Qualifying Examinations

Real Analysis

Suggested Syllabus for Real Analysis Qualifying Examination

[O]  Metric Spaces

Distances and metric spaces.  Open sets, closed sets, cluster points, closure of a set.  Dense subsets, separable spaces.  Cauchy sequences, complete spaces.  Compact spaces.  Continuous mappings, uniform continuity.

[I]  Lebesgue Integral on the Real Line

1. Lebesgue Measure on the Real Line
   Measure on a s-field.  Construction of the Lebesgue measure space via outer measure.  Lebesgue measurable sets and Borel sets.  Lebesgue measurable functions.  Convergence a. e., convergence in measure, Egorov's theorem.  Approximation of Lebesgue measurable functions by continuous functions and step functions.
    
   
2. Lebesgue Integral on the Real Line
   Integration of Lebesgue measurable functions.  Dominated convergence Riemann integrability.  Approximation by truncation, approximation by continuous functions and step functions.
    
   
3. Differentiation and Integration
   Functions of bounded variation.  Absolutely continuous functions and singular functions. Indefinite integrals.
    
   
4. The Lp-space
   Normed linear spaces, Banach spaces. Representation theorem for bounded linear functions on Lp -spaces.

[II]  Abstract measure and integration

Signed measures, Radon-Nikodym and Lebesgue decomposition theorems, Outer measures, extension theorem, Lebesgue-Stieltjes integral, product measures, Fubini-Tonelli theorem.

References---Real Analysis

Lectures on Real Analysis, J. Yeh

Real Analysis, by H.L. Royden, 3rd edition 1988

• Chapters 3, 4 pp. 54-96; Chapter 5 sections 1, 2, 3, 4 pp. 97-112;
• Chapter 6 pp. 118-135; Chapter 7 section 1, 2, 3, 4, 5, 6, 7 pp. 139-157;
• Chapter 11 pp. 253-281; Chapter 12 pp. 288-312

Or Measure and Integral, by R.L. Wheeden and A. Zygmund 1977

• Chapters 1, 2, 3, 4, 5 pp. 1-85; Chapter 6 sections 1, 2, 3 pp. 87-97;
• Chapter 7 sections 1, 2, 3, 4 5 pp. 98-118; Chapter 8 sections 1, 2, 3, 4 pp. 125-135;
• Chapter 10 sections 1, 2, 3 pp. 161-181; Chapter 11 sections 1, 2, 3 pp. 193-201
  

Complex Analysis

Suggested Syllabus for Complex Analysis Qualifying Examination

I. Complex Numbers and Functions

The field of complex numbers, geometry of the complex plane, polar representation, the extended plane and spherical representation, analytic functions, power series, rational functions, elementary functions (exponential, trigonometric and logarithmic), Cauchy-Riemann equations, M`` obius transformations, cross ratio.

II. Complex Integration and Cauchy's Theorem

Line integrals, power series representation of analytic functions, Cauchy's estimate, Cauchy's theorem.

III. Applications of Cauchy's Theorem

Liouville's theorem, Fundamental theorem of Algebra, identity (=uniqueness) theorem, maximum modulus theorem, Schwarz's lemma, Morera's theorem, index (=winding number) of a closed curve, Cauchy's integral formula, argument principle, open mapping theorem.

IV. Singularities

Removable singularities, poles, order and singular part of a pole, Laurent expansions, essential singularities, Casorati-Weierstrass theorem, residues, residue theorem, evaluation of real integrals, Rouche's theorem.

V. Normal families, Montel theorem, the Riemann mapping theorem, Automorphism groups of the unit disc, punch disk, etc.  Conformal mappings (or angle preserving maps) between two given regions.

VI. Harmonic functions

Mean value property, Maximum principles, Jensen's formula, Poisson's formula, Dirichlet problem for disk, and Harnack's theorem.

References--Complex Analysis

Functions of One Complex Variable, by J. B. Conway 2nd edition, 1978

• Chapter 1 pp. 1-10;
• Chapters 3, 4, 5 pp. 30-127;
• Chapter 6, sections 1, 2 pp. 128-133;
• Chapter 7, sections 1, 2, 4 pp. 142-154, 160-163;
• Chapter 10, sections 1, 2 pp. 252-263.

Complex Analysis, by J. Bak and D.J. Newman 1982

• Chapters 1, 2, 3, 4, 5, 6, 7 pp. 1-85;
• Chapters 9, 10 pp. 96-118;
• Chapter 11 section 1 pp. 119-127;
• Chapter 14, pp. 169-174;
• Chapter 16 pp. 184-190

Function Theory of One Complex Variable, by R.E. Greene and S.G. Krantz

Complex Analysis, by Lars V. Ahlfors

Algebra

Suggested Syllabus for Algebra Qualifying Examination

Linear Algebra:

Vector spaces and bases; linear transformations and their matrix representations; characteristic and minimal polynomials; eigenspaces and eigenvalues; diagonalization; rational and Jordan canonical forms; inner product spaces; orthonormal bases; isometric diagonability (that is, diagonalization via unitary or orthogonal matrices).

I.    Groups

Groups and group homomorphisms and isomorphisms; cyclic groups; cosets; Lagrange's Theorem; normal subgroups; quotient groups; the isomorphism theorems; groups acting on sets; Sylow theory; semi-direct products, free groups; permutation groups; solvable groups.

II.  Rings

Rings, ideals and homomorphisms; quotient rings; isomorphism theorems for rings; polynomial rings; principal ideal domains; unique factorization; Gauss's Lemma.

III.  Modules

Modules and module homomorphisms; free modules and direct sums of modules; computing kernel and range of a matrix over a PID and its applications to canonical forms of a matrix over a PID.

IV.  Fields

Field extensions; presentation of algebraic and transcendental extensions: splitting fields and normal extensions; separable extensions; Galois extensions and finite fields; transcendence bases; rings of integers of quadratic extensions of Q

References-Algebra

Linear Algebra, C.W. Curtis, Chapters 2-7, pp. 16-227

or

Intro. to Matrices and Linear Transformations (3rd edition), D.T. Finkbeiner, Chapters 1-9, pp. 1-305

Graduate Topics:

Basic Algebra I, N. Jacobson, Chapters 1-4, pp. 26-270, 287-290

Algebra (2nd edition), or S. Lang, Chapters I, II, III, VII, pp. 3-93, 265-334

Algebra, Michael Artin