Syllabi for Core Qualifying Exams
Algebra
Courses: MAS 5107 (Advanced Linear Algebra), MAS 5311 (Algebra I),
MAS 5312 (Algebra II).
Topics: Group theory up to Sylow's theorems; elementary theory of rings
and modules, including the structure of finitely generated modules over a
Euclidean Domain; basic results on finite dimensional vector spaces; the algebra
of linear transformations; eigenvalues; the basic canonical forms; field theory up
to the fundamental theorem of Galois theory; finite fields. The examination is
usually divided into four parts: group theory, theory of rings and modules, field
theory, and linear algebra. Students are expected to work problems in each
part.
References:
- K. Hoffman and R. Kunze, Linear Algebra, second edition,
Prentice-Hall, 1971, all except 5.6, 5.7, Chapters 9 and 10.
- I. Herstein, Topics in Algebra, second edition, Xerox College
Publishing, 1975, Chapters 1-6 and 7.1.
Analysis
Courses: MAA 5306 (Introduction to Real Analysis), MAA 5307 (Real
Analysis I), MAA 6616 (Real Analysis II).
Topics: Differentiation, Riemann-Stieltjes integrals, uniform
convergence, Fourier series, special functions; Lebesgue measure and integration
on the real line, classical Banach spaces; Banach spaces, measure and integration,
Riesz Representation Theorem, Radon-Nykodym Theorem.
References:
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1964 and
1976, Chapters 5-8.
- H. L. Royden, Real Analysis, Macmillan, 1988, Chapters 1-6, 10-14.
- J. McDonald and N. Weiss, A Course in Real Analysis, Academic Press,
1999, Chapters 1-6, and 9.
Mathematical Statistics
Courses: STA 5326 (Mathematical Statistics I), STA 6326 (Mathematical
Statistics II).
Topics: Sampling distributions; point estimation, interval estimation;
hypothesis testing; order statistics and applications. Sequential analysis;
sequential sampling, estimation and testing. Decision theory and Bayesian
analysis; utility and loss; prior information; Bayesian inference and hypothesis
testing; empirical Bayes analysis; robustness. Min-max analysis, Bayesian
sequential analysis.
References:
- V. K. Rohatgi, An introduction to probability theory and statistics,
second edition, Wiley, 2001, Chapters 7-11.
- J. O. Berger, Statistical decision theory and Bayesian analysis,
second edition, Springer Series in Statistics, Springer-Verlag, 1985.
- V. P. Savchuk and C. P. Tsokos, Bayesian statistical methods with
applications to reliability, World Federation Publishers, 1996, Chapters
1-7.
Topology
Courses: MTG 5316 (Topology I), MTG 5317 (Topology II).
Topics:
- Point Set Topology: topological spaces, continuity, product topology,
quotient topology, metric topology, connectedness and compactness, countability
and separation axioms, Urysohn Lemma and applications, Tychonoff's theorem,
compactifications, paracompactness.
- Algebraic Topology: compact 2-manifolds, fundamental group, homotopy,
van Kampen theorem, covering spaces, homology theory (singular or simplicial),
exact sequences, Mayer-Vietoris theorem.
References:
- Point Set Topology: J. R. Munkres, Topology, second edition,
Prentice-Hall, 2000, Chapters 1-6.
- Algebraic Topology: J. R. Munkres, Topology, second edition,
Chapters 9, 11, 12, 13, and J. R. Munkres, Elements of Algebraic
Topology, Addison-Wesley, 1984, Chapters 1-3, or W. S. Massey,
A Basic Course in Algebraic Topology, Springer-Verlag, 1991, Chapters
I-VIII.