skip to content
 

Timetable (GRAW01)

Introductory/instructional workshop

Monday 6th January 2020 to Friday 10th January 2020

Monday 6th January 2020
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (Isaac Newton Institute)
10:00 to 11:00 Olivier Dudas
Finite reductive groups and their irreducible characters - 1
In these talks I will present some of the results of Lusztig from the 80's about the classification of irreducible characters of finite reductive groups. Starting from the example of GL(n,q) and Sp(4,q), I will explain the relation between Lusztig's classification and the geometry of conjugacy classes in the Langlands dual group.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Radha Kessar
Introduction to fusion systems-I
Fusion systems crystallize the phenomenon of p-localisation in finite group theory, modular representation theory and homotopy theory. The two talks will provide an introduction to the basic features of the theory of fusion systems.
INI 1
12:30 to 14:00 Lunch at Westminster College
14:00 to 15:00 Ellen Henke
The classification of finite simple groups via fusion systems
After outlining some basic ideas which were used to prove the classification of finite simple groups, I will explain why one can hope to get a shorter proof by working with fusion systems. If time permits I will report on some recent developments.
INI 1
15:00 to 16:00 Chris Parker
Groups with lots of p-local subgroups of characteristic p.
I will describe recent progress towards the classification of groups of parabolic characteristic p, p a prime. I will also draw attention to where methods used in the classification might transfer easily to the classification of saturated fusion systems. Much of the work I will describe is joint with Gernot Stroth and Ulrich Meierfrankenfeld.
INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:30 Stephen Donkin
Algebraic Groups and Finite Dimensional Algebras
We will discuss the representation theory of algebraic groups in positive characteristic with particular emphasis on connections with the representation theory of finite dimensional algebras.
INI 1
17:30 to 18:30 Welcome Wine Reception at INI
Tuesday 7th January 2020
09:00 to 10:00 Radha Kessar
Introduction to fusion systems-II
INI 1
10:00 to 11:00 Olivier Dudas
Finite reductive groups and their irreducible characters - 2
In these talks I will present some of the results of Lusztig from the 80's about the classification of irreducible characters of finite reductive groups. Starting from the example of GL(n,q) and Sp(4,q), I will explain the relation between Lusztig's classification and the geometry of conjugacy classes in the Langlands dual group.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Markus Linckelmann
Blocks of finite group algebras I
This is the first of two introductory talks on the modular representation theory of finite group algebras. After briefly introducing p-modular systems, we will review basic properties of algebras over complete discrete valuations rings as far as needed to describe some of the classic material on blocks of finite group algebras, much of which is due to Brauer and Green. This includes the notions of defect groups, the Brauer correspondence, relative projectivity, and the Green correspondence. We describe how to associate fusion systems to blocks, based on work of Alperin and Broue.
INI 1
12:30 to 14:00 Lunch at Westminster College
14:00 to 15:00 Frank Lübeck
Computing with reductive groups
I will sketch how we can model connected reductive groups on a computer via an efficient description of a root datum. It will be mentioned how some data about the reductive groups can be computed from this simple data structure. Some explicit data about conjugacy classes and representations of these groups will be presented and explained.
INI 1
15:00 to 16:00 Derek Holt
Low dimensional cohomology of finite groups
Algorithms for computing the 1- and 2-cohomology of finite groups acting on finite dimensional modules will be discussed.

Applications to finding subgroups of finite groups and to classifying transitive permutation groups of low degree will be described.

We also describe briefly methods for computing the projective indecomposable modules of finite groups over finite fields, and their application to computing higher cohomology groups.
INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:30 Jay Taylor
Representations and Unipotent Classes
In the representation theory of finite reductive groups unipotent classes play a prominent role. In this talk I will illustrate why this is the case. Specifically I will present work of Kawanaka and Lusztig and explain to some extent how these fit together. Examples will be our guide, with the focus being on GL_n(q) and Sp_4(q).
INI 1
Wednesday 8th January 2020
09:00 to 10:00 Markus Linckelmann
Blocks of finite group algebras II
This is the second of two introductory talks on the modular representation theory of finite group algebras. We investigate the interplay between the invariants of the module categories of block algebras and the invariants of the associated fusion systems, leading up to statements of some of the conjectures which drive this area.
INI 1
10:00 to 11:00 Colva Roney-Dougal
Aschbacher's theorem
This talk will introduce Aschbacher's theorem, which classifies the subgroups of (most of the) classical groups into nine classes. I will briefly describe the classes, and give some indication of how the theorem is proved.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Aner Shalev
Connections with other Areas of Mathematics
It is well known that groups occur naturally in many branches of mathematics, which thereby benefit from Group Theory.
In my talk I plan to show how various branches of mathematics contribute to classical as well as recent advances in Group Theory,
and help solving various open problems and conjectures. These branches include Probability, Lie Theory, Representation Theory,
Number Theory and Algebraic Geometry.
INI 1
12:30 to 14:00 Lunch at Westminster College
14:00 to 18:00 Free afternoon
19:30 to 22:00 Formal Dinner at Emmanuel College (Hall)
Thursday 9th January 2020
09:00 to 10:00 Martin Liebeck
The subgroup structure of almost simple groups, I
In these two talks I will discuss the subgroup structure of almost simple groups, the emphasis being mainly on maximal subgroups.
The first talk will focus on alternating and classical groups, and the second on exceptional groups of Lie type.
INI 1
10:00 to 11:00 Eamonn O'Brien
Algorithms for matrix groups I
Most first-generation algorithms for matrix groups defined over
finite fields rely on variations of the Schreier-Sims algorithm,
and exploit the action of the group on an set of vectors or subspaces
of the underlying vector space. Hence they face serious practical limitations.

Over the past 25 years, much progress has been achieved on
developing new algorithms to study such groups.
Relying on a generalization of Aschbacher's theorem
about maximal subgroups of classical groups,
they exploit geometry arising from the natural action
of the group on its underlying vector space to
identify useful homomorphisms. Recursive application of these
techniques to image and kernel now essentially allow us to
construct in polynomial time the composition factors
of the linear group. Using the notion of standard generators,
we can realise effective isomorphisms between a final simple group
and its "standard copy".

In these lectures we will discuss the "composition tree" algorithm
which realises these ideas; and the "soluble radical model" which
exploits them to answer structural questions about the input group.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Britta Späth
Reduction Theorems for global-local Conjectures
This talk should give an overview of the reduction theorems proven in the last years. The aim is to prove certain long-standing conjectures relating representations of finite groups to the ones of certain subgroups by reducing them to statements about representations of finite quasi-simple groups and use the classification of finite simple groups.
Key to the proof of the reduction theorems is a refinement of Clifford theory. These results have also put forward numerous questions on the representations of finite simple groups.
INI 1
12:30 to 14:00 Lunch at Westminster College
14:00 to 15:00 Klaus Lux
Basic algorithms in the representation theory of finite groups and algebras
The aim of this talk is an introduction to the basic algorithms in computational representation theory. We will start with the fundamental MeatAxe algorithm for proving the irreducibility of a representation. Based on the MeatAxe one can give algorithmic solutions to more advanced tasks such as determining the endomorphism ring of a representation or the decomposition into a direct sum of indecomposable representations. Finally, we will discuss the condensation method, an important tool used in these applications.
INI 1
15:00 to 16:00 Peter Cameron
Finite permutation groups: the landscape post-CFSG

In the 40 years since the Classification of Finite Simple Groups was first (tentatively) announced, our understanding of finite permutation groups has been radically transformed: previously untouchable classical problems have been solved, many new classifications produced, and applications to a wide area of mathematics, from number theory to computer science, have been developed. In this lecture I will outline the theory of finite permutation groups and explain how, using the O'Nan--Scott Theorem, CFSG can be brought to bear on various questions. In some cases, proofs avoiding CFSG have subsequently been found; but there are some interesting challenges where this has not so far proved possible.

INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:00 Ana Bernal
On self-Mullineux and self-conjugate partitions
The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this talk we will see details on this, on the Mullineux map and I will show an explicit bijection between the two mentioned families of partitions in terms of the Mullineux symbol.
INI 1
17:00 to 17:30 Cesare Giulio Ardito
Classifying 2-blocks with an elementary abelian defect group

Donovan's conjecture predicts that given a $p$-group $D$ there are only finitely many Morita equivalence classes of blocks of group algebras of finite groups with defect group $D$. While the conjecture is still open for a generic $p$-group $D$, it has been proven in 2014 by Eaton, Kessar, Külshammer and Sambale when D is an elementary abelian 2-group, and in 2018 by Eaton, Eisele and Livesey when D is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classification up to Morita equivalence over a complete discrete valuation ring $\mathcal{O}$ has been achieved when $p=2$ for abelian $D$ with rank $3$ or less, and for $D=(C_2)^4$.In my PhD thesis I have done $(C_2)^5$, and I have partial results on $(C_2)^6$. I will introduce the topic, give some definitions and then describe the process of classifying these blocks, with a focus on the process and the tools needed to produce a complete classification. All the obtained data is available on https://wiki.manchester.ac.uk/blocks/.

INI 1
Friday 10th January 2020
09:00 to 10:00 Martin Liebeck
The subgroup structure of almost simple groups, II
In these two talks I will discuss the subgroup structure of almost simple groups, the emphasis being mainly on maximal subgroups.
The first talk will focus on alternating and classical groups, and the second on exceptional groups of Lie type.
INI 1
10:00 to 11:00 Eamonn O'Brien
Algorithms for matrix groups II
Most first-generation algorithms for matrix groups defined over
finite fields rely on variations of the Schreier-Sims algorithm,
and exploit the action of the group on an set of vectors or subspaces
of the underlying vector space. Hence they face serious practical limitations.

Over the past 25 years, much progress has been achieved on
developing new algorithms to study such groups.
Relying on a generalization of Aschbacher's theorem
about maximal subgroups of classical groups,
they exploit geometry arising from the natural action
of the group on its underlying vector space to
identify useful homomorphisms. Recursive application of these
techniques to image and kernel now essentially allow us to
construct in polynomial time the composition factors
of the linear group. Using the notion of standard generators,
we can realise effective isomorphisms between a final simple group
and its "standard copy".

In these lectures we will discuss the "composition tree" algorithm
which realises these ideas; and the "soluble radical model" which
exploits them to answer structural questions about the input group.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Charles Eaton
Conjectures in local representation theory
I will survey the principal conjectures concerning the local determination of information about blocks of finite groups. Local determination, meaning determination from information about p-subgroups, their normalizers and related subgroups, has been a significant strand of the subject since Brauer. In some cases we can already tell a great deal about a block even from just the isomorphism type of its defect groups. Much of the recent activity around block theory revolves around a number of conjectures predicting precisely how invariants are determined from local subgroups. These invariants range from simple numerical invariants such as the dimension of the centre to the equivalence class of the module category and related categories.
INI 1
12:30 to 14:00 Lunch at Westminster College
14:00 to 15:00 Peter Cameron
Finite permutation groups: applications to transformation semigroups and synchronization

For the last 10 years, I have been working with João Araújo and others on exploring how our new understanding of finite permutation groups can be used to advance the theory of finite transformation semigroups. In particular, I will talk about regularity and idempotent generation of semigroups, and synchronizing automata. The pioneering work had been done by semigroup theorists assuming that the transformation semigroup contains the symmetric or alternating group; but this assumption can be substantially weakened in many cases. (The first such result was a classification of the permutation groups G with the property that, for any non-permutation t, the semigroup generated by G and t is (von Neumann) regular. I will mention many open problems in this area.

INI 1
15:00 to 15:30 Surinder Kaur
Conjugacy Classes and the Normal Complement Problem in Group Algebras
The study of structure of the unit group is one of the fundamental problems in theory of group algebras. In this direction, we mainly address the conjugacy class problem and the normal complement problem in certain classes of modular group algebras. First, we discuss the conjugacy class problem in the unit group of modular group algebras of some finite p-groups. Then we establish the role conjugacy classes play in resolving the classical normal complement problem in modular group algebras of some classes of groups whose order is divisible by two primes.
INI 1
15:30 to 16:00 Aluna Rizzoli
Finite Singular Orbit Modules for Algebraic Groups

We determine all irreducible modules for simple algebraic groups with finitely many orbits on singular 1-spaces.

INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons