Thesis Proposals

Here is a list of subject matters for possible thesis at the level of bachelor or master.

  • Sard's Theorem

  • following: Section 3. of Abate and Tovena, Curves and Surfaces

  • Math involved: basic differential geometry.

  • Level: easy

  • The elastica

  • following: Raph Levien, The elastica: a mathematical history

  • Math involved: basic differential geometry.

  • Level: easy

  • The lemniscate and elliptic integrals

  • following: starting with Wikipedia here, here, and here.

  • Math involved: basic differential geometry.

  • Level: easy

  • Ruled surfaces and minimal surfaces

  • following: Section 3.5 in Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall; 1976

  • Math involved: basic differential geometry.

  • Level: easy

  • Free Lie algebras and Lyndon words

  • following: starting with Wikipedia here and here

  • Math involved: basic algebra and combinatorics.

  • Level: easy

  • Negatively curved metric spaces and their visual boundaries

    • following: Marc Bourdon, Sur le birapport au bord desCAT(−1)-espaces, Inst. Hautes Etudes Sci. Publ. Math. (1996),no. 83, 95–104

    • and Sergei Buyalo and Viktor Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, Zurich, 2007

    • Math involved: metric geometry.

    • Level: easy

  • Complex hyperbolic spaces

  • following: Serge Lang, Introduction to complex hyperbolic spaces.

  • Math involved: Complex analysis, Linear algebra, Lie groups.

  • Level: medium

  • Quaternionic hyperbolic spaces

  • following: John Parker, Hyperbolic Spaces.

  • Math involved: Complex analysis, Linear algebra, Lie groups.

  • Level: medium

  • Group and Lie algebra cohomology

  • following: Preliminary 3 in Raghunathan, Discrete subgroups of Lie groups.

  • and Cartan and Eilenberg, Homological algebra.

  • Math involved: Lie groups, algebra.

  • Level: medium

  • Metrics of negative type

  • Math involved: linear algebra, metric geometry, theoretical computer science.

  • Level: variable

  • Strictly pseudoconvex domains

  • Math involved: complex analysis.

  • Level: variable

  • CR manifolds

  • Math involved: complex analysis.

  • Level: variable

  • Symmetric spaces of rank one

  • Math involved: Lie groups, Riemannian geometry, complex analysis.

  • Level: medium

  • Hilbert-Smith Conjecture and p-adic integers

  • Math involved: topological groups, manifolds, algebra.

  • Level: varialbe

  • Structure of nilpotent Lie algebras and Lie groups

    • following: Chapter I of Roe Goodman, Nilpotent Lie Groups.

    • Math involved: algebra, Lie groups.

    • Level: medium

  • Locally compact groups acting 2-point transitively on the spheres

  • following: Linus Kramer,Two-transitive Lie groups, J. Reine Angew. Math.563(2003), 83–113

  • Math involved: Lie groups, topology, algebra, differential geometry.

  • Level: challenging