Project description
The broad context of problems is analysis on almost Abelian groups and their homogeneous spaces. An almost Abelian group is a (connected) non-Abelian Lie group which possesses a codimension one Abelian normal subgroup. The most prominent representatives of this class are the 3-dimensional Heisenberg group
H3 , the group
ax + b of affine transformations on the line, and the isometry group
E(2) of the Euclidean plane. Applications and interpretations of other almost Abelian Lie groups and algebras can be found in
[1] and
[2]. The class of almost Abelian groups is rather diverse and representative for solvable Lie groups from the group theory perspective, which makes it a very promising context for developing new methods in non-commutative harmonic analysis. The general objective is a comprehensive study of almost Abelian groups and homogeneous spaces with natural structures on them. This appears to be an ideal context for undergraduate research supervision in the following sense. Mathematics is learnt by doing it, but when you reach the more advanced topics, explicit examples become very rare and complicated. This often results in students having only an abstract understanding of things, without a hands-on command of details. Almost Abelian groups provide explicitly tractable (non-trivial, non-textbook-classical) examples of many advanced mathematical notions; covering manifolds, non-exonential groups, PDEs with variable coefficients, self-adjoint operators on non-compact domains etc. Students are very excited to see very real examples of what they know should exist, and contribute new results to mathematics on that way. Below are some of the particular research problems addressed in collaboration with undergraduate students.
- Classification up to isomorphism of all almost Abelian homogeneous spaces and an explicit description of their homotopy types.
- Explicit description of all discrete subgroups of a connected almost Abelian group, their classification up to isomorphism or automorphisms.
- Necessary and sufficient conditions for a connected almost Abelian group to admit a faithful matrix representation, and an explicit description of such a representation when it exists.