Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

GRAW01 
6th January 2020 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.


GRAW01 
6th January 2020 11:30 to 12:30 
Radha Kessar 
Introduction to fusion systemsI
Fusion systems crystallize the phenomenon of plocalisation 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.


GRAW01 
6th January 2020 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.


GRAW01 
6th January 2020 15:00 to 16:00 
Chris Parker 
Groups with lots of plocal 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.


GRAW01 
6th January 2020 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.


GRAW01 
7th January 2020 09:00 to 10:00 
Radha Kessar  Introduction to fusion systemsII  
GRAW01 
7th January 2020 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.


GRAW01 
7th January 2020 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 pmodular 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.


GRAW01 
7th January 2020 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.


GRAW01 
7th January 2020 15:00 to 16:00 
Derek Holt 
Low dimensional cohomology of finite groups
Algorithms for computing the 1 and 2cohomology 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. 

GRAW01 
7th January 2020 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).


GRAW01 
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.


GRAW01 
8th January 2020 10:00 to 11:00 
Colva RoneyDougal 
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.


GRAW01 
8th January 2020 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. 

GRAW01 
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. 

GRAW01 
9th January 2020 10:00 to 11:00 
Eamonn O'Brien 
Algorithms for matrix groups I
Most firstgeneration algorithms for matrix groups defined over finite fields rely on variations of the SchreierSims 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. 

GRAW01 
9th January 2020 11:30 to 12:30 
Britta Späth 
Reduction Theorems for globallocal Conjectures
This talk should give an overview of the reduction theorems proven in the last years. The aim is to prove certain longstanding conjectures relating representations of finite groups to the ones of certain subgroups by reducing them to statements about representations of finite quasisimple 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. 

GRAW01 
9th January 2020 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.


GRAW01 
9th January 2020 15:00 to 16:00 
Peter Cameron 
Finite permutation groups: the landscape postCFSG 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'NanScott 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. 

GRAW01 
9th January 2020 16:30 to 17:00 
Ana Bernal 
On selfMullineux and selfconjugate 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 selfMullineux partitions is equal to the cardinality of a distinguished subset of selfconjugate 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.


GRAW01 
9th January 2020 17:00 to 17:30 
Cesare Giulio Ardito 
Classifying 2blocks 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 2group, and in 2018 by Eaton, Eisele and Livesey when D is any abelian 2group. 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/. 

GRAW01 
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. 

GRAW01 
10th January 2020 10:00 to 11:00 
Eamonn O'Brien 
Algorithms for matrix groups II
Most firstgeneration algorithms for matrix groups defined over finite fields rely on variations of the SchreierSims 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. 

GRAW01 
10th January 2020 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 psubgroups, 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.


GRAW01 
10th January 2020 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 nonpermutation t, the semigroup generated by G and t is (von Neumann) regular. I will mention many open problems in this area. 

GRAW01 
10th January 2020 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 pgroups. 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.


GRAW01 
10th January 2020 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 1spaces. 