1. Algebra (A. Groups and Rings, B. Modules
and Fields)

2. Analysis (A. Real Analysis, B. Complex Analysis)

3. Differential Equations (A. Ordinary Differential Equations, B. Partial
Differential Equations)

4. Geometry - Topology (A. Geometry, B. Topology)

5. Numerical Analysis (A. Numerical Analysis 1, B. Numerical Analysis 2)

**A.
****Groups and Rings : **

Groups, quotient groups, isomorphism theorems, alternating
and dihedral groups, direct products, free groups, generators and relations,
actions, Sylow theorems, nilpotent and solvable groups, normal and subnormal
series. Rings, ring homomorphisms, ideals, factorization in commutative rings,
rings of quotients, localization, principal ideal domains, Euclidean domains,
unique factorization domains, polynomials and formal power series,
factorization in polynomial rings.

**Main Reference :**

T.
W. Hungerford, *Algebra*, Springer - Verlag, 1974. Sections 1.1, 1.2, 1.3,
1.4, 1.5, 1.6, 1.8, 1.9, 2.4, 2.5, 2.6, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6.

**Other References :**

L. C. Grove,
*Algebra*

N. Jacobson,
*Basic Algebra*

S. Lang, *Algebra*.

**B.
****Modules
and Fields : **

Modules, homomorphisms,
exact sequences, free modules, vector spaces, tensor products, modules over a
PID. Fields, field extensions, the fundamental theorem of Galois theory,
splitting fields, algebraic closure and normality, the Galois group of a
polynomial, finite fields.

**Main Reference :**

T. W. Hungerford, *Algebra*,
Springer - Verlag, 1974. Sections 4.1, 4.2, 4.5, 4.6, 5.1, 5.2, 5.3, 5.4, 5.5,
including the appendices.

**Other References :**

L. C.
Grove, *Algebra*

N.
Jacobson, *Basic Algebra*

S.
Lang, *Algebra*

**A.
****Real Analysis : **

1. Measures: Sigma algebras, the Concept of
Measure, Outer measures, Caratheodory's Theorem, Borel measures

2. Integration: Measurable Functions,
Integration of Non-Negative Functions and the Monotone Convergence Theorem,
Integration of Complex Functions and the Dominated Convergence Theorem, Modes
of Convergence, Egoroff's Theoerem, Product Measures, the Fubini-Tonelli
Theoerem, the Lebesgue Integral in R^{n}

3. Decomposition of Measures: Signed
Measures, The Hahn Decomposition Theorem, the Lebesgue-Radon-Nikodym Theorem,
Complex Measures

4. L^{p} Spaces: Holder's and
Minkowski's Inequalities, the Dual of L^{p} , Convolutions

5. Radon Measures: Positive Linear
Functions on C_{c}(X), the Riesz Representetion Theorem

**Main Reference :**

G.Folland, *Real analysis* , Chapters 1; 2 ; Sections : 3.1,3.2,3.3 ; 6.1,6.2,6.3
; 7.1,7.2,7.3 ; 8.2 .

**Other References : **

H.Royden, *Real
Analysis*

W.Rudin*, Real and Complex Analysis,* McGraw-Hill Inc., 1966.

** B. Complex Analysis : **

1.
Elementary Properties of Analytic Functions: Power series expansions, Complex
line integrals, Complex differentiation, Cauchy-Riemann equations, Cauchy's
theorem and Integral Formula, Open mapping theorem, Classification of isolated
singularities, Laurent expansions, Calsulus of residues.

2.
The Argument Principle: The index of a closed curve, The general form of
Cauchy's theorem, Residue theorem, The Argument Principle, Rouche's theorem.

3.
The Maximum Modulus Principle: The Maximum Modulus Principle, Schwarz Lemma,
One-to-one holomorphic mappings of the unit disc onto itself, Mobius
transformations.

4.
Zeros and Poles of Analytic Functions: Runge's theorem, Meromorphic functions,
Infinite products, Weierstrass Factorization theorem.

5.
Analytic Continuation: Analytic continuation along a path, Monodromy theorem.

6.
Riemann Mapping Thoerem: Normal Families, Riemann mapping theorem.

**References: **

L.
V. Ahlfors, *Complex Analysis*,
McGraw-Hill Inc., 1966. Chapters 1-5, Sec. 6.1, Sec. 8.1.

J. B. Conway, *Functions
of One Complex Variable*, Springer - Verlag, 1978. Chapters 1-5, Sec. 6.1, 6.2, Chapter 7, Chapter 8, Sections
9.2, 9.3.

W. Rudin, *Real
and Complex Analysis*, McGraw-Hill Inc., 1966. Chapter 10, Sec. 12.1-12.6, Chapter
13, Sec. 14.3-14.9, Sec. 15.1-15.5, Sec. 16.9-16.16.

**A.
****Ordinary Differential Equations: **

**Please note that the contents and
references of this section were changed on 29 June 2004.**

1. Initial Value Problem: First order systems of equations;
Peano's existence theorem; Euler's method of approximation; Uniqueness of
solutions; The method of successive approximations; Differential inequalities
and comparison method; Gronwall inequality; Continuous and differential dependence
of solutions on parameters; The maximal interval of existence; Continuation of
solutions.

2. Sturmian theory: Sturm-Picone theorem and its
consequences.

3. Linear systems: Linear homogeneous and nonhomogeneous
systems of differential equations with constant and variable coefficients;
Fundamental matrices; Abel's formula; The matrix Exponential; Structure of
solutions of systems with constant Coefficients; Floquet theory; Adjoint
system.

4. Higher order linear differential equations: Fundamental
set; Abel's formula; Adjoint equation.

5. Stability: Lyapunov stability and instability; Basic
definitions on stability; Stability by linearization.

6. The topics in Math 254 are included.

**Main References:**

R.K. Miller
and A.N. Michel, *Ordinary Differential
Equations* ( Chapters 1.1-1.2, 2.1-2.8, 3.1-3.3,3.4-3.6, 5.1-5.3, 5.5-5.6,
6.1,6.2).

Boyce and
DiPrima, *Elementary Differential
Equations and Boundary Value Problems*.

**Other References:**

Corduneanu, *Principles of Differential and Integral Equations* (Chapters
1.1-1.5, 2.1-2.5, 3.1-3.5, 4.1-4.6, 5.1-5.3, 8.5, 9.3, 9.4).

J. Cronin, *Differential Equations: Introduction and Qualitative Theory* ,
Chapters 1, Chapter 2 (except Inhomogeneous Linear Systems), Chapter 4 (except
Some stability theory for autonomous nonlinear systems and Phase asymptotic
stability for periodic solutions).

Ş. Alpay, E. Akyıldız, A. Erkip, *Lectures on Differential Equations*.

**B. Partial Differential Equations : **

1. First order equations:
Introduction, quasi-linear equations, characteristic curves (method of
Lagrange), Cauchy problem for
quasi-linear equations.

2. Second order
equations: Linear (almost linear) second order equations, auxiliary conditions,
normal (canonical) forms, Cauchy problem for second order equations,
Cauchy-Kowalewski theorem, Green's identity.

3. Elliptic equations
(Laplace equation): Harmonic functions, fundamental solutions, maximum
principle and its applications, solution of Dirichlet problem (by use of
Green's function and of seperation of variables), smoothness of solutions.

4. Hyperbolic Equations
(Wave equation): Initial value problems, d'Dalembert's solution, domain of
dependence and influence, well-posedness, n dimensional wave equation (use of
spherical means), initial and boundary value problems.

5. Parabolic equations
(Heat equation): Initial value problems, initial-boundary value problem,
maximum principle.

**References: **

F.John, *Partial Differential Equations* (Fourth
Edition), Springer-Verlag,

Sections:
1.1-1.5,1.7,1.8,2.1,2.3,2.4,3.2,3.3(d),3.4,4.1-4.3,5.1,6.1(a,b,c).

H. F. Weinberger, *Partial Differential
Equations*,

Sections:
Chapter 1-Chapter 3, 4.14,4.18,4.22-4.24,4.26.

**A. Geometry :**

1. Differentiable Manifolds, Differentiable
Functions and Mappings: Differentiable manifolds, differentiable functions and
mappings, rank of a mapping, immersions, submersions, submanifolds and
imbeddings

2. Vector Fields on Manifolds: Tangent
space at a point of amanifold, the differential of a differentiable mapping,
vector fields, Lie bracket of vector fields

3. Tensors and Tensor Fields on Manifolds:
Tensors, tensor fields and differential forms, pull back of a differentiable
mapping by differentable mapping, exterior differentiation, Riemannian metric
on manifolds, orientation on manifolds, volume element

4. Integration on Manifolds: Integration on
manifolds, manifolds with boundary, boundary orientation of the boundary of a
manifold, Stokes's Theorem.

**References:**

W. Boothby, *An Introduction to Differentiable manifolds and Riemannian Geometry*,

Academic
Press, (sections: III.1-III.5, IV. 1-
IV-2, V.1-V.8, VI.1, VI.2, VI.4, VI.5 ).

F. Warner, *Foundations of Differentiable Manifolds and
Lie Groups*,

(sections: 1.1-1.42, 2.1-2.23, 4.1- 4.9).

**B. Topology** **:**

1. Topological spaces and
continuous functions: Topological spaces, basis and subbasis, subspace
topology, continuous functions, product topology, metric topology, quotient
topology.

2. Compactness : Compact
spaces, compact sets in Rn, Heine Borel Theorem, Tychonoff Theorem, limit point
compactness, sequential compactness, compactness in metric spaces, local
compactness and one-point compactification.

3. Connectedness:
Connected spaces, path connected spaces, components, local connectedness, local
path connectedness.

4. Separation and
Countability Properties: T0, Hausdorff, regular, normal spaces, Uryshon Lemma,
Tietze Extension Theorem, countability properties, Lindelöf, separable, countably
compact spaces

**References :**

J.
Munkres, *Topology, A First Course*,

(sections:
2.1-2.10, 3.1-3.8, 4.1- 4.3).

S.
Willard, *General Topology*

(sections: 2, 3, 5, 6, 7,
8, 9, 13, 14, 15, 16, 17, 18, 19, in section 19 only one point compactification
is included).

**A. Numerical Analysis 1: **

1. Computational Preliminaries: Absolute
and relative errors, loss of significance, roundoff errors, stable and unstable
computations, conditioning.

2. Solution of Linear Systems of Equations:
Gaussian elimination, LU-decomposition, pivoting and scaling in Gaussian
elimination, condition numbers, error analysis and stability in Gaussian
elimination. Basic iterative methods (Jacobi, Gauss-Seidel and Successive over
relaxation methods), convergence of Jacobi, Gauss-Seidel and successive over
relation methods. Conjugate gradient method.

3. Linear Least Square Problems: Matrix
factorizations that solve the linear least-squares problems, normal equations,
QR decomposition and solving least square problems using QR decomposition,
orthogonal matrices, Householder transformation.

4. Singular Value Decomposition.

5. The Algebraic Eigenvalue Problem: The
power method, the inverse power method, localization of eigenvalues, reduction
to Hessenberg form, Householder transformation, QR algorithm for eigenvalue
problems, estimation of eigenvalues.

**References: **

J.W. Demmel,
*Applied Numerical Linear Algebra*,
SIAM, 1997.

Sections: 1.7, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2
(3.2.1, 3.2.2), 3.2.3.,3.4 (3.4.1), 3.5 (3.5.2), 4.4 (4.4.1, 4.4.2, 4.4.3,
4.4.4, 4.4.5, 4.4.6), 6.1, 6.4, 6.5 (6.5.1, 6.5.2, 6.5.3, 6.5.4, 6.5.5 (up to
Definition 6.12)), 6.6.3.

J. Stoer and
R. Bulirsh*, Introduction to Numerical
Analysis*, Third Edition,

Springer-Verlag, 2002.

Sections: 1.2, 1.3, 1.4, 4.1, 4.2, 4.3,
4.4, 4.7, 4.8 (4.8.1, 4.8.2, 4.8.3), 6.0, 6.1, 6.5.1, 6.5.4, 6.6.1, 6.6.2, 6.6.3, 6.6.6, 6.9 (up to
corollary 6.9.5), 8.0, 8.1, 8.2, 8.3. , 8.7.1.

**B.
****Numerical
Analysis 2 : **

1. Interpolation and
Approximation: Polynomial interpolation (Lagrange and Hermite interpolations),
divided difference and the Newton form of the interpolating polynomial, error
of polynomial interpolation, interpolation by cubic splines. Bsplines.
Trigonometric interpolation. Interpolation at the zeros of orthogonal
polynomials. Orthogonal polynomials and least-squares approximations.
Interpolation using Chebyshev polynomials.

2. Numerical
Differentiation: Numerical differentiation based on polynomial interpolation.

3. Numerical Integration
(Quadrature): Interpolatory numerical integration, Newton-cotes formulas,
Gaussian Quadrature, errors of quadrature formulas. Extrapolation, adaptive
quadrature.

4. Solution of Nonlinear
Equations: Newton's method, fixed-point algorithm, convergence of the
fixed-point algorithm and Newton's method. Root finding algorithms for
polynomials.

**References : **

J.
Stoer and R. Bulirsh, *Introduction to
Numerical Analysis*, Third Edition,

Springer-Verlag, 1980.

Sections: 2.1 (2.1.1,
2.1.3, 2.1.4, 2.1.5), 2.3, 2.4, 2.4.4., 3.1, 3.2, 3.4, 3.5., 3.6, 5.1, 5.2,
5.3., 5.5, 5.6 .

L. W. Johnson and R. D.
Riess, *Numerical Analysis*, Addison Wesley, 1982.

Sections: 4.1, 4.3, 4.5,
5.1, 5.2 (5.2.1, 5.2.2, 5.2.3, 5.2.4, 5.2.5, 5.2.6), 5.3 (5.3.1, 5.3.2, 5.3.3),
6.1, 6.2 (6.2.1, 6.2.2, 6.2.3, 6.2.4), 6.3., 6.4 (Numerical differentiation),
6.5 (6.5.1, 6.5.2).