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?

General Algebra and Applications (Catarina Carvalho)

An algebraic structure is a set together with operations on that set. General, or universal, algebra studies algebraic structures as a whole, e.g. the variety of groups, or the variety of inverse semigroups, and not specific groups or inverse semigroups.
Some topics that we are particularly interested in:

Applications of general algebra in computational complexity, namely in the constraint satisfaction problem
Connections with graph theory, namely polymorphisms of graphs and graph homomorphism.
Connections with logic, namely finite model theory