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Mathematics & Statistics

**Courses:** 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.**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:**

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

**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.

**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, Pearson, 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.