## I. Algebra

This exam has three sections. There are nine questions, three from each section and students are expected to answer a total of six questions (two from each section).

### i. Group Theory

Basic properties of groups, subgroups, normal subgroups, quotient groups, group homomorphisms, fundamental isomorphism theorems, cyclic groups, permutation groups, direct sums, automorphism groups, group actions, Sylow theorems and their applications, simple groups, solvable groups.

### ii. Field and Galois Theory

Field extensions, splitting fields, cyclotomic fields, Galois group of a polynomial, fundamental theorem of Galois theory, applications of Galois theory, solvability by radicals.

### iii. Ring and Module Theory

Basic properties of rings, subrings, ideals, quotient rings, ring homomorphisms, fundamental isomorphism theorems for rings, principal ideal domains, unique factorization domains, Euclidean domains, polynomial rings, direct sums of rings, Artinian and Noetherian rings, rings of fractions.

Basic properties of modules, submodules, quotient modules, module homomorphisms, fundamental isomorphism theorems for modules, modules of fractions (localization), length and composition series, simple modules, free and projective modules, direct sums and products of modules, chain conditions (Artinian and Noetherian modules), exact sequences of modules, and diagram chasing.

**References:**

Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote (main text)
Chapters 1-4, 6.1, 7-9, 10.1-10.3, 10.5, 13, 14, 15.1, 15.4

Algebra, 3rd edition, Graduate Texts in Mathematics, by Serge Lang (supplementary
text)

Basic Algebra I, 2nd edition, by Nathan Jacobson (supplementary text)

**Examiners:**
Ela Celikbas,
Olgur Celikbas

## II. Real Analysis

This exam covers Lebesgue Measure and associated topics through the L p spaces. Questions from advanced calculus also form a significant part of the exam.

### Advanced Calculus

- Concept of limit for sequences and functions
- Including sup/inf and limsup/liminf
- Axiom of real numbers, and its equivalent properties
- Sets and properties
- Open sets, closed sets, compact sets
- Continuity, uniform continuity, absolute continuity
- Continuous functions on bounded closed intervals
- Infinite series of numbers and functions
- Uniform convergence, pointwise convergence, power series
- Basic theory of differentiation and Riemann integration

### Real Analysis

- Lebesgue measure and measurable functions
- Littlewood’s three principles
- The development of the Lebesgue integral
- Convergence theorems for integrals of sequences of functions
- Monotone functions and functions of bounded total variation
- Differentiation and absolute continuity, the fundamental theorem of calculus
- Contrasting Riemann and Lebesgue integration
- L p(E) spaces and their properties
- Minkowski and Holder inequalities, completeness, approximation by step functions and continuous functions

**References:**

Elementary Analysis: The Theory of Calculus (2nd Ed) by Kenneth A. Ross

Real Analysis (3rd Ed. Chap. 1-6 or 4th Ed. Chap. 1-8) by H.L. Royden and P.M.
Fitzpatrick

**Examiners:**
Harumi Hattori,
Dening Li

## III. Differential Equations

This exam covers intermediate ordinary differential equations at about the level of Brauer and Nohel. Below is a list of topics covered on the exam.

- Linear homogeneous systems with constant coefficients
- The general solution, the fundamental matrix, the matrix exponential, phase diagrams in 2-dimensions, stability of equilibrium solutions.
- General linear systems
- The fundamental matrix, nonhomogeneous systems.
- Linearization about critical points of nonlinear autonomous systems
- Phase plane portraits in 2-dimensions.
- Theory of Existence, uniqueness, and continuity
- Stability analysis by perturbation (or linearization) and Liapunov method
- Existence and stability of periodic solutions in 2-dimensions
- Poincare-Bendixson Theorem. Floquet theory.
- Bifurcation analysis and center, stable, unstable manifolds
- Saddle-node, transcritical, pitchfork.

**References:**

Nonlinear Ordinary Differential Equations, Third Edition, by D. W. Jordan and
P. Smith Oxford Applied Math and Computing Science Series (1999) Chapters 1, 2,
4, 8-12, and Appendix A.

Qualitative Theory of Ordinary Differential Equations by F. Brauer and J. Nohel
W. B. Benjamin, Inc, New York, (1969) Chapters 1-6

Differential Equations and Dynamical Systems by Lawrence Perko. Springer-Verlag
Chapters 1, 2 (except2.13), 3 (3.1 to 3.9), 4 (4.1 to 4.4).

**Examiners:**
Casian Pantea,
Harry Gingold

## IV. Topology

- Methods of defining topological spaces
- Bases and subbases, local bases
- Continuous functions
- Various equivalent conditions for continuity
- Techniques of producing new spaces
- Subspaces, product spaces
- Separation axioms
- T0, T1, Hausdorff, regular, T3, completely regular, Tychonoff, normal, T4
- Compact spaces, connected spaces
- Statement and applications of Tychonoff theorem
- Metric spaces
- Completeness, compactness in the context of metric spaces

**References:**

Topology (2nd edition) by J. Munkres Chapters I-7

General Topology by S. Willard Sections 2–8, 10, 13–17, 24 and 26

General Topology by J. Kelley All Chapters except 2, 6 and Appendix

**Examiners:**
Krzysztof Ciesielski,
Jerzy Wojciechowski

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