Each requirement has two parts:

**1. Examination**

The examinations are offered at least twice a year-during the week before classes begin in the Fall Quarter, and again about two weeks before the end of classes in the Spring Quarter. The examinations are designed to test whether students have adequate knowledge of relevant material - not necessarily whether they have the ability to do research. Normally a grade of A- on the exam will be adequate to pass at the Ph.D. level and a grade of B to pass at the M.A. level. It is advantageous for a Master's student to pass at the Ph.D. level if the student wishes ultimately to transfer to the Ph.D. program. Students are allowed unlimited attempts for each exam. Any student or admitted applicant may request copies of old exams from the Staff Graduate Advisor.

**2. Completion of a designated one-year graduate course with good grades**

Normally, an average grade of A- for Ph.D. students and of B for Master's students will be adequate to satisfy part 2 of an area requirement. Courses taken S/U will not satisfy the requirement.

Whether or not a student has fulfilled an area requirement will be determined by the Graduate Committee in consultation with the student's graduate-course instructor(s) and the faculty members who prepared and graded the examination. The total performance in the examination and course work will determine whether or not the area requirement is satisfied. The Graduate Committee will determine how any deficiencies, if present, may be corrected. Complete descriptions of the various area requirements follow. Students who have successfully completed graduate level course work at another institution can petition the Graduate Committee to allow that course work to satisfy the course component of an area requirement.

## Area Requirement in Algebra

This requirement consists of an examination outlined below and a one year graduate course in Modern Algebra: Math 220ABC.

**Algebra Examination**

The algebra examination is based on work covered in good undergraduate algebra courses. At UCSB such courses would be 108ABC and 111ABC. The examination will permit a limited selection in the choice of questions to answer. Topics for which students are always responsible, and which are therefore likely to appear, include:

**Groups:**Lagrange's theorem, permutation groups, Cayley's theorem, cyclic groups, morphisms, quotient groups, automorphism groups, direct product representation for finitely generated abelian groups, Sylow theorems, groups of small order.**Rings and Fields:**Integer and polynomial rings, factorization theory, subrings and ideals, morphisms, quotient rings, fields of quotients, Euclidean rings, prime and maximal ideals, algebraic and transcendental field extensions, prime fields, finite fields, Galois theory over the rational numbers.**Linear Algebra:**Vector spaces, linear transformations, matrices, system of linear equations, eigenvalues and eigenvectors, inner product spaces, normal linear transformations, similarity, elementary divisors and invariant factors, canonical forms.

**References**

- Herstein, I., Topics in Algebra, third edition
- Jones, Burton W., An Introduction to Modern Algebra
- Stewart, I., Galois Theory
- Birkhoff, G., Maclane, D., A Survey of Modern Algebra
- Strang, G., Linear Algebra and its Applications
- Halmos, P., Finite Dimensional Vector Spaces
- McCoy, N., Introduction to Modern Algebra
- Hoffman, K., Kunze, R., Linear Algebra
- Gantmacher, F.R., The Theory of Matrices, Vol. I

## Area Requirements in Analysis

This requirement consists of an examination based on undergraduate material such as that found in UCSB courses 118ABC and 122AB and a one-year graduate course taken from the following list:

- Real Analysis: Math 201ABC
- Complex Analysis: Math 202ABC

**Analysis Examination**

**Topics in Real Analysis: **The real number system, topology of Rn, continuity, differentiability, Riemann integration, sequences and series, convergence processes including uniform convergence, functions of several variables, and introductions to metric spaces and to measure and integration.

**References**

- Rudin,
*Principles of Mathematical Analysis*, Third Ed. (covers all topics and more. The exam excludes 7.28-7.33, 8.15-8.22, and all of Chapter 10. From Chapter 6 the exam will cover only ordinary Riemann integration, not the more general Riemann-Stieltjes integral. - Andrew Browder,
*Mathematical Analysis*(Chapters 1-10). - Robert Strichartz,
*The Way of Analysis*(excluding Chapters 11,12,and 15).

**Topics in Complex Analysis: **Complex numbers and functions, Cauchy integral theorem and Cauchy's integral formula and consequences, Residue calculus, elementary conformal maps, power series and Laurent series, elementary properties of analytical continuation, zeros and singularities of analytic functions.

**References (Most undergraduate complex variables texts are OK.)**

- Brown & Churchill,
*Complex Variables and Applications*, 6th ed. (Sections 1-80 cover these topics) - Spiegel,
*Complex Variables*, Schaum's Outline Series (covers all topics) - Knopp and Konrad,
*Problem Book in the Theory of Functions*, Vol. 1 & 2 (an old book but still an excellent source for problems)

A detailed description of the area requirement in Analysis can be found by clicking here [PDF].

## Area Requirement in Applied Mathematics

This requirement consists of an examination in either Numerical Analysis or Differential Equations, based on undergraduate material such as that found in UCSB courses 104ABC, 124AB, and a one-year graduate course taken from the following list:

- Ordinary Differential Equations: Math 243ABC
- Partial Differential Equations: Math 246ABC
- Numerical Analysis: Math 206ABCD (only 3 quarters are required)

**Numerical Analysis Examination**

**Root Finding, Interpolation and Numerical Integration**- Solution of Nonlinear Equations: Fixed-point iteration, Newton's Method and steepest descent.
- Interpolation: Lagrange polynomials, Newton's divided differences and Neville's algorithm.
- Numerical Integration and Differentiation: Newton-Cotes quadratures and composite rules, Richardson extrapolation. Finite differences.

**Numerical Linear Algebra**- Direct methods: Gaussian elimination, LU, LDL, Cholesky factorizations.
- Iterative methods: Jacobi, Gauss-Seidel and SOR.
- The conjugate gradient method.
- Convergence of iterative methods. Condition number of a matrix.
- Approximation of eigenvalues: Gerschgorin circle theorem, the power method, the QR algorithm.

**Numerical Methods for Ordinary Differential Equations**- Basic existence theory for ODEs: existence, uniqueness, and well-posedness.
- One-step difference methods: Euler's method and Runge-Kutta methods. Consistency and order of one-step methods.
- Linear multi-step methods: Consistency, order, zero-stability (root condition), and convergence. Adams-Bashforth, Adams-Moulton, and predictor-corrector methods.
- Absolute stability.
- Finite difference methods for linear and nonlinear boundary-value problems for ODEs.

**Finite Difference Methods for PDE's**- Order of accuracy and consistency. The CFL condition.
- Von Neumann analysis and stability.
- Finite difference methods for hyperbolic, elliptic, and parabolic PDEs. Implicit and explicit methods.

**Approximation Theory**- Least squares approximation.
- Orthogonal polynomials. Chebyshev polynomials and trigonometric approximation.
- The fast Fourier transform.

**References**

*Numerical Analysis*, Seventh Edition, by Burden and Faires.*Finite Difference Schemes and Partial Differential Equations*, by J. C. Strikwerda.

**Differential Equations Examination**

**Ordinary**- Existence and uniqueness for the initial value problem
- Continuous dependence on initial conditions
- Linear systems: constant coefficients, periodic systems, fundamental matrices
- Variation of parameters formula
- Phase plane analysis
- Stability of critical points
- Planar autonomous systems

**Partial Differential Equations**- Well-posedness
- The wave equation: D'Alembert's formula, energy, method of characteristics, finite propagation speed, weak solutions
- Laplace equation: Green's identities, Poisson kernel, maximum principle, harmonic functions
- Heat equation: Fourier transform, maximum principle, dissipation
- Boundary value problems via Fourier series
- Eigenvalue problems via minimization

**References**

*Differential Equations, Dynamical Systems, and Linear Algebra,*, by Hirsch and Smale.*Partial Differential Equations, An Introduction*, by Strauss.

## Area Requirement in Geometry/Topology

This requirement consists of an examination in either Geometry or Topology, based on undergraduate and graduate material such as that found in UCSB courses 147 AB and 240 AB (Geometry), or 145 and 221 A (Topology), and a one-year graduate course taken from the following list:

- Math 221ABC, Foundations of Topology, Homotopy Theory, and Differential Topology
- Math 232ABC, Algebraic Topology
- Math 240ABC, Introduction to Differential Geometry and Riemannian Geometry

**Geometry Examination**

Arc length of curves, curvature and torsion, surfaces, first and second fundamental forms, curvature of surfaces, geodesics, Gauss-Bonnet formula.

**References**

- do Carmo,
*Differential Geometry of Curves and Surfaces*, Chapters 1-4 - McCleary,
*Geometry from a Differential Viewpoint*

**Topology Examination**

Metric spaces, topological spaces, continuous functions, product spaces, compactness, connectedness, path-connectedness, completeness, quotient spaces, fundamental group, covering spaces.

**References**

- Croom,
*Principles of Topology* - Sutherland,
*Introduction to Metric and Topological Spaces* - Kahn,
*Topology, Intro. to the Point-Set and Algebraic Areas* - Munkres,
*Topology: A First Course* - Hatcher, online book

## Probability and Statistics

Mathematics Ph.D. students with secondary interests in Probability or Statistics may, with the approval of the Graduate Committee, substitute course work (or Area Requirements) in these areas. Such students should get course and exam information from the Department of Statistics and Applied Probability and discuss their plans with their advisor.