MATH 488: Hung Le Pham's Projects

Banach algebra and automatic continuity

Linear mappings between finite dimensional Euclidean spaces are always continuous, but linear mappings between Banach spaces (infinite dimensional generalisations of Euclidean spaces) are no longer guaranteed to be continuous. However, if one imposes more algebraic structure on the spaces and the mappings, such as replacing Banach spaces by Banach algebras and linear mappings by algebra homomorphisms or derivations, then there is a wealth of results that say that in many cases, such mappings are automatically continuous. The aim of this project is to develop sufficient basic theory of Banach algebras, and then to survey these automatic continuity results.  

Paradoxical decompositions and amenable groups

Amenable groups is a nice class of groups that generalises solvable groups (in particular, abelian groups) and finite groups. This class of groups arise from the work of Hausdorff, Banach, Tarski, and von Neumann on paradoxical decompositions in early 20th century: the reason why Banach and Tarski can mathematically divide an orange into six pieces and rearrange them to get two new oranges that are exact replica of the original one is because the group of rigid motions in the space is not amenable. Since then amenability has become an important concept both in geometric group theory and in functional analysis. The aim of this project is to understand this concept and its properties.