Periodic Reporting for period 1 - STABRT (Stability conditions in Representation Theory and beyond)
Reporting period: 2021-09-01 to 2023-08-31
In science, researchers typically select a subject of study, measure one or several features of the subject, and then attempt to derive information from the collected data. The obtained data can be broadly categorized into two types. The first category consists of continuous data, usually quantitative, capable of taking infinitely many values that transition smoothly from one to another. Examples include the weight of newborn babies or the temperature of a certain material.
In contrast, the second category involves discrete data, generally more qualitative, taking specific, distinct values where transitions represent a notable change in the feature. For instance, whether a lamp is on or off or the number of eggs a bird has laid in its nest are examples of discrete data.
An intriguing problem arises when attempting to understand how discrete and continuous data from the same subject interact. A classic example of this interaction can be observed in the study of water. Under normal circumstances, water can exist in three states: solid, liquid and gas. We have come to understand that, under typical conditions near sea level, water is solid below 0 degrees, liquid between 0 and 100 degrees, and a gas above 100 degrees. Notably, within the spectrum of all possible water temperatures, 0 and 100 degrees are special points, representing walls of some sort where abrupt changes occur.
This project is situated in pure mathematics, specifically in the representation theory of Artin algebras. In this context, researchers typically work with discrete objects, such as isomorphism classes of modules or torsion classes in the module category. However, recent developments have revealed that associated with every algebra in a significant family is a geometric object of continuous nature known as the scattering diagram of the algebra. This diagram was hypothesized to encode much of the discrete information of the algebra and its module category. Subsequently, it was demonstrated that a substantial portion of the scattering diagram of an algebra can be constructed using modules satisfying a specific homological condition known as τ-rigid modules.
This project has several objectives. Firstly, it aims to comprehensively understand scattering diagrams of algebras from both homological and representation theoretic perspectives. Secondly, it seeks to leverage the geometric nature of scattering diagrams to establish new connections and strengthen existing ones between representation theory and other mathematical branches such as geometry and topology. Finally, it intends to utilize the scattering diagram of algebras to obtain new insights into the behavior of module categories that are currently poorly understood.
In contrast, the second category involves discrete data, generally more qualitative, taking specific, distinct values where transitions represent a notable change in the feature. For instance, whether a lamp is on or off or the number of eggs a bird has laid in its nest are examples of discrete data.
An intriguing problem arises when attempting to understand how discrete and continuous data from the same subject interact. A classic example of this interaction can be observed in the study of water. Under normal circumstances, water can exist in three states: solid, liquid and gas. We have come to understand that, under typical conditions near sea level, water is solid below 0 degrees, liquid between 0 and 100 degrees, and a gas above 100 degrees. Notably, within the spectrum of all possible water temperatures, 0 and 100 degrees are special points, representing walls of some sort where abrupt changes occur.
This project is situated in pure mathematics, specifically in the representation theory of Artin algebras. In this context, researchers typically work with discrete objects, such as isomorphism classes of modules or torsion classes in the module category. However, recent developments have revealed that associated with every algebra in a significant family is a geometric object of continuous nature known as the scattering diagram of the algebra. This diagram was hypothesized to encode much of the discrete information of the algebra and its module category. Subsequently, it was demonstrated that a substantial portion of the scattering diagram of an algebra can be constructed using modules satisfying a specific homological condition known as τ-rigid modules.
This project has several objectives. Firstly, it aims to comprehensively understand scattering diagrams of algebras from both homological and representation theoretic perspectives. Secondly, it seeks to leverage the geometric nature of scattering diagrams to establish new connections and strengthen existing ones between representation theory and other mathematical branches such as geometry and topology. Finally, it intends to utilize the scattering diagram of algebras to obtain new insights into the behavior of module categories that are currently poorly understood.
All the work conducted in this project is interconnected.
To thoroughly comprehend the associated scattering diagram of an algebra, we've pursued two approaches. First, we characterized the stability space of string and band modules. This characterization allows us to completely determine the wall-and-chamber structure of special biserial algebras. Second, we studied the space of chains of torsion classes in abelian categories, particularly in module categories. Our main result in this line of research can be stated as follows: the scattering diagram of an algebra has finitely many walls if and only if the space of chains of torsion classes of the algebra is compact.
We've also explored the connection between the representation theory of algebras and algebraic topology and algebraic geometry through their wall-and-chamber structures. Concerning algebraic geometry, we've demonstrated that the deformation theory of the wall-and-chamber structure of a cluster-tilted algebra of Dinkyn type includes as a subvariety the universal cluster algebra of the same type. Independently, a bridge between representation theory and algebraic topology has emerged, where we have shown that the tau-cluster morphism category of an algebra is determined by their wall-and-chamber structure.
In recent years, higher homological algebra has played a central role in representation theory, with significant efforts to generalize classical notions to this new setting. One of our aims is to develop wall-and-chamber structures and scattering diagrams in higher homological algebra. To achieve this, we begin by studying the concept of higher torsion classes and their relationship with classical torsion classes. We have demonstrated that every d-torsion class inside a d-cluster tilting subcategory is the intersection of a classical torsion class with the said d-cluster tilting subcategory and we have provided an internal characterization of d-torsion classes.
Concurrently, we've been exploring abelian categories with enough projective objects, a generalization of the category of finitely presented modules over an algebra. In this context, we've introduced and studied the notion of support τ-tilting subcategories.
Finally, we've utilized the tools developed in the study of scattering diagrams to address internal problems within the representation theory of algebras. Notably, we've shown that an algebra is τ-tilting finite if and only if the dimension of all bricks (a special type of module) is bounded.
To thoroughly comprehend the associated scattering diagram of an algebra, we've pursued two approaches. First, we characterized the stability space of string and band modules. This characterization allows us to completely determine the wall-and-chamber structure of special biserial algebras. Second, we studied the space of chains of torsion classes in abelian categories, particularly in module categories. Our main result in this line of research can be stated as follows: the scattering diagram of an algebra has finitely many walls if and only if the space of chains of torsion classes of the algebra is compact.
We've also explored the connection between the representation theory of algebras and algebraic topology and algebraic geometry through their wall-and-chamber structures. Concerning algebraic geometry, we've demonstrated that the deformation theory of the wall-and-chamber structure of a cluster-tilted algebra of Dinkyn type includes as a subvariety the universal cluster algebra of the same type. Independently, a bridge between representation theory and algebraic topology has emerged, where we have shown that the tau-cluster morphism category of an algebra is determined by their wall-and-chamber structure.
In recent years, higher homological algebra has played a central role in representation theory, with significant efforts to generalize classical notions to this new setting. One of our aims is to develop wall-and-chamber structures and scattering diagrams in higher homological algebra. To achieve this, we begin by studying the concept of higher torsion classes and their relationship with classical torsion classes. We have demonstrated that every d-torsion class inside a d-cluster tilting subcategory is the intersection of a classical torsion class with the said d-cluster tilting subcategory and we have provided an internal characterization of d-torsion classes.
Concurrently, we've been exploring abelian categories with enough projective objects, a generalization of the category of finitely presented modules over an algebra. In this context, we've introduced and studied the notion of support τ-tilting subcategories.
Finally, we've utilized the tools developed in the study of scattering diagrams to address internal problems within the representation theory of algebras. Notably, we've shown that an algebra is τ-tilting finite if and only if the dimension of all bricks (a special type of module) is bounded.
The work conducted in this project has brought us to the verge of establishing a coherent scattering diagram for every Artin algebra, a substantial achievement in itself. This progress may also have implications for solving conjectures related to skew-symmetrizable cluster algebras. Furthermore, the enhanced understanding of wall-crossing phenomena in representation theory developed in this project could be leveraged in the mid- to long-term to address longstanding conjectures in algebraic geometry.
We anticipate that the general theory of support τ-tilting subcategories in abelian categories with enough projective modules will find numerous applications, of which we will mention two here. Firstly, in the monoidal categorifications of cluster algebras and/or the categorifications of cluster algebras arising from algebraic geometry, such as the coordinate ring of the Grassmanian or its positroid variety. In these cases, the general theory gives rise to calculations that quickly become intractable. We expect that clusters and mutations in these cluster algebras will be explicitly categorified by support τ-tilting subcategories.
The second application of support τ-tilting subcategories is more applied. In recent years, there has been significant interest in topological data analysis (TDA) and its applications to the training of neural networks. It turns out that persistence theory, the central tool in TDA, is equivalent to the study of finitely presented modules over the nonnegative real line—an abelian category with enough projective objects. Hence, a better understanding of support τ-tilting subcategories in this context could lead to applications in informatics.
Our work on the deformation theory of scattering diagrams for cluster-tilted algebras of finite type has opened the door to the development of a commutative algebra that could be associated with every τ-tilting finite algebra, capturing the combinatorial properties of cluster algebras, even if they are not defined combinatorially.
Within representation theory, the impacts are already evident. Due to the work of this project and that of other mathematicians, we have arguably established the most crucial conjecture in τ-tilting theory.
We anticipate that the general theory of support τ-tilting subcategories in abelian categories with enough projective modules will find numerous applications, of which we will mention two here. Firstly, in the monoidal categorifications of cluster algebras and/or the categorifications of cluster algebras arising from algebraic geometry, such as the coordinate ring of the Grassmanian or its positroid variety. In these cases, the general theory gives rise to calculations that quickly become intractable. We expect that clusters and mutations in these cluster algebras will be explicitly categorified by support τ-tilting subcategories.
The second application of support τ-tilting subcategories is more applied. In recent years, there has been significant interest in topological data analysis (TDA) and its applications to the training of neural networks. It turns out that persistence theory, the central tool in TDA, is equivalent to the study of finitely presented modules over the nonnegative real line—an abelian category with enough projective objects. Hence, a better understanding of support τ-tilting subcategories in this context could lead to applications in informatics.
Our work on the deformation theory of scattering diagrams for cluster-tilted algebras of finite type has opened the door to the development of a commutative algebra that could be associated with every τ-tilting finite algebra, capturing the combinatorial properties of cluster algebras, even if they are not defined combinatorially.
Within representation theory, the impacts are already evident. Due to the work of this project and that of other mathematicians, we have arguably established the most crucial conjecture in τ-tilting theory.