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Mathematics & Statistics

Syllabi for Core Qualifying Exams

Algebra

Courses: MAS 5107 (Advanced Linear Algebra), MAS 5311 (Algebra I), MAS 5312 (Algebra II).

Topics: Group Theory: basic properties; isomorphism theorems; group action; Sylow's theorems; solvable and nilpotent groups; additional topics may include free groups, group presentations by generators and relators; Rings and Modules: elementary theory of rings and modules; finitely generated modules over a PID; chain conditions; additional topics may include projective and injective modules; primary decomposition; Jacobson radical; semi-simple rings; Linear Algebra: basic results on finite dimensional vector spaces; the algebra of linear transformations; eigenvalues; rational and Jordan canonical forms; Field Theory: field extensions; the fundamental theorem of Galois theory; finite fields; Galois groups of polynomials; radical and cyclotomic extensions. The examination is divided into four parts as described above: group theory, theory of rings and modules, linear algebra and field theory. Students are expected to solve problems in each part.

References:

  1. T. W. Hungerford, Algebra, Springer-Verlag, New York-Berlin, 1980.
  2. I. M. Isaacs, Algebra: A Graduate Course, American Mathematical Society, 2009.

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:

  1. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1964 and 1976, Chapters 5-8.
  2. H. L. Royden, Real Analysis, Macmillan, 1988, Chapters 1-6, 10-14.
  3. J. McDonald and N. Weiss, A Course in Real Analysis, Academic Press, 1999, Chapters 1-6, and 9.

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, or A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.