STABRT · Stability conditions in Representation Theory and beyond

Stability conditions in Representation Theory and beyond

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.

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