Mathematics PhD

The Department offers a Ph.D. in mathematics designed for those with an undergraduate degree in Mathematics. The Ph.D. is suitable for those wishing to pursue careers in academics or industry. Possible areas of research specialization include:

  • Algebra: computational algebra, quadratic forms, division algebra, decomposability
  • Analysis/Geometry: complex, functional, quasiconformal mappings, global analysis on manifolds, microlocal analysis, geometric analysis
  • Combinatorics/Graph Theory: graph theory, random structures, ordered sets, projective planes, theory of computation
  • Computational Mathematics: numerical linear algebra, image processing, iterative methods, optimization, partial differential equations, computational fluid dynamics, high performance computing
  • Topology: low dimensional topology, knot theory, geometric topology, differential topology, hyperbolic topology

Requirements

Students admitted to the program, in full standing, should have the equivalent of an undergraduate degree in mathematics.

Pure mathematics

Students in a pure mathematics tract must complete each of the following five areas.

  1. The following courses:
    • Math 511 & 512: Complex Analysis & Integration Theory
    • Math 521 & 522: Algebra I & II
    • Two of the following courses:
      • Math 520: Algebra III
      • Math 543: Algebraic Topology I
      • Math 544: Algebraic Topology II
      • Math 545: Intro. to Differential Geometry I
      • Math 550: Functional Analysis
    • One the following sequences:
      • Math 515 & 516: Numerical Analysis I & II
      • Math 531 & 532: Graph Theory I & II
      • Math 535 & 536: Combinatorics I & II
      • Math 543 & 544: Algebraic Topology I & II
      • Math 545 & 546: Intro. to Differential Geometry I & II
      • Math 555 & 556: Intro. to Applied Analysis I & II
  2. Completion of written qualifying examinations in algebra and analysis as well as one area of the student's choosing.
  3. Advanced course work, including at least two courses or seminars in the student's research area.
  4. An acceptable dissertation and oral defense.

Computational mathematics

Students in the computational mathematics tract must complete each of the following five areas.

  1. The following courses:
    • Math 511 & 512: Complex Analysis & Integration Theory
    • Math 515 & 516: Numerical Analysis I & II
    • Two of the following courses:
      • Math 550: Functional Analysis
      • Math 561: Matrix Analysis
      • CS 561: Software Systems
      • CS 555: Parallel Processing
    • One the following sequences:
      • CS 555 & 561: Software Systems & Parallel Processing
      • Math 555 & 556: Intro. to Applied Analysis I & II
      • Math 771 & 772: Numerical Optimization & PDEs
  2. Completion of written qualifying examinations in analysis and numerical analysis as well as one area chosen from the following:
    • software systems & parallel processing
    • applied analysis
    • numerical optimization & numerical PDEs
  3. Advanced course work, including at least two courses or seminars in the student's research area.
  4. An acceptable dissertation and oral defense.