Study Guide for the Written Ph.D. Comprehensive Exam

Part I: Algebra

1. Vector spaces (linear algebra)
   Matrix theory, vector spaces, linear transformations, linear operators, eigensystems, inner products and norms, orthogonality and projection, bilinear and quadratic forms, spectral theory in finite dimensions, and basic properties of linear groups (such as GL(n), SL(n), the orthogonal and unitary groups).

2. Groups
   Basic facts about groups, including cyclic, dihedral, and symmetric groups, as well as linear (matrix) groups (see above). Also homomorphisms, cosets and quotients, mapping properties and isomorphism theorems, free groups, group operations (actions), applications of group actions to geometric and combinatorial symmetry (e.g. symmetries of regular polyhedra, counting formulas, Burnside's formula), the class equation, the Sylow theorems, the structure theorem for finitely generated abelian groups.

3. Rings
   By a ring we mean a commutative ring with identity. Topics include rings, homomorphisms, ideals, quotients, mapping properties and isomorphism theorems, polynomial rings, adjoining relations, integral domains, fraction fields, prime and maximal ideals, unique factorization domains, modular arithmetic.

4. Fields
   Subfields of the complex numbers, function fields (basic notions only), finite fields, field extensions, basic properties of finite extensions, algebraic vs. transcendental extensions, algebraic closure.

   Suggested courses: Math 4107, Math 4305, and Math 6121. Math 4108 and Math 6122 are also relevant. Suggested reference: Michael Artin, Algebra, especially chapters 1-8, 10, 13.

Part II: Real analysis

1. Set Theory
   Cardinality, cartesian products and the Axiom of Choice, partial orders and Zorn's lemma, well-ordering

2. Measure Spaces
   Sigma-algebras; the sigma-algebras of Borel and Lebesgue measurable sets; measures, including the counting measure; Lebesgue measure; finite and sigma-finite measures; signed measures, complex measures, and product measures

3. Integration Theory
   The Lebesgue integral; integration with respect to a measure or signed measure; convergence theorems, including Fatou's lemma, the monotone convergence theorem, Levi's theorem, Lebesgue's dominated convergence theorem; product measures and the theorems of Fubini and Tonelli; absolutely continuous measures and the Radon-Nikodym Theorem; singular measures and the Lebesgue decomposition

4. Function Theory
   Various modes of convergence; the theorems of Lusin and Egorov; monotone functions and functions of bounded variation; differentiation, absolutely continuous functions and the Fundamental Theorem of Calculus

5. Topological Spaces
   The real number system; metric spaces, including completeness, the Baire category theorem and its consequences; separation axioms; compact spaces and the Tychonoff theorem

6. The Classical Function Spaces
   L^p and l^p spaces for 1 <= p <= infinity; C(K) spaces; Holder's and Minkowski's inequalities, bounded linear functionals on C(K) and on l^p and L^p for 1 <= p < infinity; Ascoli's theorem; the Stone-Weierstrass theorem

7. Elementary Functional Analysis
   Hilbert space, the projection theorem, applications to approximation, completeness and orthonormal bases, representation of bounded linear functionals on Hilbert space, the Hahn-Banach, closed graph and open mapping theorems

   Suggested courses: Math 6337. Math 6338 and Math 6580 are also relevant.
   Suggested references include:
   Folland, Real Analysis: Modern Techniques and their Applications, 2nd edition
   Halmos, Measure Theory
   Hewitt and Stromberg, Real and Abstract Analysis
   Kreyzig, An Introduction to Functional Analysis
   Royden, Real Analysis, 3d edition
   Rudin, Real and Complex Analysis