This is a list of suggested topics for the independent study component of Math 443. These are only suggestions. Each student is welcome to propose other topics which are of interest to them. Regardless of the origin of the chosen topic, each student is expected to meet periodically with the instructor to discuss the topic. These meetings will serve to define and focus the chosen topic as well as to provide an opportunity to discuss content. Many of the topics below require familiarity with either the material from Math 441 or Math 442. Some of the topics may require familiarity with other subjects, for example, complex variables or partial differential equations (students without this background should avoid these topics). References are given parenthetically and listed at the bottom of this page. Students are encouraged to go to the library and read about the topics of interest before making a choice and meeting with the instructor.

Each student project will cumulate in 1-2 50-minute lectures given by the student to the class and a 5-10 page written report on their topic. Both the presentation and the written report will be graded with attention to the depth and understanding of the selected topic as well as the clarity of presentation.

The deadline for choosing a topic and having had the initial meeting with the instructor is

• Fundamental Group and Covering Spaces [P, M]

• Complete Metric Spaces and Functions Spaces [P, M]

• Differential Forms and Applications [doC]

• Minimal Surfaces [O, S-IV and others]
  • Existence of isothermal parameters
  • Second variation formula
  • Bernstein's Theorem
  • Weierstrass Representation
  • Plateau Problem - calculus of variations

• Surfaces of (non-zero) constant mean curvature [S-IV, H]
  • Variational Formulation
  • Hopf's Theorem

• Hyperbolic Geometry [B]
  • Discrete groups and the hyperbolic plane
  • geodesics in the hyperbolic plane
  • geometry of hyperbolic triangles/area/distance

• Topics in Knot Theory
  • Invariants and elementary moves [?]
  • Total curvature of curves [S-III]

• Towards General Relativity [N and others]
  • Geometry of Minkowski Space
  • Curves and Surfaces in Minkowski Space

[B] The geometry of Discrete Groups by Alan F. Beardon

[doC] Differential Forms and Applications by Manfredo P. do Carmo

[H] Differential Geometry in the Large by Heinz Hopf

[M] Topology: a first course by James R. Munkres

[N] Spacetime and Singularities: An Introduction by Gregory L. Naber

[O] A Survey of Minimal Surfaces by Robert Osserman

[P] Foundations of Topology by C. Wayne Patty

[S-III] A Comprehensive Introduction to Differential Geometry, Vol. III by Michael Spivak

[S-IV] A Comprehensive Introduction to Differential Geometry, Vol. IV by Michael Spivak