Unipotent characters of finite groups

我的研究方向是群论。群是被广泛研究的抽象数学对象。在某种意义上，群论是对对称性的研究。小组在大多数数学领域都有应用，也在物理、化学和其他科学领域有应用。我将研究有限群表示的某些方面。组可以被视为可能成为对称性的抽象元素的集合。表示法是将混凝土结构的对称性与这些元素相关联的一种方式。表象理论已被证明是研究群的一种非常有价值的工具。事实上，表示法使数学家能够相对容易地找到关于群的深层定理的证明。其中一个定理早在1901年就由Georg Frobenius证明了，尽管付出了巨大的努力，但从那以后没有人找到不使用表示法的证明。表示本身是重要的，并且有许多应用，例如，在化学中的分子对称性的研究中。两种主要类型的有限群对这个项目特别重要：李型群和可解群。20世纪70年代末，Pierre Deligne和George Lusztig在李型群的表示理论方面取得了突破性的发现。他们使用代数几何的思想来描述表示法，代数几何是数学的一个非常不同的分支。另一方面，可解群的表示已经被成功地用更直接的方法来研究。这个项目的基本主题是将这两种方法结合在一起。我将研究中间群的表示，每个中间群由两个分量组成，一个是可解的，另一个是Lie类型的。其主要目的是将Deligne和Lusztig的几何思想推广到更广泛的群类，从而在中间群的表示理论上取得进展。

My research is in the theory of groups. Groups are widely studied abstract mathematical objects. In some sense, group theory is the study of symmetry. Groups have applications in most areas of mathematics, as well as in physics, chemistry and other sciences. I will investigate certain aspects of representations of finite groups. A group may be seen as a collection of abstract elements that can potentially become symmetries. A representation is a way to associate symmetries of a concrete structure to those elements. Representation theory has proved to be an extremely valuable tool for investigating groups. Indeed, representations allowed mathematicians to find relatively easy proofs of deep theorems about groups. One of these theorems was proved by Georg Frobenius as early as 1901, and despite significant efforts, nobody has found a proof that does not use representations ever since. Representations are important in their own right and have numerous applications, for instance, in the study of molecular symmetry in chemistry.Two major types of finite group are of particular importance to this project: groups of Lie type and solvable groups. In the late 1970s Pierre Deligne and George Lusztig made ground-breaking discoveries in representation theory of groups of Lie type. They described representations using ideas from algebraic geometry, a very different branch of mathematics. On the other hand, representations of solvable groups have been successfully investigated by more direct methods.The underlying theme of this project is to bring these two approaches together. I will research representations of intermediate groups, each of which consists of two components, one solvable, and the other of Lie type. The key aim is to extend the geometric ideas of Deligne and Lusztig to a wider class of groups, thus making advances in representation theory of intermediate groups.