Unipotent characters of finite groups

我的研究方向是群体理论。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。分组在数学的大多数领域以及物理、化学和其他科学领域都有应用。我将研究有限群表示的某些方面。群可以看作是抽象元素的集合，这些元素有可能成为对称的。表示是将具体结构的对称性与这些元素联系起来的一种方式。表征理论已被证明是研究群体的一个极有价值的工具。事实上，表示使数学家们能够相对容易地找到关于群的深层定理的证明。早在1901年，Georg Frobenius就证明了其中一个定理，尽管付出了巨大的努力，但从那时起，没有人发现一个不使用表示法的证明。表征本身就很重要，并且有许多应用，例如，在化学分子对称的研究中。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在李型群的表征理论方面取得了突破性的发现。他们用代数几何的概念来描述表征，代数几何是数学的一个非常不同的分支。另一方面，用更直接的方法成功地研究了可解群的表示。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，每个中间群由两个组成部分组成，一个是可解的，另一个是Lie型的。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方面取得进展。

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.