Required Core Courses

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  • NOTE: this material will be updated. This is the "old" syllabi prior to Fall 2025. 
    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

    References:

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

  • NOTE: this material will be updated. This is the "old" syllabi prior to Fall 2025. 
    • Metric spaces, sequence
    • Open and Closed sets, Limits and Continuity in metric spaces
    • Connectedness, Completeness and Compactness and relation to Continuity. Uniform Continuity
    • Riemann Integration - definition, properties, sets of measure zero, Riemann-Lebesgues Theorem
    • Derivatives, Rolle's Theorem and Mean Value Theorem
    • Sequences of Functions, Pointwise versus Uniform Convergence and relation to continuity and derivatives
    • Series of Functions, Weierstrass M test, relation to continuity, integration and derivatives. 

    References:

    • Richard Goldberg, Methods of Real Analysis, 2nd edition
    • Marsden and Hoffman: Elementary Classical Analysis
    • Apostol: Mathematical Analysis

  • NOTE: this material may be updated. This is the "old" syllabi prior to Fall 2025. 

    1. Rootfinding
      • Existence and uniqueness of roots
      • Bisection
      • Newton's method
      • Fixed-point iteration
      • Determining if an approximation is sufficiently accurate
    2. Finite difference approximations and partial differential equations
      • Derivative approximation formulas
      • Explicit and implicit methods for the heat equation and related PDEs
    3. Linear systems - Direct methods
      • Gaussian elimination
      • LU Decomposition and back substitution
      • Positive definite matrices and Choleski
      • Banded/sparse systems
    4. Vector and matrix norms
    5. Linear systems - Iterative methods
      • Jacobi's method
      • Gauss-Seidel
      • General matrix splitting

    References:

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

  • NOTE: this material will be updated. This is the "old" syllabi prior to Fall 2025. 

    Differential Equations:

    1. Power series solutions
    2. Laplace transforms
    3. Homogeneous and non-homogenous systems of linear differential equations
    4. Fourier series
    5. Matrix exponential

    References:

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

Syllabi from Exams PRIOR to Fall 2025

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    1. Holomorphic (or Analytic) Functions of a Complex Variable
    2. Cauchy-Riemann Conditions and Harmonic Functions
    3. Elementary Complex Functions ( ez, zn, z1/n, logz)
    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

    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

    • Formulating linear programming models
    • Solving linear programming problems using the simplex method
      (and using the two-phase simplex method when appropriate)
    • The theory of the simplex method; convergence
    • The geometry of linear programming; convexity
    • Duality theory, including the complementary slackness theorem
    • Sensitivity analysis
    • The Dual simplex method
    • The transportation problem

    References:

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