The following requirements are in addition to, or further elaborate upon, those requirements outlined in the Current Students section of this Web site.

Coursework Requirements

Courses: Units

• Group I:
  Two one-year sequences from courses listed below by end of third year: 6
  MATH 412-1,2,3 Introduction to Modern Analysis
  MATH 426-1 Basic Differential Equations
  MATH 427-1,2 Partial Differential Equations
  MATH 441-1 Geometry and Topology
  MATH 443-1,2 Introduction to the Geometry of Manifolds
  MATH 444-1,2 Introduction to Functional Analysis
  MATH 447-1,2,3 Dynamical Systems
  MATH 450-1,2 Probability
  MATH 462-1,2,3 Stochastic Processes Applied mathematics courses (approved by Graduate Committee)
• Group II:
  Two one-year sequences from courses listed below by end of third year: 6
  MATH 441-1 Geometry and Topology
  MATH 442-1,2 Algebraic Topology
  MATH 465-1,2,3 Algebraic Topology II
  MATH 470-1,2,3 Algebra
  MATH 477 Commutative Algebra
  MATH 478 Group Theory (one quarter)
  MATH 480 Homological Algebra (one quarter)
  MATH 482-1 Algebraic Number Theory
  MATH 483-1,2,3 Algebraic Geometry
  MATH 484 Lie Algebras
  MATH 486-1,2,3 Algebraic K-theory
  Total Required Units: 12
  (A passed preliminary exam in a subject may be substituted for a course.)

Other PhD Degree Requirements

• Examinations: three written preliminary examinations drawn from the following: algebra, algebraic topology, dynamical systems, geometry of manifolds, modern analysis (real and complex), partial differential equations and probability; oral qualifying examination
• Research/Projects: see PhD Dissertation below
• PhD Dissertation: original, independent research
• Final Evaluations: defense of dissertation