Quasi-isolated blocks and Brauer's height zero conjecture.

群是对称的数学抽象。认识到不相关的系统和对象可以具有相同的对称性，导致19世纪中叶的数学家们为群发展了一套内在公理，这些公理不涉及它们所描述的基本对象的对称性。这种抽象将组定义为对象的集合，其中任意两个对象可以遵循一组规定的规则--组公理--以一种方式相乘。这种抽象方法的优点是，为某一特定群体发现的任何事实都可以立即适用于该群体出现的所有背景。更重要的是，现在可以提出问题和证明适用于所有群的定理。群的公理简短且易于陈述，但群具有丰富的结构，并且它们的研究一直是过去150年来数学的中心主题。理解有限群的一种方法是通过它们的线性表示。表示法由群的基准以及群通过线性变换作用于其上的向量空间组成。由于与群相比，向量空间具有简单的结构，因此线性化方法将一个复杂的问题分解为一组更简单的问题。本课题研究有限群的模表示理论。这一理论由R.Brauer在20世纪30年代创立，它同时研究了零特征域(如复数)和正特征域(如模素数的整数域)上有限群的表示。这项工作的目的是完成李型有限群的L块的分类，并证明布劳尔高度为零猜想的正向性，这是一个近50年来一直悬而未决的问题。

Groups are the mathematical abstraction of symmetry. The realization that unrelated systems and objects can have the same symmetry led mathematicians in the mid-nineteenth century to develop a set of intrinsic axioms for groups which make no reference to the underlying objects whose symmetry they describe. This abstraction defines a group as a collection of objects in which any two objects can be multiplied in a way which follows a prescribed set of rules-the group axioms. This abstract approach had the advantage that any fact discovered for a particular group could then be applied at once to all the contexts in which this group appeared. Even more, it now became possible to formulate questions and prove theorems which held true for all groups.The axioms for groups are short and easy to state, but groups have a rich structure, and their study has been a central theme of mathematics for the past 150 years. One approach to understanding finite groups is through their linear representations. A representation consists of the datum of a group along with a vector space on which the group acts by linear transformations. Since vector spaces have a simple structure compared to groups, the linearisation approach breaks up a complex problem into a collection of simpler problems. This project deals with the modular representation theory of finite groups. In this theory, founded by R. Brauer in the 1930s, one studies simultaneously the representations of finite groups over fields of zero characteristic, such as the complex numbers and over fields of positive characteristic such as the field of integers modulo a prime number. The aim of this project is to complete the classification of l-blocks of finite groups of Lie type and to prove the forward direction of Brauer's height zero conjecture,a problem that has been open for nearly fifty years.