Preliminary Exam Contents

PRELIMINARY EXAM CONTENTS

TOPICS:
  • Groups and Commutative Algebra
  • Modules and Homological Algebra
  • Complex Analysis
  • Geometry
  • Numerical Analysis
  • Numerical Methods for Differential Equations
  • Method of Mathematical Physics
  • Real Analysis
  • Partial Differential Equations

 

1. GROUPS AND COMMUTATIVE ALGEBRA
1.Groups.
a) Groups and subgroups. Cyclic groups. Lagrange’s theorem. Fermat’s little theorem. Euler’s theorem. Permutation groups.
b) Normal Subgroups. Homomorphisms of Groups. Quotient Groups.
c) Factor Theorem. The Isomorphism Theorems. Correspondence Theorems
d) Direct product of groups. Finitely generated abelian groups.

2. Commutative algebra.

a) Commutative rings. Subrings. Polynomial rings. Unique factorization. domains.
b) Ideals. Radical of ideal. Principal ideals. Factor rngs. Isomorphism theorems. Extensions and contractions of ideals.
c) Prime and maximal ideals. Jacobson radical. Nilradical. Prime avoidance theorem.
d) Primary ideals. Primary decompositions. Uniqueness theorem for primary decompositions. Rings of fractions.
e) Modules over commutative rings. Chain conditions. Hilbert’s basis theorem. Krull’s intersection theorem. Nakayama’s theorem.

Main References:

1. R. Ash , Abstract Algebra: The Basic Graduate Year

http://www.math.uiuc.edu/~r-ash/Algebra.html

Chapter 1: Sections 1.1 – 1.5.

Chapter 2: Sections 2.1 – 2.8.

Chapter 3: Sections 3.1 – 3.5.

Chapter 4: Sections 4.1, 4.2, 4.3, 4.7.

2. R.Y. Sharp, Steps in Commutative Algebra.

Chapters 1-9.

Other References:

1. T. Hungerford, Algebra, 1974.

2. P. M. Cohn, Basic Algebra: Groups, Rings, Fields. Springer 2003.

3. M. F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, 1969.

4. I. Kaplansky, Commutative rings, 1974.

 

2. MODULES AND HOMOLOGICAL ALGEBRA
1.Modules.
a) Modules. Submodules. Modular law. Maximal and minimal submodules. Finitely generated and finitely cogenerated modules.
b) Factor modules. Factor rings. Homomorphisms. Factor theorem. The isomorphism theorems for Modules. Composition Series. Theorem of Jordan-Hölder-Schreier.
c) The endomorphism ring of a module. Direct Sums. Direct Products. Free modules.
d) Noetherian and Artinian Modules. Endomorphisms of Noetherian and Artinian Rings.
e) Big and small submodules. Radical. Semisimple Modules. Socle. Characterizations of finitely generated and finitely cogenerated modules.

2. Homological Algebra
a) Exact sequences and diagram chasing. Direct and inverse limits. Pullback and pushout diagrams.
b) Projective modules. Injective modules. Baer criterian.
c) Tensor product of modules. Flat modules.
d) Injective envelopes. Projective covers.
e) Resolutions. Chain maps. Derived functors. Functors ExtRn, TorRn.

Main References:

1. F. Kasch, Modules and Rings Academic Press, 1982.

Chapter 2: Sections 2.1 – 2.5,

Chapter 3: Sections 3.1, 3.2, 3.4, 3.5, 3.7,

Chapter 4: Sections 4.1 – 4.4,

Chapter 5: Sections 5.1 – 5.7

Chapter 6: Sections 6.1, 6.2, 6.4.

Chapter 8 : Section 8.1.

Chapter 9 : Sections 9.1-94.

2. L.R.Vermani, An elementary approach to homological algebra, 2000.

Chapter 1: Sections 1.1 – 1.7

Chapter 2: Sections 2.1 – 2.3

Chapter 3: Sections 3.1 – 3.4

Chapter 4: Sections 4.1 – 4.3

Chapter 5: Sections 5.1 – 5.3

Chapter 6: Sections 6.1 – 6.3

Chapter 7: Sections 7.1 – 7.3.

Other References:

1. T. Hungerford, Algebra, 1974.

2. P.M. Cohn, Basic Algebra: Groups, Rings, Fields. Springer 2003.

3. J. Lambek, Lectures on Rings and Modules, New York, London, 1966.

4. M. Scott Osborne, Basic Homological Algebra, 2000. 5. R. Alizade, A. Pancar, Homoloji Cebire Giris, 1999. 

 

3. COMPLEX ANALSIS
1. Analytic functions; Cauchy Riemann Equations, Harmonic Functions, The Cauchy Integral Formula and the Cauchy Integral Theorem, Power series Expansion for an anlytic function, the Cauchy Estimates and Liouville’s Theorem.
2.Meromorphic functions; Behaviour of Holomorphic functions near a singularity, Expansion around singular points, Laurent expansions, Calculus of residues, singularities at infinity.
3. Zeros of a Holomorphic function; Counting zeros and poles, Local geometry of holomorphic functions, Maximum modulus principle, Schwarz Lemma.
4. Mappings; Biholomorphic mappings, Linear fractional transformations, Riemann mapping theorem
5. Harmonic Functions; Basic properties, Maximum principle and the mean value property, Poisson integral formula, Schwarz reflection principle, Dirichlet problem.
6. Analytic Continuation; Analytic continuation along a curve, Riemann Surfaces.

References:

1. R.E. Greene & S.G. Krantz, Function theory of one complex variable, 2002.

chapters 1.1, 1.2, 1.3, 1.4, 1.5; 2.1,
2.2, 2.3, 2.4, 2.5, 2.6; 3.1, 3.2, 3.3,
3.4, 3.5, 3.6; 4.1, 4.2, 4.3, 4.4, 4.5,
4.6, 4.7; 5.1, 5.2, 5.3, 5.4, 5.5 ; 6.1
,6.2, 6.3, 6.4, 6.6, 7.1, 7.2, 7.3, 7.4,
7.5, 7.6, 7.9; 10.1, 10.2, 10.3, 10.4

2. J. B. Conway, Functions of one complex variable, 1978.

chapters 1.1, 1.2, 1.3,
1.4, 1.5, 1.6, 3.1, 3.2, 3.3, 4.2, 4.3,
4.4, 4.5, 4.6, 4.7, 4.8, 5.1, 5.2, 5.3,
6.1, 6.2, 7.4, 9.1, 9.2, 9.3, 10.1, 10.2,
10.3, 10.4, 10.5

 

4. GEOMETRY


1.Curves and Surfaces in Euclidean 3-space:

a) Natural parameter, curvature and torsion of a curve. The Serret-Frenet formulae.
b) The first and the second fundamental form of a surface. The Gaussian and mean curvature.
c) Complex coordinate changes. Surfaces in complex space. Conformal transformations. The conformal form of the metric on a surface. Surfaces of constant curvature.

2. Tensors and Differential Forms. 

a) Symmetrical and skew-symmetrical tensors.
b) Skew-symmetric tensors and differential forms
c) The exterior product of differential forms and the exterior algebra. The Hodge * operator.
d) The Exterior Derivative of a Form.

3. Covariant Differentiation and Riemann curvature tensor:

a) Covariant Differentiation and the Metric. The Christoffel symbols.
b) Geodesics. Examples of geodesics in plane, sphere, the Lobachevskiy plane, the surface of revolution.
c) The Riemann Curvature Tensor. The Ricci tensor and the scalar curvature. The Gauss theorem relating scalar and the Gaussian curvature.

Main References:

1. Dubrovin, B. A., Fomenko, A. T., Novikov, S. P. Modern Geometry – Methods and Applications, Part I, Graduate Texts in Mathematics, vol.93, Springer-Verlag
New York. 1992

Part I

Chapter 1: Sections 1.1-1.2, 2.1, 3.1, 
5.1-5.2,

Chapter 2: Sections 7.1-7.3, 8.1-8.3, 9, 11.1-11.3, 12.1-12.3, 13.1-13.3,

Chapter 3: Sections 17.1-17.2, 18.1-18.3, 19.1-19.4

Chapter 4: Sections 25.1-25.2, 28.1-28.2, 29.1-29.3, 30.1-30.4

2.Weintraub Steven H. Differential Forms, Academic Press, 1997

3.Novikov S.P. and Fomenko A.T. Basic Elements of Differential Geometry and Topology. Kluver, 1990.

4.Novikov S.P. and Taimanov I.A. Modern Geometric Structure and Fields. AMS, Graduate Studies in Mathematics vol. 71, 2006.

 

5. NUMERICAL ANALYSIS
0. Basics: Vector spaces. Normed Linear Space .Vector norms and inner product spaces. Matrix norms.
1. Rootfinding:Bisection method, Newton’s method, Secant method, fixed-point algorithms, convergence of algorithms, Solution of Nonlinear System of equations, Newton’s method for Nonlinear Systems.
2.Solutions of Linear System of Equations: Direct methods, Gauss elimination, pivoting, LU factorization, Cholesky factorization, error analysis.  Iterative Methods, Jacobi and Gauss-Seidel method, convergence of iterative methods.
3.Interpolation: Polynomial interpolation (Lagrange and Hermite interpolations),  error of polynomial interpolation, splines ( piecewise interpolation), Chebychev interpolation
4.Approximation: Approximation in L-inf norm, Existence and and Uniqueness of best polynomial approximation in L-inf L_2, Oscillation theorem. Minimax approximation to x^(n+1). Gram-Schmidt orthogonalization, Orthogonal polynomials and least-squares approximations. Construction of best approximation  polynomials in L-inf  and L_2.
5.Numerical Integration: Interpolatory numerical integration, Newton-Cotes formulas, Gaussian Quadrature, errors of quadrature formulas.
6. Numerical Differentiation:Approximating  to Function Derivatives, Finite Difference operators and their truncation error. Difference Equations and their solution
7.Numerical Solution of Ordinary Differential Equations: Numerical methods for initial value problems: One-step methods (Taylor Series method, Runge-Kutta methods), Linear multistep methods (explicit and implicit methods),  Absolute and zero stability. Convergence analysis. Two point boundary value problems.

Main References:

1. Kendall E. Atkinson, “ An Introduction to Numerical Analysis” 2rd Ed., John Wiley  & Sons. Suggested Sections: 2.2-2.5, 3.1-3.2-3.3-3.4-3.6-3.7-4.3-4.4-4.5, 5.2-5.3- .4, 6.2 to 6.6, 6.8, 6.10, 8.1 to 8.7

2. Endre Suli and David F. Mayers, “ An Introduction to Numerical Analysis”, Cambridge University Press, 2003

3. A. Quarteroni, R. Sacco, F. Valeri, “Numerical Mathematics”, 2000 Springer-Verlag New York, Inc.

Suggested References:

1.K.Atkinson and W.Han, “Elementary Numerical Analysis”, 3rd Ed., Wiley 2004.

2. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007.

3.D. Kincaid and W. Cheney, “Numerical Analysis: Mathematics of Scientific Computation” 3rd Ed., Brooks/Cole, 2002. Suggested Sections: 3.1-3.4, 4-0 to 4.6, 6.0-6.4, 6.7-6.9, 6.12, 7.1-7.4, 8.0-8.5

 

6. NUMERICAL METHODS FOR DIFFERENTIAL EQNS.
1.Finite Difference Schemes:Stability, consistency, convergence. CFL condition.
2.Analysis of Finite Difference Schemes: Fourier’s method. Von Neumann analysis. Order of accuracy. Dissipation and dispersion.
3. Finite difference schemes for parabolic and hyperbolic PDEs: Heat and convection-diffusion equations. Wave equation. Von Neumann analysis. Stability conditions. Order of accuracy. Treatment of boundary conditions.
4.Finite difference schemes for eliptic PDEs: Regularity estimates. Maximum principles. Convergence.
5.Finite element method for eliptic PDEs: Weak formulation. Galerkin FEM. Triangular and rectangular elements. Implementation. Cea’s lemma. Poincare and inverse inequalities.  Interpolation error. Bramble-Hilbert lemma. Best approximation property. Error bounds. Aubin trick.
6. Finite element method for convection-diffusion problems:Weak formulation. Galerkin FEM.  Streamline upwind Petrov-Galerkin method. Stabilization parameters. Relation to the bubble functions. Error bounds. Main References:

1. J. Strikwerda, “ Finite difference schemes and partial differential equations” 2nd Ed., SIAM. Suggested Sections: Ch1, Ch2, 3.1-3.4, 5.1-5.2, Ch6, 8.1-8.3,12.1-12.5

2. H.Elman, D.Sivester and A. Wathen, “Finite elements and fast iterative solvers”,OUP 2005.Suggested Sections:1.1,1.2,1.3.1,1.3.2,1.4,1.5.1,3.1-3.3,3.4.1

Suggested References:

1.K.Atkinson and W.Han, “Elementary Numerical Analysis”, 3rd Ed., Wiley 2004.

2. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007.

3.Endre Suli and David F. Mayers, “ An Introduction to Numerical Analysis”, Cambridge University Press, 2003

4. K.W.Morton,D.Mayers,”Numerical solution of PDEs”, Cambridge UP 2005

 

7. METHODS OF MATHEMATICAL PHYSICS
1. Differential Equations of Mathematical Physics:
a) Wave equation in one dimension. Boundary Value Problem. D’Alembert’s Solution.
b) Heat Equation in one dimension. Initial Value Problem.
c) Laplace Equation.
d) Schrodinger Equation.
e) Cylindrical and Spherical Coordinates. Separation of Variables. Sturm-Liouville theory.
f) Singular Points. Series Solutions.
g) Nonhomogeneous Equation. Green’s Function

2. Special Functions.

a) Bessel Functions
b) Legendre Functions
c) Hermite Functions
d) Laguerre Functions
e) Hypergeometric and confluent hypergeometric functions.

3. Fourier Series and Fourier Transform

a) General Properties. Applications.
b) Integral transforms. Development of the Fourier Integral.
c) Fourier Transform. Inversion Theorem.
d) Fourier Transform of Derivatives.
e) Convolution Theorem.
f) Momentum Representation 

Main References:

1. Ordinary Differential Equations Theory and Applications, M. Rama Mohana Rao.

Chapters 1.1-1.6, 2.1-2.5, 3.1

2. W. E. Boyce and R. C. DiPrima , Elementary Differential Equations and Boundary Value Problems, J. Willey, 2001

 

8.REAL ANALYSIS
9.PARTIAL DIFFERENTIAL EQUATIONS