#### Syllabi for Qualifying Examination Courses

# Qualifying Examinations for a Doctoral Degree

(For students enrolled from the second semester of the 2017 school year)

#### Summary

These rules below will apply for entering first-year doctoral candidates starting from the second semester of the 2017 academic year.

* Existing rules are applied to matters not mentioned below.

**Details on the 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 pass written examinations of the following two courses: Algebra and Analysis. The department shall write a syllabus for each course and announce it. If a student signed up for general QE and does not show up (except for unavoidable circumstances such as the student’s illness, death of a family member, etc.), the student will not be allowed to take any QE for one year.

The evaluation committee for each course deals with exam questions and grading.

**○ ****Forming and Operating an Evaluation Committee**

The Evaluation Committee consists of 3 members per course, and the members’ term is 3 years. One of the members changes each year and the Head Professor of the Department appoints the new member. The first committee members’ term is 1 year, 2 years, and 3 years.

The first Evaluation Committee since execution of the policy makes a syllabus of the QE.

**○ ****QE Schedule**

In principle, the QE is on the last Thursday and Friday of every January and July.

The Evaluation Committee decides the length of the exam ranging from 3 hours to 5 hours.

**○**** Grading and Pass/Fail**

The student’s name must not be visible when grading.

In principle, the same person must grade the same questions.

An Evaluation Committee composed of the Head Professor of the Department, Chair of the Graduate Studies Committee, and Chair of Examinations Committee holistically decides pass/fail based on the results. (There is no pass/fail per course.)

**▣ Advanced QE Courses**

Each professor announces the major’s advanced QE’s range.

Students who passed general QE must choose their tentative academic advisor.

Tentative academic advisor establishes a committee composed of 3 professors and should be the Chair of the Committee. The tentative academic advisor develops the plan for advanced QE, submits it to the department, and notifies the student.

**○**** Grading and Pass/Fail**

Students take exams for the courses designated by their advisor and take QE with all members of Advanced QE Committee present.

Advanced QE Committee decides the results (pass vs. fail) in 3 to 6 months after the approval by the department.

**Management of Academic Records after Passing the QE**

Once the student passed 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.

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

**Algebra**

◆ Group Theory

– Basic definitions and examples

– Dihedral and symmetric groups

– The quotient group

– Homomorphisms and isomorphisms

– Group actions

– Subgroups and normal subgroups

– Subgroups generated by subsets of a group

– Lagrange theorem

– The isomorphism theorems

– Cayley’s theorem and the class equation

– Automorphisms

– Sylow’s theorems

– The simplicity of

– Direct and semi-direct product

– The fundamental theorem of finitely generated abelian groups

◆ Ring Theory

– Basic definitions and examples

– Ring homomorphisms and quotient rings

– Properties of ideals

– Ring of fractions

– Chinese remainder theorem

– Euclidean domains

– Principal ideal domains

– Unique factorization domains

– Polynomial rings over UFDs

– Eisenstein criterion

◆ Modules and vector spaces

– Basic definitions and examples of modules

– Quotient Modules and module homomorphisms

– Direct sums and free modules

– Exact sequences of modules

– Projective, injective and flat modules

– Basic definitions and examples of vector spaces

– The matrix of a linear transformations

– Determinants

– Modules of PIDs

– Characteristic and minimal polynomials

– eigenvalues and eigenvectors

– Rational canonical forms

– Jordan canonical forms

◆ Field Theory

– Basic theory of field extensions

– Finite and algebraic extensions

– Splitting field and algebraic closures

– Cyclotomic polynomials and extensions

– Fundamental theorem of Galois theory

– Finite fields

– Simple extensions

– Galois groups of polynomials

– Cyclotomic extensions

– Solvable and radical extensions

– Insolvability of the quintic

**Analysis**

1. Basic Topology on a Metric Space: Limit Point, Interior Point, Open Set, Closed

Set, Compact Space (Finite Intersection Property, Heine-Borel Theorem,

Sequentially Compact, Countably Compact, Lebesgue Number), Separable Space,

Second Countable Space, Connected Space, Perfect Set

2. Countable Set, Uncountable Set, Cantor Set

3. Sequence & Series: Convergent Sequence, Cauchy Sequence, Monotone Real

Sequence, Limsup & Liminf, Complete Metric Space, Baire Category Theorem,

Convergent Series, Comparison Test, Integral Test, Ratio Test, Root Test, Absolute

Convergence, Conditional Convergence, Radius of Convergence

4. Continuity: Continuous Function, Uniformly Continuous Function, Intermediate

Value Theorem, Discontinuity, Properties of a continuous function on a compact

(connected) set, Convex Function, Extension of a Uniformly Continuous Function

5. Differentiability: Differentiable Function, Critical Point, Mean Value Theorem,

L’Hospital’s Rule, Taylor Theorem

6. Riemann-Stieltjes Integral: Integrable Function, Properties of the Integral,

Fundamental Theorem of Calculus

7. Sequence of Functions: Pointwise versus Uniform Convergence, Uniform

Convergence and Continuity, Uniform Convergence and Differentiability, Uniform

Convergence and Integration, Stone-Weierstrass Theorem, Ascoli-Arzela Theorem,

Continuous but Nowhere Differentiable Function

8. Analytic Function, Gamma Function, Trigonometric Polynomial, Convergence of

Fourier Series in L_2

9. Functions of Several Variable: Inverse Function Theorem, Implicit Function

Theorem

10. Integration of Differential Forms: Green Theorem, Stokes‘ Theorem, Closed

Form, Exact Form

11.Lebesgue Integral: Lebesgue Measure, Measurable Function, Comparison with the

Riemann Integral, Monotone Convergence Theorem, Lebesgue Dominated

Convergence Theorem, L_2 Space

12. Properties of Holomorphic Functions: Cauchy-Riemann Equation, Elementary

Functions, Branches of Log z, Power Series, Taylor Series, Uniqueness Theorem

13. Complex Integration: Cauchy-Goursat Theorem, Cauchy Integral Formula,

Calculus of Residue, Cauchy’s Residue Theorem, Contour Integration, Evaluation of

Definite Integrals

14. Applications of Complex Integration: Morera’s Theorem, Liouville’s Theorem,

Analytic Continuation, Schwarz Reflection Principle, Maximum Modulus Principle,

15. Meromorphic function: Zeros and Isolated Singularities, Laurent Series,

Casorati-Weierstrass Theorem, Argument Principle, Rouche’s Theorem, Open

Mapping Theorem

16. Conformal mapping: Mapping by Elementary Functions, Fractional Linear

Transformation, Schwarz Lemma, Riemann Mapping Theorem

(Ordinary Differential Equations

17. First Order ODE, Second Order ODE, Higher Order ODE, Systems of ODE,

Applications

# Qualifying Examinations for a Doctoral Degree

(For students enrolled until the first semester of the 2017 school year)

#### 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

**MATH501 Algebra I **

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

**MATH502 Algebra II **

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

**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.]

**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

**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

**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.

**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

**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

** 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).