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.
Every year, the exams will be given in January and August. Each exam will
be 3 1/2 hours long.
Topics students are expected to know for each exam are listed below. Past exams and tips are available.
- 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,
- Thomas W. Hungerford, Algebra, Springer Verlag, 8th ed.
(1997) ISBN: 0387905189, Chapters 1-3 and 5.
- 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
- 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
- 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
Also useful are the following Schaum's Outlines:
- 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
- Mechanical vibrations: Newton's laws, spring-mass systems, two-mass
oscillators, friction, damping, pendulum, linear stability and equilibria,
energy analysis, phase plane analysis, nonlinear oscillations, control oscillations,
inverse probllem.  pp 1-114 and  Chapters 0-2.
- Traffic flow: Velocities and velocity fields, trafffiic flow and
density, conservation laws, linear and nonlinear car-following models, steady
state, first order partial differenntial equations (the method of characteristics),
green light mooddels and rarefaction solution, shock waves (effect of rred
light annd slower traffic ahead), highway with entrance (inhomogenneous problem),
traffic wave propagation, optimiization problem.  pp. 259-394.
- Dynamical systems: Nonlinear systems in the plane, interacting species,
limit cycles, Hamiltonian systems, Liapunov functions and stability, bifurcation
theory, three-dimensional autonomous systems and chaos, Poincare maps and
nonautonomous systems in the plane, linear discrete dynamical systems. 
Chapters 3-9, 13.
-  Richard Haberman, Mathematical Models, Mechanical Vibrations,
Population Dynamics and Traffic Flow, SIAM Classics in Applied Mathematics
(1987) ISBN: 978-0898714081.
-  Stephen Lynch, Dynamical Systems with Applications Using
Maple, Springer-Verlag (2000) ISBN: 0-8176-4150-5.
- Probability Spaces: basic axioms, conditional
probability and independence
- Denumerable Probabilities: limit sets,
- 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
- 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
- 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
-  Billingsley,
Patrick. Probability and Measure. Wiley, Anniversary Edition, 2012.
- differentiation and Riemann integration of functions of one real variable,
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,
- AC and BV functions, fundamental theorem of calculus, Lebesgue differentiation
- Hilbert spaces, Lp spaces, lp spaces, Hölder
and Minkowski inequalities, completeness.
- 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
- Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill
Science/Engineering/Math, 3rd ed., (1976) ISBN: 007054235X.
- • 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
• Analysis of variance models: fixed effects, nested models, random effects, mixed effects
- 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.
- 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
- 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
- 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.