Department of Mathematics and Statistics

Real and Complex Analysis

Real Analysis:

The real number system

Elementary metric space theory

Sequences and series

Differential calculus

Integral calculus (the Riemann integral)

Sequences and sums of functions (Weierstrass Approximation theorem, uniform convergence, Arzela-Ascoli theorem)

The Lebesgue integral (on the real line)

Complex Analysis:

Complex functions: limits, continuity, properties of elementary functions including branches

Differentiability: derivatives Cauchy-Riemann equations, analyticity, harmonic functions

Integration: Cauchy theorems, Residue theorem, Morera's theorem, Maximum-Modulus theorem

Series: Taylor's theorem, Laurent series expansions

Mappings: elementary functions, properties of conformal maps

Further properties of analytic functions: singular points, zeros, analytic continuation, residues, evaluation of real and complex integrals using residues

Some references:

Real Analysis:

Rudin, Principals of Mathematical Analysis, McGraw-Hill

Goldberg, Methods of Real Analysis, Blaisdell

Complex Analysis:

Churchill, Complex Variables and Applications, McGraw-Hill

Saff & Smider, Fundamentals of Complex Analysis, Prentice-Hall

Abstract and Linear Algebra

Abstract Algebra:

Basic concepts of groups and rings including Lagrange's theorem, normal subgroups, factor groups, homomorphisms and isomorphisms, permutations, the field of quotients of an integral domain, polynomial rings, factorization in integral domains.

Linear Algebra:

Vector spaces, linear transformations and matrices, determinants, canonical forms.

Some references:

Abstract Algebra:

Herstein, Topics in Algebra, Wiley

Fraleigh, A First Course in Abstract Algebra, Addison-Wesley

Linear Algebra:

Hoffman and Kunze, Linear Algebra, Prentice-Hall

Curtis, Linear Algebra, Springer-Verlag

Friedberg, Insel, Spence, Linear Algebra, Prentice-Hall

One of:

1. Topology

   Axiomatization of topological spaces

   Different ways of introducing a topological structure.

   Continuous maps

   Characterizations of continuity, initial sources and final sinks, discrete and indiscrete spaces.

   Fundamental constructions

   Basis for open (or closed) sets, subbase, subspaces, products, quotients, sums. Lattice of topologies on a set.

   Convergence

   Sequences; filters and ultrafilters.

   Countability

   First and second axiom. Lindelof spaces.

   Separation

   Hausdorff, regular, completely regular, normal spaces; $T_i$-spaces, $i=0,1,2,3\frac{1}{2},4$. Urysohn's Lemma.

   Compactness

   In Euclidean spaces. Tychonoff Theorem, Stone-Cech compactification. Local compactness in $T$-spaces, Alexandroff compactification.

   Connectedness

   Components, local connectedness, path connectedness.

   Metric spaces

   Cauchy sequences, completeness. Uniform continuity. Baire's theorem.

   Metrization theorems and paracompactness

   The classical theorems of Urysohn and of Nagata-Smirnov-Bing. Stone's theorem.

   Function spaces

   Pointwise and compact convergence. The compact-open topology, Ascoli's theorem. Uniform convergence for metric spaces.

   Approximation

   Stone-Weierstrass theorem.

   Some References

   J. Dugundji, Topology, Allyn and Bacon, Inc., 1966

   R. Engelking, General Topology, PWN-Polish Scientific Publishers, 1977

   James R. Munkres, Topology, Prentice-Hall, Inc., 1975

   Stephen Willard, General Topology, Addison-Wesley, 1968

2. Differential Equations

   Ordinary Differential Equations:

   Generalities and soluble classes of first order ODE's.

   Second order linear ODE's (General theory and explicit solutions in case of constant coefficients).

   Systems of linear ODE's with constant coefficients.

   Series solutions of second order linear ODE's with analytic coefficients near an ordinary point and near a regular singularity.

   Non-linear autonomous ODE's of second order. Phase plane analysis; in particular, equilibrium solutions, their classifications and their stability.

   Existence and uniqueness theorems. Nature of dependence on initial conditions.

   Partial Differential Equations:

   First order linear (and quasi-linear) equations.

   Classification of second order equations and their canonical forms.

   Method of separation of variables to solve boundary value problems, in particular the heat equation, the wave equation and the Laplace equation. In this connection: the Convergence theorem for Fourier series. Also Fourier integrals.

   Wave equation (or other hyperbolic equations). The Initial (Boundary) value problem. D'Alembert's Principle. (Huygens' principle.)

   Heat equation (or other parabolic equations). The Initial (Boundary) value problem. Existence and uniqueness theorem in one-dimensional heat equation.

   Laplace equation (or other elliptic equation). Basic properties of harmonic functions, in particular, the Maximum Principle. Boundary value problems. The Dirichlet problem, Green's function and Poisson's formula.

   Some References:

   ODE's:

   Boyce and DePrima, Elementary Differential Equations, Wiley.

   Simmons, Ordinary Differential Equations, McGraw-Hill.

   Birkhoff and Rota, Ordinary Differential Equations.

   PDE's:

   Zachmano glu and Thoe, Introduction to Partial Differential Equations, William & Willkins Comp., Baltimore.

   Colton, Partial Differential Equations: An Introduction, Random House Birkhaeuser Math Series.

   Berg and McGregor, Elementary Partial Differential Equations, Holden-Day, San Francisco.

3. Probability and Statistics

   This will be based upon a knowledge of the material in Mathematics 5680 and 5690.

Real and Complex Analysis

Real Analysis:

Completeness of $\mathbb{R}^n$, Sequences and Series.

Compactness of $\mathbb{R}^n$, Connectedness.

Uniform Convergence.

Riemann integral, existence of the integral, uniform convergence and the integral.

Improper integrals.

Complex Analysis:

Analytic functions and the Cauchy-Riemann Equations.

Elementary conformal mappings.

Cauchy-Goursat theorem, Cauchy integral formula, residue calculus.

Taylor and Laurent series.

Differential Equations

Ordinary Differential Equations:

Linear systems, calculation of fundamental matrices.

Variation of parameters for systems.

Boundary value problems, eigenvalue problems, Sturm-Liouville theory.

Plane autonomous systems, Liapunov stability.

Partial Differential Equations:

Method of characteristic for first order equations.

Boundary value problems for the Laplace, Wave and Heat equations, separation of variables.

Greens functions for the Laplace, Wave and Heat equations, Poisson's kernel, Dirichlet problem, method of images.

Differential Equations

Ordinary Differential Equations

General theory for first order equations

Existence and uniqueness of solutions

Continuous dependence on parameter and initial conditions

Infinite series solutions and method of majorants

Linear Systems

General theory of linear systems

Linear periodic systems

Second Order linear equations

Boundary value problems, Green's functions, Sturm-Liouville theory

Comparison theorems

Qualitative theory of ordinary differential equations in the plane

Limit cycles, Poincaré-Bendixson Theorem

Stability, Liapunov's method

Partial Differential Equations

Cauchy-Kowalevski Theorem

Hyperbolic systems

Existence and uniqueness of solutions

Method of characteristics

Energy estimates

Fourier transforms, Green's functions

Second order elliptic equations

Application of the Maximum Principle

Elementary Sobolev space theory:

$H^1(\mathbb{R}^n)$ and $H^2(\mathbb{R}^n)$

Existence and uniqueness of solutions

Dirichelet Principle

Perron's Method

Eigenvalues of the Laplace operator

Heat equation

Existence and uniqueness of solutions

Fundamental solutions

Energy estimates

Maximum principle

Algebra

Background material

Linear Algebra

Vector spaces and linear transformations

Determinants

Canonical forms

Diagonalization of quadratic forms

General

Homomorphism theorems

Jordan-Hölder Theorem

Fittings's Lemma

Krull-Schmidt Theorem

Groups

Group actions

Fundamental counting theorem

Permutation groups

Transitivity and primitivity

Simplicity of $A_n$ for $n \ge 5$

Class equation

Frattini Argument

Sylow's theorems and $p$-groups

Group constructions

Direct products

Semidirect products

Structure of finitely generated Abelian groups

Derived series and central series

Solvable groups and nilpotent groups

Fields

Simple extensions (algebraic and transcendental)

Galois Group of an extension

Algebraic closure

Separable and inseparable extensions

Normal extensions

Fundamental theorem of Galois Theory

Finite fields

Rings

Projective and Injective modules

Commutative Rings

Factorization

Localization with respect to multiplicatively closed sets

Simple modules and primitive rings

Jacobson radical

Jacobson Density Theorem

Artinian Rings

Wedderburn-Artin Theorems

Real Analysis

Background material

Infinite series and products

Power series

Elementary theory of the derivative

Monotone function and functions of bounded variation

Metric spaces

Topology

Completeness

Connectedness

Compactness and totally boundedness

Uniform convergence

Baire Category Theorem

Ascoli-Arzela Theorem

Stone-Weierstrass Theorem

Integration

Measure Theory on the real line

Outer measure

Relation between measure and content

Measurable function, Egoroff and Lusin Theorems

Lebesgue integral on the real line

Monotone and bounded convergence theorems, Fatou's Lemma

Riemann integral and relation to Lebesgue integral

Improper Riemann and Lebesgue integrals

Some References:

T. Apostol, Mathematical Analysis

R. Goldberg, Methods of Real Analysis

H.L. Royden, Real Analysis

W. Rudin, Principles of Mathematical Analysis

Topology

Background material

Cardinality and countability

Axiom of Choice, Well Ordering, Maximum Principle

Basic facts about the ordinals

General topology

Open set, closed set, closure, interior, and neighborhood systems

Convergence of filters and nets

Separation and countability axioms

Continuous functions

Connectedness and path connectedness

Compactness

Product spaces and Tychonov Theorem

Urysohn Lemma and Tietze Extension Theorem

Local compactness and paracompactness

Quotient spaces

Algebraic topology

Homotopy of maps and homotopy equivalence

Fundamental group

Covering spaces and classification

Seifert-van Kampen Theorem

Some references:

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

J.R. Munkres, Topology, A First Course

I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry

S. Willard, General Topology