graduate Studies Committee of the department for permission to do so."*

The exams are offered every year in January and August. Each exam is 3 1/2 hours long.

Past exams are available:

Topics students are expected to know for each exam are listed below.

## Algebra

- Groups: homomorphisms and subgroups, cyclic groups, cosets and counting, normality, quotient groups, symmetric, alternating and dihedral groups, direct product and direct sum, finitely generated abelian groups, group action, the Sylow Theorems, nilpotent and solvable groups, normal and subnormal series.

- Rings: rings and homomorphisms, ideals (prime, maximal), factorization in commutative rings, unique factorization domains, principal ideal domains, euclidean domains, polynomial rings, Eisenstein's criterion.

- Fields and Galois Theory: field extensions, splitting fields, Galois group, separability, cyclic extensions, finite fields, cyclotomic extensions, radical extensions.

#### Suggested Reference:

- Thomas W. Hungerford, Algebra, Springer Verlag, 8th ed. (1997) ISBN: 0387905189, Chapters 1-3 and 5.

## Combinatorics

- Advanced Counting: sequences, selections, distributions, partitions, lattice paths, Catalan numbers, Stirling numbers, Ferrers diagrams, Ramsey numbers, inclusion-exclusion, derangements, recurrence relations, generating functions (both ordinary and exponential), characteristic polynomial method to solve recurrences, Stirling's approximation, Polya counting

- Elementary Graph Theory: graphs, digraphs, subgraphs, degrees, adjacency matrices, incidence matrices, graph isomorphism, paths, cycles, trees, connectivity, bipartite graphs, edge contraction, subdivisions, linegraphs, independent sets, cliques

- Flows and Related Graph Concepts: network flows, integral flows, Max-Flow/Min-Cut theorem, (edge and vertex) connectivity, vertex cuts, edge cuts, perfect matchings, Menger's theorem, Hall's marriage theorem, Tutte's theorem, factorizations, Petersen graph, Konig-Egervary theorem

- Planar Graphs: drawings in the plane, planar duals, Euler's formula, Kuratowski's theorem, convex embeddings in the plane, coloring planar graphs

- Graph Coloring: vertex and edge coloring, Brooks' theorem, Vizing's theorem, color critical graphs, perfect graphs, Lovasz's Perfect graph theorem, the Strong Perfect Graph Theorem

- Miscellaneous: Ramsey graphs, Hamiltonion cycles, random graphs, Turan's theorem, partially ordered sets, Dilworth's theorem

#### Suggested References

- Kenneth P. Bogart, Introductory Combinatorics, Harcourt-Academic Press, 3rd ed. (2000) ISBN 0121108309
- Douglas B. West, Introduction to Graph Theory, Prentice Hall, 2nd Edition (2000), ISBN: 0130144002

- V. K. Balakrishnan, Schaum's Outline: Graph Theory, McGraw Hill (1997) ISBN: 0070054894
- V. K. Balakrishnan, Schaum's Outline: Combinatorics (including graph theory), McGraw Hill (1995) ISBN: 007003575X

## Mathematical Modeling

**Basic Theory**: Linearized stability and Grobman-Hartman Theorems for maps and ODEs, Lyapunov functions, Poincare-Bendixson Theorem, existence and uniqueness of planar cycles, Bendixson’s criterion, Dulac’s criterion, planar Hamiltonian and related systems, Poincare maps**Bifurcation Theory**: Andronov-Hopf bifurcation, one-parameter bifurcations of equilibria, periodic-doubling bifurcation, planar Neimark-Sacker bifurcation, center manifolds, normal forms, two-parameter bifurcations of equilibria, homoclinic bifurcations, heteroclinic bifurcations**Applications**: Competition models, prey-predator models, infectious disease models, mechanical vibration models, reaction kinetic models

#### Suggested References

- Elements of applied bifurcation theory, Yuri A. Kuznetsov (supplemental materials https://webspace.science.uu.nl/~kouzn101/NLDV/index.html )
- Differential equations and dynamical systems, Lawrence Perko

## Probability

- Probability Spaces: basic axioms, conditional probability and independence
- Denumerable Probabilities: limit sets, Borel-Cantelli Lemma
- Existence and Extension: fields, sigma-fields
- Simple and Discrete Random Variables: expectation, Chebyshev-Markov inequalities, convergence of random variables
- The Law of Large Numbers: The Strong Law, The Weak Law
- Abstract Measures and Integration Theory: basic definitions and theorems, Fatou’s Lemma, Lebesgue Dominated Convergence Theorem, Monotone Convergence Theorem
- Random Variables and Expected Values: Random variables and distribution functions, densities, expected values, independence
- Sums of Independent Random Variables: Kolmogorov’s 0-1 Law
- Distribution on R: weak convergence, Mapping Theorem, convergence in probability and almost sure convergence, Skorokhod Representation Theorem, Helly’s Selection Theorem, characteristic functions and moments, Central Limit Theorems (classical CLT, Lindeberg’s CLT, Lyapounov’s CLT)
- Distributions on R^k: weak convergence, Cramer-Wold Device Theorem, characteristic functions, multivariate normal distributions, Multivariate Central Limit Theorem
- Conditional Expectations: conditional expectation given a sigma-field, properties, Smoothing-Towering Theorem
- Stochastic Processes: Martingales, Brownian motion

#### Suggested References

[1] Billingsley, Patrick. *Probability and Measure*. Wiley, Anniversary Edition, 2012.

## Real Analysis

- differentiation and Riemann integration of functions of one real variable, sequences of

functions, uniform convergence, Lebesgue's characterization of Riemann integrability

- topology of the line, countable and uncountable sets, Borel sets, Cantor sets and Cantor

functions, Baire category theorem

- Lebesgue measure and integration on the line, measurable functions, convergence theorems

- AC and BV functions, fundamental theorem of calculus, Lebesgue differentiation theorem

- Hilbert spaces, L
_{p}spaces, l_{p}spaces, Hölder and Minkowski inequalities, completeness.

#### Suggested References:

- H. L. Royden, Real Analysis, Prentice Hall, 3rd ed. (1988) ISBN: 0024041513 Chapters 1-7, 11, 12.
- Walter Rudin, Real and Complex Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed. (1986) ISBN: 0070542341, Chapters 4-5.

Another book that contains more elementary background information is the following.

- Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed., (1976) ISBN: 007054235X.

## Statistics

- • Families of distributions: exponential families, location and scale families

• Populations, samples, statistics, order statistics

• Sampling distributions in the normal case (t, chi-squared, F)

• Data reduction: sufficiency, factorization theorem, completeness

• Parametric point estimation: maximum likelihood, method of moments, Bayes estimation

• Unbiasedness and variance: Cramér-Rao lower bound, Rao-Blackwell theorem, Lehmann-Scheffé theorem, uniform minimum variance unbiased estimators

• Hypothesis testing: likelihood ratio tests, size, level, power, Neyman-Pearson lemma, uniformly most powerful tests

• Interval estimation: confidence, inverting a test, pivotal quantities

• Asymptotic properties of sequences of estimators: consistency, efficiency, asymptotic normality of MLE, asymptotic behavior of LRT

• Simple linear regression model: estimation, hypothesis testing, confidence estimation

• Regression diagnostics: residuals, outliers, influence

• Variance stabilizing transforms and weighted least squares

• Regression on functions of several variables: estimation, hypothesis testing, confidence estimation, special design matrices (polynomial, qualitative predictors), diagnostics

• Analysis of variance models: fixed effects, nested models, random effects, mixed effects

#### Suggested References:

- Berger, George and Berger, Roger L. Statistical Inference, second edition. Duxbury, 2001.

Hocking, Ronald R. Methods and Applications of Linear Models: Regression and the Analysis of Variance, third edition. Wiley, 2013.

## Mathematical Physics

- Particles and fields, recapitulation of quantum mechanics of 1930's, general principles of quantum mechanics including: linear operators, states and observables, fundamental principles of quantum mechanics, compatibility of observations, representations and transformations, time-dependent equation, and the density matrix. Central forces: orbital angular momentum, and motion under central forces. Approximation methods: the method of variation, the method of perturbation, the method of perturbation-variation, and the method of Wentzel-Kramers-Brillouin.

- Rotation and angular momentum, coupling of two angular momenta, scalar, spinor, and vector fields, spin-dependent interactions, polarization of particles with spin, system of particles, LS-coupling and jj-coupling, two-fermion system in LS-coupling, the helium atom, configuration mixing, systems of more than two fermions, calculation of single-particle wave function (Hartree-Fock approximation), time-dependent perturbation theory, scattering, and the method of partial waves.

#### Suggested References

- R. Shankar, Principles of Quantum Mechanics (2nd Ed.), Kluwer Academic/Plenum Publishers, 1994, ISBN: 0306447908.
- N. Zettili, Quantum Mechanics: Concepts and Applications , John Wiley & Sons, 2001, ISBN: 0471489441.