QUALIFYING EXAMINATIONS STUDY GUIDEAugust 2012

1. Abstract Algebra

(a) Integers

i. Euclidean Algorithm

ii. Greatest common divisor and least common multiple

iii. Fundamental Theorem of arithmetic

iv. Euler phi-function

(b) Groups

i. Definition of groups and subgroups, Lagrange’s Theorem

ii. Cyclic groups dihedral groups

iii. Symmetric groups alternating groups, factorization into a product of disjointcycles

iv. Direct products

v. Matrix groups - General linear group and other groups involving matrices;

vi. Homomorphisms, monomorphisms, epimorphisms, and isomorphisms

vii. Normal subgroups quotient groups

viii. Cayley’s Theorem (every group is isomorphic to a permutation group)

ix. Definition of simple groups

x. Knowledge of all groups of small order - say up to 10.

(c) Rings and Fields

i. Definition of rings and subrings

ii. Homomorphisms

iii. Ideals and quotient rings

iv. Definition of integral domain and field

v. Fields Z/pZvi. Field of fractions of an integral domain

vii. Principal ideal domains

viii. Polynomial rings R[x], irreducible polynomials

ix. Euclidean domains and unique factorization domains

x. Calculations in (Z/pZ)[x] and reduction modulo n

2. Linear Algebra

(a) Linear Equations, Vector Spaces, Linear Transformations

i. Systems of linear equations, matrices, elementary row operations, reducedrow echelon form of a matrix

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ii. Matrix algebra.

iii. Vector spaces, subspaces

iv. Bases and dimension

v. Coordinate matrix relative to a basis

vi. Linear transformations/linear mappings

vii. Representation of transformations by matrices

viii. Bilinear forms, symmetric bilinear forms (scalar product), orthogonality, pos-itive definite case, orthonormal bases

ix. Sylvester theorem

x. Bilinear maps, general orthogonal bases, quadratic forms

xi. Inner product spaces, linear functionals, normal operators

(b) Determinants and Canonical Forms

i. Definition and properties of determinants

ii. Eigenvalues, characteristic and minimal polynomials, Cayley-Hamilton The-orem

iii. Invariant subspaces, simultaneous triangularization and diagonalization

iv. Cyclic subspaces and decompositions

v. Rational and Jordan forms, computation of invariant factors

3. FUNCTIONS OF ONE VARIABLE

(a) Basic properties of N, Z, Q, R(b) Sequences of real numbers

i. Limits

ii. Bolzano-Weierstrass theorem

iii. Cauchy sequences

iv. Divergent sequences and monotone sequences

v. The number e

(c) Limits of functions

i. One-sided and two-sided limits

ii. Limits at infinity and infinite limits

(d) Continuity

i. Continuous functions

ii. Extreme value theorem

iii. Intermediate value theorem

iv. Inverse functions and continuity

v. Uniformly continuous functions and their characterization on bounded inter-vals

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(e) Differentiation

i. Derivative of a function

ii. Basic properties of differentiable functions (including: differentiability impliescontinuity, algebraic properties, chain rule)

iii. Rolle’s theorem and the mean value theorem

iv. L’Hopital’s rule

v. Differentiable functions and monotonicity

vi. Differentiable functions and convexity

vii. Differentiability of inverse functions

viii. Taylor’s theorem

(f) The Riemann integral

i. Upper and lower Riemann sums and integrals

ii. Riemann integrable functions, Riemann integrals and their basic properties

iii. Characterization of Riemann integrable functions and Riemann integrals interms of general Riemann sums

iv. The fundamental theorems of calculus

v. Improper Riemann integrals

(g) Series of real numbers

i. Sum of a series, convergents series and their basic properties

ii. Series of nonnegative terms: comparison theorems, integral test

iii. Absolute convergence

iv. Root and ratio tests

v. Alternating series test

(h) Sequences and series of functions

i. Pointwise and uniform convergence of sequences and series of functions

ii. Pointwise and uniform Cauchy’s conditions

iii. Uniform convergence and continuity

iv. Uniform convergence and differentiation

v. Uniform convergence and Riemann integration

vi. M-test for uniform convergence of series

vii. The Weierstrass approximation theorem

4. FUNCTIONS OF SEVERAL VARIABLES

(a) Metric and topological properties of Rn

i. The Euclidean spaces Rn

ii. Open and closed subsets of Rn

iii. The boundary of a set

iv. Sequences of points in Rn

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v. Relatively open and relatively closed sets

vi. Compact sets, connected sets

(b) Continuity

i. Limits of functions of several variables

ii. Continuous functions

iii. Continuous functions on compact sets and the extreme value theorem

iv. Continuous functions on connected sets

v. Uniformly continuous functions

vi. Linear mappings

(c) Differentiation

i. Differentiability in several variables

ii. The chain rule

iii. The mean value theorem

iv. Gradient, Divergence and Curl: definition and properties

v. Higher order derivatives

vi. Taylor’s formula

vii. Extrema of functions of several variables

(d) The implicit and inverse function theorems

i. The contraction principle

ii. The implicit function theorem

iii. The inverse function theorem

iv. The Lagrange multiplier method

(e) Integration

i. Riemann integration on n−dimensional rectangles

ii. Jordan content and Jordan regions

iii. Riemann integration on Jordan regions

iv. Continuity and Riemann integrability

v. Basic properties of Riemann integrable functions

vi. Fubini’s theorem

vii. Change of variables in Rn

viii. Green’s and divergence theorems

5. COMPLEX ANALYSIS

(a) Complex numbers

i. Basic properties of complex numbers and complex functions

ii. Powers, roots, exponential function, logarithmic function, trigonometric andhyperbolic functions

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(b) Analytic functions

i. Limits and continuity of complex functions

ii. Analytic functions: derivative, differentiation formulas, Cauchy-Riemann Equa-tions

iii. Harmonic functions: definition and their connection with analytic functions;harmonic conjugate

(c) Complex integration

i. Contours and paths in the complex plane. Simple paths, simply connectedregions, multply connected regions

ii. Integral of complex-valued functions and contour integrals: properties, inde-pendence of paths, antiderivatives as path integrals

iii. Cauchy’s integral theorem: version for deformable (homotopic) paths, versionfor simply connected regions/paths, version for multiply connected regions

iv. Cauchy’s integral formula

v. Cauchy’s integral formula for derivatives and applications: derivatives of an-alytic functions are analytic, estimates on moduli of derivatives, Morera’sTheorem, Liouville’s theorem, Fundamental Theorem of Algebra

vi. Maximum and minimum modulus Principles; Schwarz’s Lemma

(d) Complex series

i. Sequences and series of complex numbers/complex functions and basic prop-erties

ii. Sequences and series of analytic functions

iii. Power series: radius of convergence (formula for the radius), uniform conver-gence, term-wise integration/differentiation, Taylor coefficients, uniqueness.

iv. Taylor/Maclaurin Series: Taylor series of analytic functions, Maclaurin Seriesof elementary functions

v. Laurent series of analytic functions

vi. Applications of Taylor and Laurents series: zeros of analytic functions, iden-tity principle, isolated singularities (removable, pole, essential) and their char-acterizations, singularities at ∞

vii. Casorati-Weierstrass Theorem

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(e) Residue theory

i. Residues: formulas for residues at poles, Residue Theorem, computation ofdefinite and improper integrals

ii. Rouche’s Theorem

iii. Inverse Function Theorem

iv. Open Mapping Theorem

(f) Conformal Mappings

i. Conformal mappings and their boundary behavior

ii. Linear fractional transformations: basic properties and examples

RECOMMENDED READING

First and foremost we recommend that you contact the teachers of the current orprevious Math 8210 (Basic Algebra) and Math 8220 (Basic Analysis) classes.

i. Abstract and Linear Algebra

Math 4720 (Introduction to Abstract Algebra I):

A. Herstein: Abstract Algebra 3rd. ed.Sections: 1.1-7, 2.1-7, 2.9, 3.1-3, 4.1-7, 5.1, 5.3-4

B. Fraleigh: A First Course in Abstract Algebra 5th. ed.Sections: 1.1-4, 2.1-4, 3.1-4 5.1-6, 6.1-2, 7.1-3, 8.1-3, 8.5

Math 4920 (Introduction to Abstract Linear Algebra):

A. Lang: Linear Algebra 3rd. ed. (+ Solutions Manual by Shakarchi)Chapters: I-XI

B. Hoffman and Kunze: Linear Algebra 2nd. ed.Chapters: 1-9 and Appendices: A.4-A.5

ii. Real and Complex Analysis

Math 4940 (Introduction to Complex Variables):

A. Asmar: Applied Complex Analysis with Partial Differential Equations,2nd ed.

B. Brown and Churchill: Complex Variables and Applications 6th. ed.

C. Marsden and Hoffman: Basic Complex Analysis 2nd. ed.

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Math 4700 (Advanced Calculus I):

A. Wade: Introduction to Analysis

B. Bartle and Sherbert: Introduction to Real Analysis, 2nd. ed.

C. Rudin: Principles of Mathematical Analysis, 3rd. ed.

Math 4900 (Advanced Calculus II):

A. T. Apostol, Calculus, Vol II

B. Rudin: Principles of Mathematical Analysis, 3rd. ed.

C. Wade: Introduction to Analysis

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