Pure Mathematics projects

Topological Algebra (Yann Peresse)

Many important groups, semigroups and rings etc. can be given a natural topology that is compatible with their algebraic operations. Studying this combination of algebraic and topological structure can lead to a nice interplay of Topology and Algebra and a deeper understanding of these objects. For example, a topological group is a group equipped with a topology under which the group operation and inversion are continuous; an easy example is the group of real numbers under the usual addition and topology. Questions and problems in this area include:

• Given a specific group (semigroup, ring) specify what sort of topologies it admits. For example, the group S∞ of all permutations of a countably infinite set admits a Polish (completely metrizable and separable) topology called the pointwise topology. The pointwise topology is known to be the unique second countable Hausdorff topology on S∞ . Many other interesting groups and semigroups also admit unique Polish topologies.
• Given a group (semigroup, ring) with a fixed topology we can study subgroups with certain topological properties (open, closed, dense, compact. ...). For example, the closed subgroups of S∞ under the pointwise topology are precisely automorphism groups of relational structures.
• Study the class of all groups (semigroups, rings) that have some topological property. For example, there is a rich body of research into groups admitting compact Hausdorff topologies.

Key words:

• Topological groups;
• Semigroup Theory;
• Polish spaces, Polish groups, Polish semigroups;