## Ph.D. Qualifying - Written Exams in Mathematics - Regulations

### Ph.D. Qualifying - Written Exams in Mathematics - Regulations

Students should complete the written part consisting of three subject exams during the first five semesters of their Ph.D. program.

The exam subjects are:

Group A:

1. Real Analysis
2. Complex Analysis
3. Ordinary Differential Equations
4. Partial Differential Equations

Group B:

1. Algebra
2. Finite Fields and Applications
3. Function Fields

The student is required to take a written exam on Real Analysis from Group A and Algebra from Group B. S/he decides on another exam subject from the above list, One of the exam subjects may be from another University Program; in this case the approval of the advisor and the Program coordinator are required.

### Exam Subjects

#### A1) Real Analysis

Topics:

1. Metric spaces; completeness, compactness and connectedness. Continuous functions. The contraction mapping theorem. Ascoli- Arzela theorem.
2. The Lebesgue measure. General measures. Convergence theorems. Decomposition theorems, Hahn decomposition, Radon-Nikodym theorem. Product measure, Fubini and Tonelli's theorems.
3. Normed spaces. Open mapping, Closed graph theorems, Hahn Banach theorem. Uniform Boundedness principle, weak topologies. Linear Operators. Hilbert spaces.

Sources

• Classical Analysis. J. Marsden, M. Hoffman, Freeman.( 1)
• Real Analysis. W. Rudin, (1, 2 and 3)
• Real Analysis. H. L. Royden (1, 2 and 3)
• Introductory Real Analysis. A. N. Kolmogorov, S. V. Fomin (Dover Books) (1, 2 and 3)
• Real Analysis. G. Folland (1, 2 and 3)

#### A2) Complex Analysis

Topics:

1. Elementary Properties of Analytic Functions: Power series expansions, Complex line integrals, Complex differentiation, Cauchy-Riemann equations, Cauchy's theorem and Integral Formula, Open mapping theorem, Classification of isolated singularities, Laurent expansions, Calculus of residues.
2. The Argument Principle: The index of a closed curve, general form of Cauchy's theorem, Residue theorem, The Argument Principle, Rouche's theorem.
3. The Maximum Modulus Principle: The Maximum Modulus Principle, Schwarz Lemma, One-to-one holomorphic mappings of the unit disc onto itself, Mobius transformations.
4. Zeros and Poles of Analytic Functions: Runge's theorem, Meromorphic functions, Infinite products, Weierstrass Factorization theorem.
5. Analytic Continuation: Analytic continuation along a path, Monodromy theorem

Sources:

• L. V. Ahlfors, Complex Analysis, McGraw-Hill Inc., 1966.
• J. B. Conway, Functions of One Complex Variable, Springer - Verlag, 1978.
• W. Rudin, Real and Complex Analysis, McGraw-Hill Inc., 1966.

#### A3) Ordinary Differential Equations

Topics:

1. Existence-Uniqueness theorem
2. Nonlinear systems and stability
3. Limit Sets;The Poincare-Bendixon theorem
4. Bifurcation: Bifurcation of Fixed Points
5. Boundary value problems
6. The Sturm comparison theorem
7. Eigenvalues and Eigenfunctions for Linear Differential Operators

Sources:

• Ordinary Differential Equations: E. Coddington, N. Levinson
• Ordinary differential equations: R. K. Miller and A. N. Michel,
• Differential Equations, Dynamical Systems, and Linear Algebra: M. W. Hirsh and S. Smale

#### A4) Partial Differential Equations

Topics:

1. First order PDE's, characteristics. The Cauchy-Kowalevski theorem. Classification.
2. Hyperbolic equations (the wave equation);solution formulas, characteristics, Cauchy and initial/boundary value problems, energy method.
3. Elliptic equations (Laplace equation); maximum principles, fundamental solutions, Green's function, Poisson's formula, solution of the Dirichlet's problem.
4. Parabolic equations (the heat equation); fundamental solution, maximum principle, energy methods.

Sources:

• Partial Differential Equations: F. John
• Partial Differential Equations: L. Evans
• Partial Differential Equations: J. Wloka

#### B1) Algebra

Topics:

1. Groups: Groups, subgroups, normal subgroups, cosets, quotient groups, Lagrange’s theorem, cyclic groups, homomorphisms, isomorphism theorems, symmetric, alternating and dihedral groups, direct products, free abelian groups, finitely generated abelian groups, action of a group on a set, Sylow theorems.
2. Rings: Rings, subrings, homomorphisms, ideals, prime and maximal ideals, quotient rings, isomorphism theorems for rings, direct products and Chinese remainder theorem, ring of quotients and localization, factorization in commutative rings, unique factorization domains, Euclidean domains, polynomial rings, factorization in polynomial rings.
3. Fields: Field extensions: algebraic and transcendental extensions, simple extensions and their characterization, Galois extensions and the fundamental theorem of Galois theory, splitting fields, algebraic closure, separability, normality, fundamental theorem of Galois theory, structure of finite fields, cyclic extensions, cyclotomic extensions.

Sources:

• Algebra: T. W. Hungerford
• Algebra: S. Lang
• Topics in Algebra: I. N. Herstein
• Abstract Algebra: D. S. Dummit, R. M. Foote

#### B2) Finite Fields and Applications

Topics:

1. Structure of Finite Fields: Characterization of finite fields, roots of irreducible polynomials, trace, norm, bases, roots of unity and cyclotomic polynomials, representation of elements of finite fields.
2. Polynomials over Finite Fields and their Factorization: Order of polynomials, primitive polynomials, irreducible polynomials, factorization of polynomials over small and large finite fields, calculation of roots of polynomials.
3. Bases: Polynomial bases, normal bases and their existence, arithmetic in normal bases representation, the complexity of normal bases, dual bases, self-dual bases.
4. Coding Theory: Linear block codes, Hamming codes, bounds on codes and their asymptotic versions: singleton bound, Plotkin bound, Gilbert-Varshamov bound, sphere packing bound, cyclic codes: generator polynomial, check polynomial, zeros of a cyclic code, BCH codes, Reed-Solomon codes.

Sources:

• Introduction to Finite Fields and Their Applications: R. Lidl, H. Niederreiter
• Finite Fields: Structure and Arithmetics: D. Jungnickel
• Applications of Finite Fields: A. J. Menezes
• Introduction to Coding Theory: J. H. van Lint
• The Theory of Error-Correcting Codes: F. J. MacWilliams, N. J. A. Sloan

#### B3) Function Fields

Topics:

1. Foundations: Algebraic function fields of one variable, places, valuation ring of a place, discrete valuations, the rational function field and its places, weak approximation theorem, divisors, genus of a function field, canonical divisors, Riemann-Roch theorem, strong approximation theorem, Weierstrass gap theorem, Clifford’s theorem.
2. Extensions of function Fields: Algebraic extensions of function fields, ramification index, relative degree, subrings of function fields, local integral bases, Kummer’s theorem, Hurwitz genus formula, the different and Dedekind’s different theorem, constant field extensions, Galois extensions: Kummer and Artin-Schreier extensions, function fields over finite fields, Hasse-Weil Theorem.

Sources:

• Algebraic Function Fields and Codes: H. Stichtenoth
• Rational Points on Curves over Finite Fields: H. Niederreiter, C. Xing
• Algebraic Curves over a Finite Field: J. Hirschfeld, G. Korshmaros, F. Torres.