- Marat Akhmetov
- Mehpare Bilhan
- Bülent Karasözen
- Murat Yurdakul
- Hurþit
Önsiper

The Qualifying Exam is given twice every year,
in January and September, in five main areas (Algebra, Analysis, Differential
Equations, Geometry-Topology and Numerical Analysis).

Generally, each student is
expected to choose one area.

**Algebra :**

Must Courses: None

Elective Courses: 511, 523,
736, Rings and Modules*

Number of must + elective
courses that each student has to select: 0+2

**Analysis :**

Must Courses: 570**

Elective Courses:502**, 558,
566, 571

Number of must + elective
courses that each student has to select: 1+1

**Differential Equations :**

**PDE :**

Must Course: 584**

Elective Courses: 580**, 702

Number of must + elective courses that each student has to select: 1+1

**ODE :**

Must Course: 588

Elective Courses: 711, 723

Number of must + elective courses that each student has to select: 1+1

**Geometry – Topology :**

Must Course: 537

Elective Courses: 538, 545, 551

Number of must + elective courses that each student has to select: 1+1

**Numerical Analysis :**

Must Courses: 593, 677

Elective Courses: None

Number of must + elective
courses that each student has to select: 2+0

(*): This course will be opened when there is demand.

(**): Note the change in contents.

**ALGEBRA
**

**As minor topic:** Abelian groups; torsion, divisible, torsion-free groups,
pure subgroups, finitely generated abelian groups. Solvable and nilpotent
groups, Hall (pi)- subgroups. Permutation groups. Representations.
Fixed-point-free automorphisms. **As major topic:** All of the above and
"Locally nilpotent groups, locally solvable groups. Finiteness properties.
Infinite solvable groups."

**Note:** The extra topics in the major category can be replaced by
other topics. However, the approval of all the faculty members in the area of group
theory is necessary.

**Main Reference: **

D. J. Robinson, *A Course In The Theory of Groups*,
Springer-Verlag. (For minor: 4, 5, 7.1, 7.2.1 - 7.2.4, 8.1, 9.1, 10.5.) (Extra
topics for majors: 12, 14, 15)

**Other References: **

Martin Dixon, *Sylow Theory Formations and Fitting Classes
of Locally Finite Groups*.

O. Kegel and B.
Wehrfritz, *Locally Finite Groups*.

**B. Algebraic Number Theory (Math 523): **

Ring of integers of an algebraic number field. Integral
bases. Norms and traces. The discriminant. Factorization into irreducibles.
Euclidean domains. Dedekind domains. Prime factorization of ideals. Minkowski's
Theorem. Class-group and class number.

**Main Reference: **

I.N. Stewart and D.O.
Tall, *Algebraic Number Theory*, Second
Edition, 1987. (Chapters 1, 2, 3, 4, 5, 9, 7.1, 10, 12, especially questions of
these chapters and sections).

**Other References: **

A. Fröhlich and M.J.
Taylor, *Algebraic Number Theory*,
Chapters I-IV.

J. Neukirch, *Algebraic Number Theory*, Chapter I.

**C. Rings and Modules: **

Categories. Universal Algebra. Modules. Basic Structure
Theory of Rings. Elements of Homological Algebra.

Commutative Ideal Theory: General Theory and Noetherian
Rings.

**Main Reference: **

N. Jacobson, *Basic Algebra II*.

**Other References:**

Propositional and first-order logic. The compactness theorem
and consequences. Theories that are: complete, model-complete,
quantifier-eliminable, categorical. Structures that are: prime, minimal,
universal, saturated, stable.

**Main References: **

Bruno Poizat, *A Course in Model Theory*.

David Marker, *Model Theory: An Introduction*.

**Other References: **

Chang & Keisler, *Model Theory*.

Wilfred Hodges, *Model Theory*.

Compact operators, compact operators in Hilbert Spaces,
Banach algebras, the spectral theorem for normal operators, unbounded operators
between Hilbert spaces, the spectral theorem for unbounded self-adjoint
operators, self-adjoint operators, self-adjoint extensions.

**Main Reference: **

R. Meise, D. Vogt, *Introduction to Functional Analysis, *Oxford
Sci. Pub. 1997.

**Other References:**

Review of metric spaces, Normed Linear Spaces, Dual Spaces and
Hahn-Banach Theorem, Bidual and Reflexivity, Baire's Theorem, Dual Maps,
Projections, Hilbert Spaces, The spaces Lp(X,m), C(X), Locally Convex Vector
Spaces, Duality Theory of Ics, Projective and Inductive topologies.

**Main Reference: **

R. Meise&D. Vogt*, Introduction to Functional Analysis*,
Clarendom Press, Oxford, Sci. Pub. 1997.

**Other References: **

W. Rudin, *Real and Complex Analysis*, 1987.

W. Rudin, *Functional Analysis*, 1973.

M. Jarchow, *Locally Convex Spaces*, Teubner, 1981.

G. Köthe, *Topological Vector Spaces I, II*,
Springer-Verlag.

Holomorphic functions; comparison of one and several
variables, integral formulas, power series in several variables, plurisubharmonic
functions, pseudoconvexity, domains of holomorphy, Hormander's solution of
del-bar equation and some applications of del-bar techniques, approximation
theorems, Cousin problems.

**Main
References : **

L. Hörmander,** **Chapters
1, 2, Sections 4.1, 4.2, 4.3, 4.4, 5.5

S. Krantz, Chapter 0,
Sections 1.1, 1.2, 1.4, 2.1, 2.1, 2.3, 3.1, 3.2, 3.3, 3.4, 3.5, Chapter 4,
Sections 5.1, 5.2, 5.4, 6.1

**D. Positive Operators and Banach Lattices (Math 566):**

Vector lattices. Basic inequalities, Basic properties, Positive
operators. Extension of positive operators. Order projectives. Order continuous
operators. Lattice Homomorphisms. Orthomorphism. Chapter 1, § 1, § 2,
§ 3, § 4, § 5. Chapter
2, § 7, § 8. Banach Lattices with order continuous
norms. Weak compactness in Banach
lattices. Embedding Banach spaces. Banach lattices of operators. Capter 4, § 12,
§ 13, § 14, § 15. Compact operators. Weakly compact
operators. Chapter 5. Not: 226/5

**Main
References: **

Aliprantis and Burkinshaw,
Academic Press.

**Other References:**

Introduction to topological vector spaces, locally
convex topological vector spaces.
Inductive and projective limits. Frechet spaces. Montel, Schwartz, nuclear spaces.
Bases in Frechet spaces and the quasi-equivalence property. Köthe sequence
spaces. Linear topological invariants.

**Main Reference: **

R. Meise&D. Vogt,
*Introduction to Functional Analysis*,
Clarendom Press, Oxford, Sci. Pub. 1997.

**Other References:**

** **

**I. Ordinary Differential Equations:**

Nonlinear Periodic Systems: Limit Sets; Poincare-Bendixon
Theorem. Linearization Near Periodic Orbits. Orbital stability. Bifurcation:
Bifurcation of Fixed Points; The Saddle-Node Bifurcation; The Transcritical
Bifurcation; The Pitchfork Bifurcation; Hopf Bifurcation; Nonautonomous
Systems. Boundary Value Problems: Linear Differential Operators; Boundary
Conditions; Existence of Solutions of BVPs; Adjoint Problems; Eigenvalues and
Eigenfunctions for Linear Differential Operators; Green's Function of a Linear
Differential Operator.** **

**Main References: **

R. K. Miller and A. N. Michel, Academic Press, 1982.

M. A. Naimark : *Linear Differential
Operators*.

S. Wiggins: *Introduction to Applied
Nonlinear Systems and Chaos*.

**Other References: **

E. A. Coddington and N. Levinson , *Theory
of Differential Equations*, McGraw-Hill Book Company Inc., 1955.

J.K. Hale, *Ordinary Differential
Equations*.

M. W. Hirsh and S. Smale, *Differential
Equations, Dynamical Systems, and Linear Algebra*.

**B. Introduction to Delay Differential Equations (Math 723):**

Basic Concepts and Existence
Theorems: Classification. Statement of the basic initial value problem. The
method of steps. Existence and Uniqueness theorems for solutions of the basic
initial value problem. Integrable types of equations. Delay differential
equations as functional differential equations. Equations with piecewise
constant and with linear delays. Linear Equations: Some properties of linear
equations. Exponential estimates and stability. The characteristic equation.
The fundamental solution. The variation of constant formula. Stability Theory:
Basic concepts. Stability of solutions of stationary linear equations.
Lyapunov's second method and it's application for delay equations.

**Main References:**

L.E. El'sgol'ts. *Introduction to
the Theory of Differential Equations with Deviating Arguments*. Holden-day,
Inc. San Francisko, London, Amsterdam,1966.

J. Hale, *Functional Differential
Equations*, Springer-Verlag, New York, 1971.

**Other References: **

A. Halanay, *Differential Equations:
Stability, Oscillations, Time Lags*. Academic Pres Inc., New York, 1966.

R.D. Driver, *Ordinary and Delay
Differential Equations*. Springer-Verlag, New York, 1977.

T.A. Burton, *Stability and Periodic
Solutions of Ordinary and Functional Differential Equations*, Academic Pres,
Inc. New York, 1985.

**C. Impulsive Differential Equations (IDE) (Math 711): **

General Description of IDE: Desccription of mathematical
model. Systems with impulses at fixed times. Systems with impulses at variable
times. Discontinuous dynamical systems. Linear Systems of IDE: General
properties of solutions. Stability of solutions. Adjoint systems. Stability of Solutions of IDE: Stability
criterion based on first order approximation. Stability in systems of IDE with
variable times of impulsive effect.
Periodic Systems of IDE: Nonhomogeneous linear periodic systems.
Nonlinear perodic systems. Bounded solutions of nonhomogeneous linear systems.

**Main References: **

A.M. Samoilenko and
N. A Perestyuk, *Impulsive Differential
Equations*, World Scientific, 1995.

**Other References: **

V.
Lakshmikantham, D. Bainov, and P.S.
Simeonov, *Theory of Impulsive
Differential Equations*, World Scientific, 1989.

**II. Partial Differential Equations:**

Sobolev spaces: Weak Derivatives, Approximation by Smooth
functions, Extensions, Traces, Sobolev Inequalities, The Space ^{-1}.
Second-Order Elliptic Equations: Weak Solutions, Lax-Milgram Theorem, Energy
Estimates, Fredholm Alternative, Regularity, Maximum Principles, Eigenvalues
and Eigenfunctions. Linear Evolution Equations: Second-order Parabolic
equations (Weak Solutions, Regularity, Maximum principle), Second-order
Hyperbolic Equations (Weak Solutions, Regularity, Propagation of disturbances),
Hyperbolic Systems of First-order Equations, Semigroup theory.

**Main Reference: **

L.C. Evans, *Partial Differential Equations*, AMS
Graduate studies in Mathematics, Vol. 19, 1998.

**Other References: **

D. Gilbarg and N.
Trudinger, *Elliptic Partial Differential
Equations of Second Order*, Springer, 1983.

F. John, *Partial Differential Equations*,
Springer-Verlag.

O. A.
Ladyzhenskaya, *The Boundary Value Problems of Mathematical Physics*,
Springer-Verlag, 1985.

F. Treves, *Basic Linear Differential Equations*,
Academic Press. 1975.

J. Wloka, *Partial
Differential Equations*, Cambridge University Press, 1987.

Distributions, Review of Banach and
Hilbert spaces, Sobolev spaces (Approximation by smooth function, extension,
imbedding, compactness and trace theorems), Semigroups, Some techniques from
nonlinear analysis (Fixed point
theorems, Galerkin method, monotone iterations, variational methods).

**Main References: **

S. Kesavan, *Topics in Functional
Analysis*, John-Wiley and Sons. 1989.

D.H. Griffel, *Applied Functional
Analysis*, Ellis Horwood. 1981.

**Other Reference: **

C. W. Groetsch, *Elements of
Applicable Functional Analysis*, Marcel Dekker, 1980.

**C. Initial Value Problems in the Space of Generalized
Analytic Functions (Math 702): **Initial value
problems in Banach spaces, scales of Banach spaces, solution of IVP in scales
of Banach spaces, the classical Cauchy-Kowalewski theorem, the Homgren theorem,
basic properties of generalized analytic functions. IVP with generalized
analytic initial data.

**Main Reference: **

W. Tutschke, *Solution of Initial Value Problems in
Classes of Generalized Analytic Functions*, Springer-Verlag 1989.

**Other Reference: **Some papers and personal notes.

**4. GEOMETRY – TOPOLOGY**

Fundamental
group, Van Kampen's Theorem, covering spaces. Singular Homology: Homotopy
invariance, homology long exact sequence, Mayer- Vietoris sequence, excision.
Cellular homology. Homology with coefficients. Simplicial homology and the
equivalence of simplicial and singular homology. Axioms of homology. Homology
and fundamental group. Simplicial approximation.

**Main References: **

A. Hatcher, *Algebraic Topology* (2000)

**Other References: **

J. Munkres,* A First
Course in Topology* (Chapter
8) and

G. Bredon,* Geometry and
Topology*.

J. J. Rotman,* An
Introduction to Algebraic Topology*.

E.Spanier,* Algebraic
Topology* .

M. Greenberg, J.Harper,*
Algebraic Topology* .

W. S. Massey,* A Basic
Course in Algebraic Topology*.

Cohomology
groups, Universal Coefficient Theorem, cohomology of spaces. Products in cohomology, Künneth formula.
Poincaré Duality. Universal Coefficient Theorem for homology. Homotopy
groups.

A. Hatcher, *Algebraic Topology* (2000)

**Other References :**

J. Munkres,* Elements of
Algebraic Topology*.

G. Bredon,* Geometry and
Topology*.

J. J. Rotman,* An
Introduction to Algebraic Topology*.

E.Spanier,* Algebraic
Topology* .

M. Greenberg, J.Harper,*
Algebraic Topology* .

W. S. Massey,* A Basic
Course in Algebraic Topology*.

**C. Differential
Geometry I (Math 545)** :

Lie
derivative of tensor fields. Connections, covariant differentiation of tensor
fields, paralel translation, holonomy, curvature, torsion.

**Main Reference:**

Manfredo P. Do Carmo,*
Riemannian Geometry*, 1993 (Chapters
1-7 and 9).

**Other References:**

William M. Boothby,* An
Introduction to Differentiable Manifolds and Riemannian Geometry*, 1986.

S.Kobayashi-K.Nomizu, *Foundations
of Differential Geometry I, II* by
S.Kobayashi-K.Nomizu (1963).

T. Aubin,* A Course in
Differential Geometry*, 2000 .

T. Sakai,* Riemannian
Geometry* (1996).

**D. Algebraic Geometry
(Math 551):**

Theory
of algebraic varieties: Affine and projective varieties, dimension, singular
points, divisors, differentials, Bezout's theorem.

**Main Reference:**

R. Hartshorne, *Algebraic
Geometry* (Chapter 1), Springer-Verlag (1977).

**Other References:**

I.R. Shafarevitch, *Basic
Algebraic Geometry* (Part 1), Springer-Verlag.

K.Smith-L. Kahanpää et al.,
*An Invitation to Algebraic
Geometry* .

K. Ueno, *An Introduction to Algebraic Geometry*,
AMS.

P. Griffiths and J.Harris, *Principles of Algebraic Geometry* (Chapter 0), John-Wiley (1978).

1. Introduction to initial and
boundary value problems, numerical methods, stability and convergence analysis.

2. Numerical methods for initial
value problems: one-step methods (Taylor Series method, Runge-Kutta methods,
extrapolation methods, implicit Runge-Kutta methods), stability analysis.
Systems of differential equations, stiff problems, higher order differential
equations. Linear multistep methods (explicit and implicit methods, predictor-corrector
methods), stability and convergence, estimate of truncation error and order.
Higher order differential equations.

3. Numerical methods for boundary
value problems; Finite difference method, collocation method and shooting
method. 4. Numerical methods for Hamiltonian systems.

**Main References:**

**Other References: **

M. K. Jain, *Num. Soln. of Diff.
Eqns*., Wiley Eastern Limited, 1984.

P. Dcuflhard, F. Bornemann, *Scientific
Computing with Ordinary Differential Equations*, Springer-Verlag, 2002.

K. Brenan, S. Campbell, L. Petzold, *Numerical
Solution of Initial Value Problems and
Differential-Algebraic Equations*, SIAM 1996.

E. Hairer, Ch. Lubich, G. Warner, *Geometric
Numerical Integration: Structure Preserving Algorithms for Ordinary Differential
Equations*, Springer –Verlag, 2002.

**B. Numerical Solution of Partial Differential Equations (Math
593):**

1. Finite Difference method,
stability, convergence and error analysis for initial and boundary value
problems.

2. Parabolic Equations; explicit
and implicit methods, matrix and Von-Neumann stability analysis, truncation
error, convergence analysis, variable coefficients, derivative boundary
conditions. Two-dimensional diffusion equation, ADI method. Nonlinear
equations.

3. Elliptic Equations: Five-point
formula, irregular boundaries, solution of sparse systems.

4. Hyperbolic Equations; Explicit
and implicit methods for one and two-dimensional wave equation, stability,
convergence. First order hyperbolic equations Lax-Wendroff method, stability
analysis, CFL condition. Systems of
conservation laws.

5. Finite Volume Method.

**References :**

M. K. Jain, *Num. Soln. of Diff.
Eqns.,* Wiley Eastern Limited, 1984.

K. W. Morton, D.F. Mayers, Camb. Univ. Press, 1994.

J. W. Thomas, *Numerical Partial
Differential Equations (FDMS),*
Springer-Verlag, 1995.