# Qualifying Examinations for a Doctoral Degree

#### Summary

In order to pass the Qualifying Examination (QE), a student must score at least 60 points out of 100, and the result must be approved by the Graduate Studies Committee.

**Details on QE Course**- The QE consists of two kinds of examinations – general and advanced courses. Students must pass both examinations within four semesters of study.

**General QE Courses**: Students must select and pass written examinations 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,**,**Numerical Analysis.**ο The exams are held once per semester and the students can take up to 4 courses.

ο If the student missed the exam without cancellation, he/she will not be allowed to take QE exams for one year (except for unavoidable reasons such as illness or death in the student’s family).**General QE Courses**- ο Within the eight general QE courses listed above, the student must pass the four courses by the end of the fourth semester of graduate study

ο The department operates a token system to prevent indiscreet taking of the exam. One token can be used for one exam. POSTECH gives 9 tokens for students who have not passed any of QE courses upon the admission and 8 tokens for the rest of students (for instance, if a student registers for 2 courses, then he/she will be able to apply for 6 courses only. If a student re-register for the same course that was previously taken, then the number of the registered courses will be two.)

ο General QE are held every semester (spring: mid-July, fall: mid-January of the following year). Students may not need to pass all four courses at the same time. **QE for advanced courses**: Students must choose their academic advisor first, and then take exams for the courses designated by the advisor.**Advanced QE Courses**

ο Advanced QE result reports must be submitted to the departmental office by the end of August (Spring semester) or the end February (Fall semester). (Note: Not all 500-level courses are offered year-round. Thus, it is highly recommended to plan your coursework schedule in advance at the early stages of your graduate study.)ο Students are encouraged to discuss study ranges of advanced QE courses with their academic advisors. The results of advanced QE courses (pass or fail) will be determined after the deliberation of the Graduate Studies Committee.

ο To qualify for the advanced QE registration, you must satisfy the following requirements: during the 2 years upon the admission to POSTECH, i) you must complete at least 15 credits of the 500-level courses offered by the Department of Mathematics, ii) your GPA in those 500-level courses must be at least 3.5 or higher, and iii) you must pass the exams for the all four general QE courses.**management of academic records after passing the exams for QE Courses**

ο Once the student passes the QE, he/she will not be under any obligation to take exams. However, the Department of Mathematics will evaluate graduate students’ academic achievements once or twice a year in a faculty meeting and notify the results to students.

ο The student will receive a warning letter if his/her performance seems to be poor. If the student receives the warning letter twice, he/she will not be able to earn a doctoral degree from the Department of Mathematics at POSTECH.

#### Syllabi for Qualifying Examination Courses

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**MATH501 Algebra I **

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

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**MATH502 Algebra II **

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

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**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.]

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**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, Hlder 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

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**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

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**MATH524 Introduction to Algebraic Topology **

- Topics:
- ο Singular homology
- ο Cellular and simplicial homology
- ο Excision and Mayer-Vietoris sequences
- ο Eilenberg-Steenrod axioms and universal coefficient theorems
- ο Applications of homology theory

- Textbook: A. Hatcher, Algebraic topology, Cambridge University Press, 2002, p.97-184.
- Other references:
- ο J.W. Vick, Homology theory, Academic Press, 1973.
- ο M. J. Greenberg and J. R. Harper, Alegbraic topology: a first course, Benjamin-Cummings, 1981.
- ο J. R. Munkres, Elements of algebraic topology, Addison-Wesley, 1984.

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**MATH530 Mathematical Statistics**

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

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 like lihood
- 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

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**MATH531 Probability**

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

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

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** 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).