This workshop will consist of three lecture series of 5 hours each.

In the first part of the series, we describe the main results on fixed point ratios including some classical estimates, plus more recent work for groups of Lie type. For the classical groups, this will include a discussion of subgroup structure, conjugacy classes and also some representation theory.

We also mention analogues (and recent work) at the level of algebraic groups.

In the second half, we concentrate on applications of the work described in Part I, e.g. to generation and random generation problems for simple groups, with connections to spread and generating graphs, to problems in permutation group theory such as base sizes, to random walks on groups, and monodromy groups.

We will describe some applications of the character theory of finite groups of Lie type to the following topics: widthÂ questions, word maps, mixing times of random walks, and representation varieties.

The development of high-quality algorithms to investigate the structure of a linear group is a highly active area of research. Natural questions include:

* How do we compute the order of a given matrix?

* How do we decide if a linear group acts reducibly on its underlying vector space?

* How do we compute conjugacy classes of elements and characters of the group?

We will consider these and related topics and describe the algorithms used to answer such questions.

*Tim Burness*, Bristol University**Simple groups, fixed point ratios and applications**In the first part of the series, we describe the main results on fixed point ratios including some classical estimates, plus more recent work for groups of Lie type. For the classical groups, this will include a discussion of subgroup structure, conjugacy classes and also some representation theory.

We also mention analogues (and recent work) at the level of algebraic groups.

In the second half, we concentrate on applications of the work described in Part I, e.g. to generation and random generation problems for simple groups, with connections to spread and generating graphs, to problems in permutation group theory such as base sizes, to random walks on groups, and monodromy groups.

*Martin Liebeck*, Imperial College London**Applications of character theory**We will describe some applications of the character theory of finite groups of Lie type to the following topics: widthÂ questions, word maps, mixing times of random walks, and representation varieties.

*Eamonn O'Brien*, University of Auckland**Algorithms for linear groups**The development of high-quality algorithms to investigate the structure of a linear group is a highly active area of research. Natural questions include:

* How do we compute the order of a given matrix?

* How do we decide if a linear group acts reducibly on its underlying vector space?

* How do we compute conjugacy classes of elements and characters of the group?

We will consider these and related topics and describe the algorithms used to answer such questions.