Department of Math and Computer Science
Mathematics Comprehensive Exam Syllabus

Option I:

• Algebra
• Complex Analysis
• Real Analysis
• Topology

Option III:

• Applied Analysis
• Linear Programming
• Numerical Analysis
• Probability

Algebra

1. Group theory
   1. subgroups
   2. permutation groups
   3. homomorphisms
   4. kernels and images
   5. normal subgroups, quotient groups
   6. isomorphism theorems
2. Ring and field theory
   1. homomorphisms
   2. kernels and images
   3. ideals, quotient rings
   4. isomorphism theorems
   5. integral domains
   6. polynomial rings
   7. principal ideal domains
   8. fields
   9. algebraic field extensions
   10. Galois theory
3. Linear algebra
   1. vector spaces
   2. bases and dimension
   3. matrices and linear transformations
   4. kernels and images
   5. eigenvalues
   6. inner product spaces

References:

• Fraleigh: A First Course in Abstract Algebra
• Gallian: Contemporary Abstract Algebra
• Herstein: Topics in Algebra
• Friedberg, Insel, Spence: Linear Algebra

Complex Analysis

1. Holomorphic (or Analytic) Functions of a Complex Variable
2. Cauchy-Riemann Conditions and Harmonic Functions
3. Elementary Complex Functions ( e^z, zⁿ, z^1/n, log z)
4. Complex Integration
5. Cauchy - Goursat Theorem
6. Cauchy Integral Formula
7. Morera's Theorem
8. Liouville's Theorem
9. Fundamental Theorem of Algebra
10. Maximum Principle
11. Taylor Series of Holomorphic Functions
12. Power Series as Holomorphic Functions
13. Meromorphic Functions
14. Laurent Series
15. Residues and Contour Integration
16. Mobius (or Linear Fractional) Transformations
17. Conformal Mapping
18. Entire Functions and Picard's Little Theorem
19. Argument Principle and Rouche's Theorem

References:

• Brown and Churchill: Complex Variables and Applications
• Marsden and Hoffman: Basic Complex Analysis
• Ahlfors: Complex Analysis
• Stein and Shakarchi: Complex Analysis
• Hille: Analytic Function Theory
• Spiegel: Schaum's Outline of Complex Variables

Real Analysis

1. Metric spaces
2. Convergent sequences
3. Cauchy sequences
4. Topological ideas
   1. Open sets
   2. Closed sets
   3. Interior, closure, boundary
5. Series
6. Continuity, uniform continuity
7. Compactness
8. Connected sets, path-connected sets
9. Intermediate Value Theorem
10. Extreme Value Theorem
11. Differentiation
12. Rolle's Theorem
13. Mean Value Theorem
14. The Riemann integral
15. Fundamental theorem of calculus
16. Pointwise and uniform convergence
17. Weierstrass M Test
18. Taylor series
19. Differentiation and integration of series
20. Sets of measure zero
21. Lebesgue's theorem on Riemann integrability

• Marsden and Hoffman: Elementary Classical Analysis
• Apostol: Mathematical Analysis

Topology

1. Topological spaces
2. Interior, closure, boundary
3. Relative topology
4. Bases, subbases
5. Continuous functions
6. Homeomorphisms
7. Product spaces
8. Quotient spaces
9. Connectedness, path-connectedness
10. Compactness
11. Separation axioms

Applied Analysis

Differential Equations:

1. Solving first order and linear nth order equations; Existence, uniqueness, and applications
2. Reduction of order
3. Power series solutions
4. Laplace transforms
5. Systems of linear differential equations
6. Fourier series

References:

• Zill: Differential Equations
• Boyce and DiPrima: Elementary Differential Equations

Analysis:

1. Metric spaces
2. Convergent sequences
3. Cauchy sequences
4. Topological ideas
   1. Open sets
   2. Closed sets
   3. Interior, closure, boundary
5. Series
6. Continuity, uniform continuity
7. Compactness
8. Connected sets, path-connected sets
9. Intermediate Value Theorem
10. Extreme Value Theorem
11. Differentiation
12. Rolle's Theorem
13. Mean Value Theorem
14. The Riemann integral
15. Fundamental theorem of calculus
16. Pointwise and uniform convergence
17. Weierstrass M Test
18. Taylor series
19. Differentiation and integration of series

References:

• Marsden and Hoffman: Elementary Classical Analysis

Linear Programming

1. Formulating linear programming models
2. Solving linear programming problems using the simplex method
   (and using the two-phase simplex method when appropriate)
3. The theory of the simplex method; convergence
4. The geometry of linear programming; convexity
5. Duality theory, including the complementary slackness theorem
6. Sensitivity analysis
7. The dual simplex method
8. The transportation problem
9. The assignment problem; the Hungarian method

References:

• Thie: An Introduction to Linear Programming and Game Theory
• Winston and Venkataramanan: Introduction to Mathematical Programming

Numerical Analysis

1. Computer arithmetic, error, relative error
2. Rootfinding
   1. Existence and uniqueness of roots
   2. Bisection
   3. Newton's method
   4. Secant method
   5. Fixed-point iteration
   6. Determining if an approximation is sufficiently accurate
3. Interpolation
   1. Lagrange form
   2. Divided differences
   3. Interpolation Error Theorem
4. Numerical Differentiation
5. Numerical Integration
   1. Composite Trapezoidal Rule
   2. Composite Simpson's Rule
6. Solving linear systems by Gaussian Elimination
7. Pivoting strategies
8. LU decomposition
9. Special types of matrices
   1. Banded matrices
   2. Diagonal dominance
   3. Positive definite matrices, Choleski decomposition
10. Vector and matrix norms
11. Iterative methods for linear systems
    1. Jacobi's method
    2. Gauss-Seidel
    3. General x(k+1) = Tx(k) + c approaches (SOR and others)
12. The residual and iterative refinement
13. Condition number of a matrix
14. Gerschgorin's Theorem
15. The Power Method to approximate the dominant eigenvalue
16. Least-squares approximation of functions

References:

• Burden and Faires: Numerical Analysis
• Timothy Sauer: Numerical Analysis

Probability

1. Calculus of probability
   1. Sample space
   2. Addition rule
   3. Conditional probability
   4. Independence
   5. Bayes' Theorem
2. Random Variables
   1. Discrete and continuous univariate and multivariate distributions
   2. Derived distributions of functions of random variables
   3. Expectation
   4. Variance and covariance
   5. Chebyshev inequality
3. Limit theorems
   1. Convergence in distribution, in probability and almost sure convergence
   2. Central limit theorem
   3. Strong and weak laws of large numbers

References:

• Wackerly, Mendenhall, Scheaffer: Mathematical Statistics with Applications
• Gut: An Intermediate Course in Probability