Periodic Reporting for period 1 - FunSilting (Functorial techniques in silting theory)
Período documentado: 2019-01-01 hasta 2020-12-31
The setting of the project is an area of pure mathematics known as representation theory, which can be thought as the abstract theory of symmetry. The development of representation theory in the twentieth century has led to a beautiful theoretical framework, yielding interactions with other areas of mathematics such as geometry, combinatorics and topology. As well as being pervasive within mathematics, representation theory has various applications to real-world problems including to theoretical physics and data analysis.
The central research objective of the project was to bring together two different approaches to representation theory. The first approach, using functorial techniques, comes from mathematical logic, which is an mathematical discipline where mathematical questions are encoded into formal statements that are manipulated according to a fixed algorithm. The second approach, silting theory, comes from abstract algebra, which is the traditional setting of representation theory. By taking advantage of the synergy between these two approaches, the project aimed to make fundamental advances in the field of silting theory.
As well as these research objectives, the project aimed to provide the researcher with many opportunities to develop her management, teaching and communication skills.
The central research objective of the project was to bring together two different approaches to representation theory. The first approach, using functorial techniques, comes from mathematical logic, which is an mathematical discipline where mathematical questions are encoded into formal statements that are manipulated according to a fixed algorithm. The second approach, silting theory, comes from abstract algebra, which is the traditional setting of representation theory. By taking advantage of the synergy between these two approaches, the project aimed to make fundamental advances in the field of silting theory.
As well as these research objectives, the project aimed to provide the researcher with many opportunities to develop her management, teaching and communication skills.
The following is a summary of the activities carried out by the researcher:
-The researcher was a main speaker at five international conferences.
-The researcher gave talks in three external research seminars, including one colloquium talk.
-The researcher gave ten talks at internal seminars, including the MALGA seminar, which is a joint seminar between the Universities of Verona and Padova.
-During the project the researcher, together with the collaborators specified above, produced one publication, one preprint and has three preprints in preparation.
-The researcher wrote a features article for the London Mathematical Society (LMS) Newsletter aimed at a general audience with some mathematical background.
-The researcher was a lecturer at the LMS Autumn Algebra School introducing new PhD students to the subject area of the project.
-The researcher was a main speaker at five international conferences.
-The researcher gave talks in three external research seminars, including one colloquium talk.
-The researcher gave ten talks at internal seminars, including the MALGA seminar, which is a joint seminar between the Universities of Verona and Padova.
-During the project the researcher, together with the collaborators specified above, produced one publication, one preprint and has three preprints in preparation.
-The researcher wrote a features article for the London Mathematical Society (LMS) Newsletter aimed at a general audience with some mathematical background.
-The researcher was a lecturer at the LMS Autumn Algebra School introducing new PhD students to the subject area of the project.
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.
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.