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: algebraic groups, Galois cohomology, quadratic forms, division algebras, real algebraic geometry, symbolic computation
  • Analysis/Geometry: complex analysis, conformal and quasiconformal mappings, global analysis on manifolds, microlocal analysis, geometric analysis, partial differential equations
  • Combinatorics/Graph Theory: graph theory, random structures, ordered sets, projective planes, theory of computation
  • Computational Mathematics: high performance computing, computational fluid dynamics, image processing, inverse problems, numerical analysis (linear algebra, PDEs, optimization), scientific computation
  • Topology: low-dimensional geometric topology, combinatorial topology, geometric group theory, hyperbolic geometry

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: Analysis I & Analysis II
    • Math 521 & 522: Algebra I & II
    • Two of the following courses:
      • Math 523: Algebraic Geometry
      • Math 528: Number Theory
      • Math 543: Algebraic Topology I
      • Math 544: Algebraic Topology II
      • Math 545: Intro. to Differential Geometry I
      • Math 550: Functional Analysis
    • One of 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 557 & 558: Partial Differential Equations 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.
  5. Teaching requirements:
    • Math 590: Teaching Seminar
    • TATTO course
    • Teaching at least two one-semester courses

Computational mathematics

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

  1. The following courses:
    • Math 511 & 512: Analysis I & Analysis II
    • Math 515 & 516: Numerical Analysis I & II
    • Two of the following courses:
      • Math 550: Functional Analysis
      • Math 561: Matrix Analysis
      • CS 551: Software Systems
      • CS 555: Parallel Processing
    • One of the following sequences:
      • CS 551 & 555: Software Systems & Parallel Processing
      • Math 557 & 558: Partial Differential Equations 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
    • PDEs
    • 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.
  5. Teaching requirements:
    • Math 590: Teaching Seminar
    • TATTO course
    • Teaching at least two one-semester courses