Syllabi for Qualifying Examination Subjects

##### General QE Courses: Students must pass written exams in four out of the following eight courses: Algebra I, Algebra II, Complex Analysis, Real Analysis I, Differentiable Manifolds and Lie Groups, Introduction to Algebraic Topology, Mathematical Statistics, and Numerical Analysis.

##### Advanced QE Courses: Students must choose their academic advisor first, and then take exams in the courses designated by their advisor.

##### ->> The students who enrolled by the spring semester of 2014: your GPA in 500-level courses (which must include at least three of the following: MATH 501, 510, 514, 520, 530, 551, 561) must be 3.5 or higher.

##### ->> The students who have been enrolled since fall semester of 2014: At least 15 credits of 500-level-courses must be taken, and the courses GPA must be 3.5 or higher, and four QE courses must be passed within two years after admission.

1. MATH501 (Algebra I)

The contents of the book by T.W. Hungerford, Algebra ch.1~ch.4.

2. MATH502 (Algebra II)

The contents of the book by T.W. Hungerford, Algebra ch.5~ch.8.

3. MATH510 (Complex Analysis)

Cauchy-Riemann equations, Harmonic functions and conjugates, Elementary analytic mappings,

Complex line integrals: Cauchy's theorems, Maximum modulus principle, Open mapping theorem, unique analytic continuation.Singularities, Residues, Argument principle, Schwarz's lemma and conformal mappings, Normal families, Riemann mapping theorem, Infinite product and Weierstrass factorization, Runge's theorem, Subharmonic functions, Dirichlet problem.

[See L. Ahlfors, "Complex Analysis", Ch 1-6.]

4. MATH514 (Real Analysis I)

Lebesgue measure, Fatou Lemma, Convergence theorems, Fubini's Theorem, Approximation of the Identity and kernels, Functions of bounded variation, Absolutely Continuous Functions, H"older and Minkowski Inequalities, L^p Spaces, Fourier Series, Riesz Representation Theorem, Radon-Nikodym Theorem

Textbook: G. Folland, "Real Analysis", Wiley: Ch 1-7.

References: H.L. Royden, "Real Analysis" ; W. Rudin, "Real and Complex Analysis" ;

Wheeden and Zygmund "Real Analysis".

5. MATH520 (Differentiable Manifolds and Lie Groups)

Manifolds, Differentiable structures, immersions, submersions, diffeomorphisms, tangent and cotangent bundles, vector fields and differential forms, Orientation, Lie derivatives, Distributions and integrability (Frobenius theorem), Exact and closed forms, integration on manifolds, Lie group

Textbook: M. Spivak, "A Comprehensive Introduction to Differential Geometry", Volume I (except Riemannian geometry contents)

F. Warner : Foundations of Differentiable Manifolds and Lie Group.

Boothby: An Introduction to Differentiable Manifolds and Riemannian Geometry

6. MATH524 (Introduction to Algebraic Topology)

Topics:

- Singular homology

- Cellular and simplicial homology

- Excision and Mayer-Vietoris sequences

- Eilenberg-Steenrod axioms and universal coefficient theorems

- Applications of homology theory

Textbook: A. Hatcher, Algebraic topology, Cambridge University Press, 2002, p.97-184.

Other references:

J.W. Vick, Homology theory, Academic Press, 1973.

M. J. Greenberg and J. R. Harper, Algebraic topology: a first course, Benjamin-Cummings, 1981.

J. R. Munkres, Elements of algebraic topology, Addison-Wesley, 1984.

7. Mathematical Statistics

* Choose one between(7-1) Math530 and(7-2) Math531. (These courses are offered in alternating years.)

(7-1)MATH530 (Mathematical Statistics)

Text Book: "Mathematical Statistics: Basic Ideas and Selected Topics" by Bickel and Doksum, Holden-Day.

Chapter 1 Statistical Models: Sufficiency, Exponential family

Chapter 2 Estimation: Estimating equations, Maximum likelihood

Chapter 3 Measure of Performance: Bayes, Minimax, Unbiased estimation

Chapter 4 Testing and Confidence Regions: NP lemma, Uniformly most powerful tests, Duality,

Likelihood ratio test

Chapter 5 Asysmptotic Approximation: Consistency, First- and higher-order asymptotics, Asymptotic

normality and efficiency

(7-2)MATH531 (Probability)

Text Book: "Probability" by Breiman, Addison-Wesley.

Chapter 2 Mathematical Framework: Random variable, Expectation, Convergence

Chapter 3 Independence:

Chapter 4 Conditional Expectation:

Chapter 5 Martingales: Optimal sampling theorem, Martingale convergence theorem, Stopping times

Chapter 8 Convergence in Distribution: Characteristic function, Continuity theorem

8. MATH551 (Numerical Analysis)

Textbook: "Introduction to Numerical Analysis" by Stoer and Bulirsch, 3rd Edition, Springer

Chapter 1 Error Analysis: machine number, condition Number.

Chapter 2 Interpolation: polynomial interpolation, interpolation Error, trigonometric interpolation,

spline function

Chapter 3 Topics in Integration: numerical integration, numerical differentiation, Peano's representation,

Romberg integration, Gaussian quadrature.

Chapter 4 Systems of Linear Equations: LU-decomposition, error bounds, Householder matrix, least-squares

problem, pseudo inverse, iterative methods for linear system.

Chapter 6 Eigenvalue problems: Jordan Normal Form, Shur Normal Form, LR and QR methods, Estimation of

Eigenvalues The Gershgorin theorem).