The fundamental idea in representation theory is to understand an abstract algebraic object - usually a ring or an algebra - by investigating how it acts on simpler additive structures. These actions are known as representations or modules and are the main objects of study in representation theory. The collection of all representations associated to a particular ring can be considered as a mathematical structure in its own right and is known as the module category. Understanding the structure of module categories, as well as associated categories such as derived and stable module categories, is the central aim in the field.
There are many approaches to understanding the structure of these categories and the aim of the project was to bring together two such approaches: silting theory and functorial methods.
Silting theory is a fast-developing area of contemporary representation theory that provides a unified approach to derived representation theory, offering a vital insight into the structure of the module category and derived category.
The functorial methods used in the project grew out of an area of mathematical logic known as model theory that aims to understand the connections between formal theories of mathematics and their models. This has given rise to a wide range of foundational results and powerful techniques in representation theory: the so-called theory of purity.
The following is a summary of progress beyond the state-of-the-art made in the project:
1. We investigate the structure of some abelian categories arising in silting theory called cotilting hearts and give an explicit description of the simple objects and their injective envelopes (joint work with L. Angeleri Hügel and I. Herzog).
2. We study so-called silting and cosilting objects in a triangulated category. We define an operation on the collection of these objects called mutation and investigate the structure of the ambient triangulated category from this point of view ( joint work with L. Angeleri Hügel, J. Stovicek and J. Vitória).
3. We specialise mutation to a particularly interesting case: the setting of finite-dimensional algebras. We describe explicitly the effect of mutation on the module category, in particular on its torsion classes (joint work with L. Angeleri Hügel and F. Sentieri).
4. We classify the cosilting modules over a particular class of finite-dimensional algebras (cluster- tilted algebras of type Ã): they can be parametrised by some geometric data (asymptotic triangulations) associated to the annulus with marked points in the boundary (joint work with K. Baur).
6. We investigate silting theory in a particular setting of non commutative algebraic geometry. In particular we classify certain classes of sheaves: the cotilting sheaves and the indecomposable pure-injective sheaves (joint work with D. Kussin).
The impact of the main results in the project are likely to be wide-ranging within representation theory and beyond. The joint project with the researcher, L. Angeleri Hügel, J. Stovicek and J. Vitória is set in abstract triangulated categories, which are pervasive in representation theory, algebraic geometry and topology, and so we expect there to be broad applications in these areas. In particular, new links between this work and stability theory were identified during the project. Moreover, the project by the researcher and K. Baur suggests an innovative approach to studying the structure of the bounded derived category of certain gentle algebras, which likely to have implications for symplectic geometry due to recent connections made with the topological Fukaya categories. Both of these projects have the potential to impact the theory of cluster algebras, since the directed graph corresponding to mutation of cosilting modules over cluster-tilted algebras extends the cluster exchange graph of the corresponding cluster category. Finally, we expect the joint work with the researcher, L. Angeleri Hügel and I. Herzog to have an impact due to the methodology of its proof. This is because it introduces techniques and concepts from the model theory of modules into cotilting theory. These techniques are quite different from the usual approach to silting theory and pursuing this line of reasoning is likely to lead to new insights into the theory.
