# 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.

## Combinatorial algebra (Catarina Carvalho)

The field of combinatorial semigroup theory deals with infinite semigroups, which are natural generalisations of groups, defined by means of a presentation.

If an infinite semigroup can be defined by a finite presentation, which includes a finite generating set and a finite number of defining relations, then it can be encoded in a way that can be easily dealt with using a computer, making working with these semigroups a lot easier. These are naturally relates with computational algebra, automata theory, and the study of formal languages. Some problems in this area are:

- When is a given semigroup construction finitely generated and/or finitely presented?
- How does the finite presentability/ generation of a semigroup related with the finite presentability/generation of its group of units?