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
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
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