## Ph.D. Qualification Exam

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### Qualifying Exams Committee

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### Qualifying Exam

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.

Must Courses: None

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

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

Must Courses: 570**

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

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

**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

Must Course: 537

Elective Courses: 538, 545, 551

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

Must Courses: 593, 677

Elective Courses: None

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

**Notes:**

(*): 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*.

2. ANALYSIS

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.

3. DIFFERENTIAL EQUATIONS

**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.

**B. Applied Functional Analysis (Math 580):**

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

*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*.

**B. Algebraic Topology II (Math 538):**

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).

5. NUMERICAL ANALYSIS

**A. Numerical Methods in Ordinary Differential Equations (Math 677):**

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.