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物体可以具有相同的对称性，使得数学家在19世纪中期发展了一套关于群的内在公理，这些公理不涉及它们所描述的对称的潜在物体。这种抽象将群定义为对象的集合，其中任意两个对象可以按照一组规定的规则（群公理）进行相乘。这种抽象的方法有一个好处，那就是任何针对特定群体发现的事实都可以立即应用到这个群体出现的所有情境中。更重要的是，现在有可能提出问题并证明对所有群体都适用的定理。群的公理很短，很容易表述，但群有丰富的结构，它们的研究在过去150年里一直是数学的中心主题。理解有限群的一种方法是通过它们的线性表示。表示由群的基准和群通过线性变换作用于其上的向量空间组成。由于向量空间与群相比具有简单的结构，线性化方法将复杂问题分解为更简单问题的集合。本课题研究有限群的模表示理论。在这个由R. Brauer于20世纪30年代创立的理论中，人们同时研究了有限群在零特征域（如复数）和正特征域（如模素数的整数域）上的表示。本课题的目的是完成Lie型有限群的l块分类，并证明Brauer的高度零猜想的前进方向，这是一个已经开放了近五十年的问题。

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.