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

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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.