The Graduate Program
Written Qualifying Examination Study Guide
This document provides study guides for subjects in which written qualifying exams are given. Each guide lists topics that a student should know for the corresponding examination. An attempt has been made to put these topics in coherent order and to provide useful references. The study guides are not intended as syllabi for the corresponding graduate courses. For each of the subject areas, an introductory one-semester 8000-level course is designed to help prepare students for the qualifying exam. However, certain background material may be assumed in the 8000-level course, and the course might omit some of the topics in the study guide and include topics not appearing in the study guide. Thus it is the student's responsibility to prepare adequately for a written qualifying exam by mastering the topics on the study guide.
Written qualifying exams are offered in Algebra, Complex Analysis, Numerical Analysis, Real Analysis, Probability, and Topology. All Ph.D. candidates must pass the qualifying exams in both Complex Analysis and Real Analysis, and two other qualifying exams, including either Algebra or Topology. The Complex Analysis and Real Analysis exams are 2 hours long, and the other exams are all three hours in length. The written qualifying exams are offered every year in August before the start of fall semester classes, and in January before the start of spring semester classes. Study guides and copies of previous qualifying exams are available on the Graduate Program website.
[November 2006]
Algebra
Complex Analysis
Numerical Analysis
Real Analysis
Probabiltiy
Topology
Study Guide for Algebra Exam
Group Theory
subgroups and quotient groups
Lagrange's Theorem
fundamental homomorphism theorems
group actions with applications to the structure of groups such as the Sylow Theorems
group constructions such as:
direct and semi-direct products
structures of special types of groups such as:
p-groups
dihedral, symmetric and alternating groups, cycle decompositions
the simplicity of An, for n ≥ 5
free groups, generators and relations
solvable groups
References: [1,3,4]
Linear Algebra
determinants
eigenvalues and eigenvectors
Cayley-Hamilton Theorem
canonical forms for matrices
linear groups (GLn, SLn, On, Un)
dual spaces, dual bases, induced dual map, double duals
finite-dimensional spectral theorem
References: [1,2,4]
Foundations
Zorn's Lemma and its uses in various existence theorems such as that of a basis
for a vector space or existence of maximal ideals.
References: [1,3,4]
Theory of Rings and Modules
basic properties of ideals and quotient rings
fundamental homomorphism theorems for rings and modules
characterizations and properties of special domains such as:
Euclidean implies PID implies UFD
classification of finitely generated modules over PIDs with emphasis on Euclidean domains
applications to the structure of:
finitely generated abelian groups
canonical forms of matrices
References: [1,3,4]
Field Theory
algebraic extensions of fields
fundamental theorem of Galois theory
properties of finite fields
separable extensions
computations of Galois groups of polynomials of small degree and cyclotomic polynomials
solvability of polynomials by radicals
References: [1,3,4]
As a general rule, students are responsible for knowing both the theory (proofs) and practical applications (e.g. how to find the Jordan or rational canonical form of a given matrix, or the Galois group of a given polynomial) of the topics mentioned. A supplement to this study guide is available at Algebra PhD qual remarks.
References
[1] David Dummit and Richard Foote, Abstract Algebra, Wiley, 2003.
[2] Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice-Hall, 1971.
[3] Thomas W. Hungerford, Algebra, Springer, 1974.
[4] Roy Smith, Algebra Course Notes (843-1 through 845-3), http://www.math.uga.edu/~roy/, 1996.
[Revised November, 2006]
Study Guide for Complex Analysis Exam
I. Calculus and Undergraduate Analysis
Continuity and differentiation in one and several real variables
Inverse and implicit function theorems
Compactness and connectedness in analysis
Uniform convergence and uniform continuity
Riemann integrals
Contour integrals and Green’s theorem
Reference: [3].
II. Preliminary Topics in Complex Analysis
Complex arithmetic
Analyticity, harmonic functions, and the Cauchy-Riemann equations
Contour Integration in C
References: [1] Chapters 1, 2; [2] Chapters 1, 2, 4; [4] Chapter 1.
III. Cauchy's Theorem and its consequences
Cauchy's theorem and integral formula, Morera’s theorem, Schwarz reflection
Uniform convergence of analytic functions
Taylor and Laurent expansions
Maximum modulus principle and Schwarz’s lemma
Liouville's theorem and the Fundamental theorem of algebra
Residue theorem and applications
Singularities and meromorphic functions, including the Casorati-Weierstrass theorem
Rouche’s theorem, the argument principle, and the open mapping theorem
Estimates using Cauchy Integral Formula: Cauchy inequalities and, more generally, bounds on holomorphic functions and their derivatives on compact sets
References: [1] Chapters 4, 5, 6; [2] Chapters 5, 7, 8, 9; [4] Chapters 2, 3, 5, 8 (§2,3).
IV. Conformal Mapping
General properties of conformal mappings
Analytic and mapping properties of linear fractional transformations
Automorphisms of the disk, plane, and Riemann sphere
References: [1] Chapters 3, 8; [2] Chapters 3, 4; [4] Chapter 8 (§1,2).
References
[1] L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill.
[2] E. Hille, Analytic Function Theory, Vol. 1, Ginn and Company.
[3] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill.
[4] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.
[Revised June 2007]
Study Guide for Numerical Analysis Exam
Number systems and errors in digital computation, machine unit round off error.
References: [1,2]
Numerical solution of nonlinear equations. References: [1,2]
Interpolation theory and applications. References: [1,2]
Numerical integration in one or more dimensions. References: [1,2]
Spline theory and applications in computer graphics. References: [1,2,3,4]
Numerical differentiation. References: [2,5]
Remainder theory and Peano's Theorem. References: [2,5]
Approximation theory and applications. References: [1,2]
Direct and iterative methods for linear systems. References: [1,2]
Algebraic eigenvalue problem. References: [1,2]
Numerical solution of systems of ordinary differential equations.
References: [1,2]
Numerical methods for boundary value problems involving ordinary differential equations. Reference: [1]
Solution of systems of nonlinear equations. References: [1,2]
Optimization and nonlinear least squares techniques. References: [1,2]
References
[1] Burden, R.L. and Faires, J.D., Numerical Analysis, 4th edition, PWS Publishers, 1985
[2] Atkinson, K.E., An Introduction to Numerical Analysis, 2nd edition, John Wiley and Sons, 1989
[3] Brodies, K.W. (ed.), Mathematical Methods in Computer Graphics and Design, Academic Press, 1980
[4] Swan, T., Mastering Turbo Pascal 5.5, Hayden Books, 1989
[5] Davis, P., Interpolation and Approximations, Blaisdell, 1965
Study Guide for Probability Theory Exam
MATHEMATICAL FOUNDATION OF PROBABILITY IS ASSUMED:
Random variables (r.v.s), expectation and higher moments of r.v.s, Fatou's lemma, monotone and dominated convergence theorems; inequalities of Markov, Chebyshev, Holder, Minkowski, and Jenson.
Convergence; Distribution Functions and Characteristic Functions:
Weak convergence of probability measures, Alexandrov theorem, tightness and weak compactness, Prohorov theorem.
Infinitely divisible distribution and Levy-Khintchine representation.
References: [1,3,4,5]
Laws of Large Numbers
Sums of independent r.v.s, Khintchine-Kolmogorov theorem
Kolmogorov's Three-series and Two-series theorems
Weak and Strong laws of large numbers
References: [1,2,3,4,5]
Central Limit Theorems
Various central limit theorems and rates of convergence
Convergence in distribution to infinitely divisible distributions
References: [1,2,4,5]
Discrete-time Martingales
Martingales and semimartingales
Doob's inequalities (including upcrossing inequality)
Optional sampling and convergence theorems
References: [1,2,4,5]
References
[1] K.L. Chung: A Course in Probability Theory, 2nd Edition, Academic Press, N.Y., 1978.
[2] Y.S. Chow and H. Teicher: Probability Theory, 2nd Edition, Springer-Verlag, N.Y., 1988.
[3] B.V. Gnedenko and A.N. Kolmogorov: Limit Distributions for Sums of Independent Random Variables, 2nd Edition, Addison-Wesley, Massachusetts, 1961.
[4] R.G. Laha and V.K. Rohatgi: Probability Theory, John Wiley, N.Y., 1979.
[5] A.N. Shiryayev: Probability, Springer-Verlag, N.Y., 1984.
Study Guide for Real Analysis Exam
I. Calculus and Undergraduate Analysis
Continuity and differentiation in one and several variables
Compactness and connectedness in analysis
Sequences and series
Uniform convergence and uniform continuity
Taylor's Theorem
Riemann integrals
Reference: [2]
II. Measure and Integration
Measurability:
Measures in Rn and on σ-algebras
Borel and Lebesgue measures
Measurable functions
Integrability:
Integrable functions
Convergence theorems (Fatou’s lemma, monotone and dominated
convergence theorems)
Characterization of Riemann integrable functions
Fubini and Tonelli theorems
Lebesgue differentiation theorem and Lebesgue sets
References: [1] Chapters 1, 2, 3; [3] Chapters 3, 4, 5, 11, 12; [4] Chapters 1, 2, 3, 6.
III. Lp and Hilbert Spaces
Lp space: Holder and Minkowski inequalities, completeness, and the dual of Lp
Hilbert space and L2 spaces: orthonormal basis, Bessel’s inequality, Parseval’s identity,
Linear functionals and the Riesz representation theorem.
References: [1] Section 5.5, Chapter 6; [3] Chapter 6; [4] Chapter 4.
References
[1] G. Folland, Real Analysis, 2nd edition, John Wiley & Sons, Inc.
[2] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill.
[3] H. L. Royden, Real Analysis, 3rd edition, Macmillan.
[4] E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press.
[Adopted May 2006]
Study Guide for Topology Exam
General Topology
Topological spaces, continuous functions, product and quotient topology [1, ch. 2]
Connectedness and compactness [1, ch. 3]
Countability and separation axioms, Urysohn lemma, Tietze theorem [1, ch. 4, except §36]
Complete metric spaces and function spaces [1, §43, 45]
Algebraic Topology
Classification of surfaces [2, ch. I]
Fundamental group [2, ch. II], [3, §1.1]
van Kampen’s theorem [2, ch. III, IV], [3, §1.2]
Classification of covering spaces [2, ch. V], [3 §1.3]
Homology:
simplicial, singular, cellular; computations and applications [3, ch. 2], [4, ch. 4]
Degree of a map of Sn [3, p. 134], [4, §21]
Euler characteristic [3, p. 146], Lefschetz fixed point theorem [3, p. 179], [4, §22]
The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.
References
[1] J. Munkres, Topology, second edition, Prentice-Hall, 2000.
[2] W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.
[3] A. Hatcher, Algebraic Topology, Cambridge U. Press, 2002.
(Revisions and corrections http://www.math.cornell.edu/~hatcher/AT/ATpage.html )
[4] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
[Revised May 2006]
