Modular character sheaves

本项目旨在完成模群表示的基本数学理论，即有限数系统对称性的代数描述。群表示理论可以应用于任何涉及对称性的线性问题。然而，模块化情况下，底层字段的特征是素数，比实标量或复杂标量更不容易理解，这种理解的缺乏阻碍了潜在的应用。这个项目将使用几何方法来回答李型有限群的模表示问题，李型有限群是有限群中最重要的一类。这个项目可以使模块化表示理论成为计算的基础，使线性代数问题的更快解决方案成为可能，并允许未来在数据传输技术等领域的应用。

This project aims to complete the fundamental mathematical theory of modular group representations, the algebraic description of symmetry over finite number systems. Group representation theory can be applied to any linear problem involving symmetry. However, the modular case, where the characteristic of the underlying field is a prime number, is less understood than real or complex scalars, and this lack of understanding blocks potential applications. This project will use geometric methods to answer questions about modular representations of the finite groups of Lie type, the most important class of finite groups. This project could make modular representation theory essential for computations, enabling faster solutions to problems of linear algebra and allowing future applications in such areas as data transmission technology.