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. 

GRA 
14th January 2020 11:00 to 12:00 
Gareth Tracey 
On the Chebotarev invariant of a finite group
Given a nite group X, a classical approach to proving that X is the Galois group of a Galois extension K=Q can be described roughly as follows: (1) prove that Gal(K=Q) is contained in X by using known properties of the extension (for example, the Galois group of an irreducible polynomial f(x) 2 Z[x] of degree n embeds into the symmetric group Sym(n)); (2) try to prove that X = Gal(K=Q) by computing the Frobenius automorphisms modulo successive primes, which gives conjugacy classes in Gal(K=Q), and hence in X. If these conjugacy classes can only occur in the case Gal(K=Q) = X, then we are done. The Chebotarev invariant of X can roughly be described as the eciency of this \algorithm". In this talk we will dene the Chebotarev invariant precisely, and describe some new results concerning its asymptotic behaviour. 

GRA 
16th January 2020 16:00 to 17:00 
George Glauberman 
A Mystery in Finite Groups of Even Order
Often in mathematics, we have a reason for believing that
something is true, but not a proof. I
plan to discuss a result that has a proof, but no clear reason.
Let x and y be elements of order two in a finite group G
that are not conjugate in G. An easy
proof shows that xy has even order. Now
take an element u that lies in a normal subgroup of odd order in the
centralizer of x in G, and an analogous
element v for y. Then (xu)(yv)
also has even order.
This result was obtained by simpleminded
manipulation of group characters, rather than by theory
or intuition about the structure of a finite group. Suggestions for reasons are welcome.


GRA 
21st January 2020 11:00 to 12:00 
Lucia Morotti 
Irreducible restrictions of representations of symmetric and alternating groups
In general the restriction of an irreducible
respresentation to a subgroup is not irreducible. There are though cases where
the restriction is irreducible. In this talk I will present a classification of
irreducible restrictions of representations of symmetric and alternating
groups, in particular about recent work joint with Alexander Kleshchev and Pham
Huu Tiep, which, up to one new family of irreducible restrictions, extends to
characteristics 2 and 3 the classification in larger characteristic, up to
considering also irreducible restrictions of spin representations when working
in characteristic 2.


GRA 
23rd January 2020 16:00 to 17:00 
Michael Giudici 
Automorphism orbits of groups and the Monster
The order of an element of a group is a natural
invariant of an automorphism. In 1992,
Zhang classified all finite groups such that for all integers $k$ the automorphism group acts transitively on the
set of all elements of order $k$. Such groups are called ATgroups. In this
talk, I will discuss recent joint work with Alexander Bors and Cheryl Praeger
that investigates two measures of how close a group is to being an ATgroup.
This includes a new interesting characterisation of the Monster simple group. 

GRAW02 
27th January 2020 10:10 to 11:00 
Meinolf Geck 
Computing Green functions in small characteristics
Green functions for finite groups of Lie type were
introduced by Deligne and Lusztig in the 1970's, using cohomological methods.
The computation of these functions is a crucial step in the more general
programme of determining the whole character tables of those groups.
Despite of a long tradition of work on Green functions,
there are still open cases for groups in small characteristic. We report on
some recent progress, which essentially relies on a combination of Lusztig's
theory of character sheaves and computer algebra methods.


GRAW02 
27th January 2020 11:30 to 12:20 
David Craven 
Constructing subgroups of exceptional algebraic groups
When trying to understand the subgroup structure of the
exceptional algebraic
groups and groups of Lie type, one often needs to
explicitly construct simple
subgroups of large finite groups. In this talk, we will
discuss some
relatively simple, yet powerful, ideas and algorithms
that can decide the
existence and number of classes of various maximal
subgroups of the larger
exceptional groups (of types E and F). We will apply
these methods to
construct some sporadic and crosscharacteristic Lie type
maximal subgroups
of E7 and E8.
This forms part of the speaker's programme to complete the classification of maximal subgroups. 

GRAW02 
27th January 2020 13:45 to 14:35 
Heiko Dietrich 
Constructive recognition of matrix groups
After decades of work by many people, the successes of
Matrix Group Recognition Projects and their
implementations in the
CompositionTree package allow, for the first time, to
compute efficiently
with large matrix groups (defined over finite fields). A
crucial ingredient
in those algorithms is the constructive recognition of
classical groups. I
will survey some of those results and then comment in
more detail on my work
(with Eamonn O'Brien and Charles LeedhamGreen) for
finding standard
generators in classical groups.


GRAW02 
27th January 2020 14:45 to 15:35 
Joanna Fawcett 
Base sizes of permutation groups
A base of a permutation group G acting on a set X is a subset of X whose pointwise stabiliser in G is trivial. The base size of G is the minimal cardinality of a base for G. Bases have proved to be very useful, both theoretically (in bounding the order of a primitive permutation group in terms of its degree) and computationally (in many algorithms for permutation groups). Recently, much progress has been made on understanding the base sizes of primitive permutation groups. This talk will survey some of these results.


GRAW02 
27th January 2020 16:05 to 16:35 
Melissa Lee 
Base sizes of permutation groups: an encore
We describe some recent work towards classifying primitive groups of affine type with base size 2, specifically those with almost quasisimple point stabilisers.


GRAW02 
27th January 2020 16:40 to 17:30 
Thomas Breuer 
Connections between Group related Databases in GAP Various databases of groups and related structures (representations, character tables, tables of marks, presentations, ...) are available in electronic form. Since it is often useful to combine information from different such databases, it is valuable to connect/integrate them tightly. The talk will report about some aspects of this integration in the computer algebra system GAP. 

GRAW02 
28th January 2020 09:10 to 10:00 
Nicolas Thiery 
Musing on implementing semigroup representation theory and software integration
Extending representation theory from finite groups to
finite semigroups brings interesting challenges, combinatorics, and
applications. Almost a decade ago, I proposed an algorithm to compute the
Cartan Matrix of a semigroup algebra  a combinatorial invariant that contains
information on how projective modules are built from simple modules. It boils
down to computing with finite semigroups, characters of groups, and
combinatorics. Despite this relative simplicity, and much to my embarrassment,
a full productiongrade implementation is only finally in reach.
In this talk, I will report on ongoing joint work with my PhD student Balthazar Charles to implement this algorithm and adapt it to modular representations, and use this occasion to illustrate the evolution of our computational landscape toward an ecosystem of interoperable software, thanks to large scale collaborations. 

GRAW02 
28th January 2020 10:10 to 11:00 
Frank Lübeck 
Computing Brauer character tables of groups of Lie type in defining characteristic
I will sketch how to compute Brauer character values of a group of
Lie type in its defining characteristic. The method uses weight
multiplicities of irreducible representation of the underlying algebraic
group, parameterizations of semisimple conjugacy classes of
the finite groups, and ad hoc arguments to relate the resulting table with
the ordinary character table of the finite group.


GRAW02 
28th January 2020 11:30 to 12:20 
Madeleine Whybrow 
An algorithm to construct dihedral axial algebras
Axial algebras are nonassociative algebras generated by semisimple idempotents, called axes, that obey a fixed fusion law. Important examples of axial algebras include the Griess algebra and Jordan algebras. Axial algebras that are generated by two axes are called dihedral and are fundamental in the study of these algebras in general. We present an algorithm to classify and construct dihedral axial algebras. This work represents a significant broadening in our understanding of axial algebras.


GRAW02 
28th January 2020 13:45 to 14:35 
Klaus Lux 
The 5modular Character Table of the Lyons Group
We will talk about the determination of the 5modular character table of the sporadic simple Lyons group Ly.
This table was computed jointly with Alexander Ryba, Queens College, CUNY, New York.
As a starting
point of our computations we will take the 111dimensional representation over the field with 5
elements, conjectured to exist by Meyer, Neutsch and constructed by Meyer,
Neutsch, and Parker. An important ingredient in our computations
will be the interplay between modular character theory and the theory of condensation of
representations, in particular condensations of tensor products and symmetrizations.


GRAW02 
28th January 2020 14:45 to 15:35 
David Stewart 
Between the sheets: rigid nilpotent elements in modular Lie algebras
(Joint with Sasha Premet) Let G be a reductive algebraic group over an algebraically closed field. Lusztig and Spaltenstein provided a method for inducing a nilpotent orbit from a Levi subgroup to the group G. Any orbit not obtained from a proper Levi subgroup is called rigid. These were classified by Kempken (for G classical) and Elashvili (for G exceptional). The latter was doublechecked computationally by De Graaf. It turns out that this classification remains valid in characteristic p. I will explain the proof of this, obtained by extending the BorhoKraft description of the sheets of the Lie algebra to positive characteristic and supported by a few computer calculations.


GRAW02 
28th January 2020 16:05 to 16:55 
Richard Parker 
10 years of meataxe development. Myself, Steve Linton and Jon Thackray have been working for nearly 10 years on a fairly major overhaul of matrix multiplication and Gaussian elimination over finite fields of order (mainly) up to 1,000 or so, aiming to make good use of modern processors  specifically the ubiquitous x8664 from Intel and AMD. With clock speeds approaching a plateau we now need to use multiple cores, utilize the various levels of cache to reduce memory bandwidth demands, use the vector registers and avoid unpredictable branches, but by doing all of these, speed improvements in excess of a factor of 100 are readily obtained over the methods of a couple of decades ago. 

GRAW02 
29th January 2020 09:10 to 10:00 
Adam Thomas 
Computational methods for exceptional groups
This talk will contain no new algorithms or methods for computation in groups. Instead, I will describe my (ab)using of existing algorithms (in Magma) to help study the structure of exceptional algebraic groups, their Lie algebras and the exceptional finite groups of Lie type. Such computational methods have guided me and often formed part of the argument in lots of my work, including joint projects with Tim Burness, Alastair Litterick and David Stewart. These applications have included representation theory, especially calculating cohomology groups, intersection of subgroups and construction of subgroups. The talk will include many openended (and impossibletoanswer) questions about the finite groups of Lie type machinery and related algorithms, which will hopefully be helpful to other users in the audience. 

GRAW02 
29th January 2020 10:10 to 11:00 
Allan Steel 
Constructing Ordinary Representations of Finite Groups via Extension
I will describe practical methods for the construction of ordinary representations of a finite group G via the extension of existing representations defined on a proper subgroup of G.
I will also describe how, using an implementation of this algorithm within Magma, I was able to construct for the first time minimaldegree faithful ordinary representations of most of the large sporadic simple groups, such as the Baby Monster.


GRAW02 
29th January 2020 11:30 to 12:20 
Willem de Graaf 
Real forms of complex embeddings of maximal reductive Lie algebras in semisimple Lie algebras
Since the work of Dynkin the reductive subalgebras of a semisimple complex Lie algebra are divided in two groups: those that are contained in a proper regular subalgebra, and those that are not (these are called Ssubalgebras). I will describe computational methods to obtain real forms of the complex embeddings of reductive Lie algebras in semisimple subalgebras. There is one algorithm for the regular subalgebras and one for the Ssubalgebras. Recently we have used these to obtain the maximal reductive subalgebras of the simple real Lie algebras of ranks up to 8. This is joint work with Heiko Dietrich, Paolo Faccin and Alessio Marrani. 

GRAW02 
30th January 2020 09:10 to 10:00 
Mohamed Barakat 
Chevalley’s Theorem on constructible images made constructive Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasiaffine. In this talk, I will present a Gröbnerbasisbased algorithm that computes the constructible image of a morphism of affine spaces, along with some applications. 

GRAW02 
30th January 2020 10:10 to 11:00 
Alexander Hulpke 
Towards a nonsolvable Quotient Algorithm [This is joint work with Heiko Dietrich from Monash U.] 

GRAW02 
30th January 2020 11:30 to 12:20 
Bettina Eick 
Conjugacy problems in GL(n,Z)
The talk describes a practical algorithm to solve the
conjugacy and the centralizer problems in GL(n,Z) in full
generality; that
is, given two matrices A and B in GL(n,Q) these
algorithms allow to check if
A and B are conjugate in GL(n,Z) and, if so, then to
determine a conjugating
element, and they allow to compute generators for the
centralizer of A in
GL(n,Z). The talk
also discusses possible extensions of this algorithm to
finitely generated abelian or nilpotent subgroups of
GL(n,Z). The latter are
open problems in computational group theory and they have
interesting
applications.


GRAW02 
30th January 2020 13:45 to 14:35 
Ulrich Thiel 
Minimal models of symplectic quotient singularities Namikawa associated to any conic symplectic singularity a hyperplane arrangement which is deeply intertwined with its geometry. For example, Bellamy proved that for a symplectic quotient singularity the cohomology of the complement of this arrangement encodes the number of minimal models of the singularity. For the symplectic singularity associated to a complex reflection group we were able to prove that the Namikawa arrangement coincides with the degenericity locus of the number of torus fixed points of the corresponding CalogeroMoser deformation. This has a series of remarkable consequences, especially it proves a conjecture by Bonnafé and Rouquier. Using representation theory and sophisticated computer algebraic methods, we could compute this arrangement explicitly for several exceptional complex reflection groups. The arrangements seem to be of a new kind, and many more are out there. This is joint work with Gwyn Bellamy (Glasgow) and Travis Schedler (London), and with Cédric Bonnafé (Montpellier). 

GRAW02 
30th January 2020 14:45 to 15:35 
Dane Flannery 
Classifying finite linear groups in prime degree
We describe the complete, computational solution of an enduring problem in linear group theory. Specifically, we describe the classification of the finite irreducible monomial groups of prime degree (over the complex numbers). An exhaustive and selfcontained solution of this problem has been obtained for all such solvable groups, and for all such groups of reasonably small degree (say, at most 23). We note obstacles that prevent a full solution of the problem for nonsolvable groups in arbitrary prime degree. This is joint work with Zolt\'an B\'acskai and Eamonn O'Brien. 

GRAW02 
30th January 2020 16:05 to 16:35 
Eilidh McKemmie 
Invariable generation of finite classical groups
We say a group is invariably generated by a subset if it forms a generating set even if an adversary is allowed to replace any elements with their conjugates. Eberhard, Ford and Green built upon the work of many others and showed that, as $n \rightarrow \infty$, the probability that $S_n$ is invariably generated by a random set of elements is bounded away from zero if there are four random elements, but goes to zero if we pick three random elements. This result gives rise to a nice Monte Carlo algorithm for computing Galois groups of polynomials. We will extend this result for $S_n$ to the finite classical groups using the correspondence between classes of maximal tori of classical groups and conjugacy classes of their Weyl groups.


GRAW02 
30th January 2020 16:35 to 17:05 
Mun See Chang 
Computing normalisers of highly intransitive permutation groups
In general, there is no known polynomialtime algorithm for computing the normaliser $N_{S_n}(H)$ of a given group $H \leq S_n$. In this talk, we will consider the case when $H$ is a subdirect product of permutation isomorphic nonabelian simple groups. In contrast to the case with abelian simple groups, where only practical improvements have been made, here we show that $N_{S_n}(H)$ can be computed in polynomial time.


GRAW02 
31st January 2020 09:10 to 10:00 
Tobias Rossmann 
Growth of class numbers of unipotent groups
This talk is devoted to the symbolic enumeration of conjugacy classes in infinite families of finite pgroups attached to a given unipotent algebraic group. I will report on recent joint work with Christopher Voll which solves this problem for unipotent groups associated with graphs. As a byproduct, we obtain polynomiality results in the spirit of Higman's conjecture on class numbers of unitriangular matrix groups.


GRAW02 
31st January 2020 10:10 to 11:00 
Jay Taylor 
Classifying Isomorphism Classes of Algebraic Groups This talk will concern connected reductive algebraic groups (CRAGs) defined over an algebraically closed field. To each CRAG one can associate a combinatorial invariant known as its root datum. A classic result of Chevalley states that the isomorphism classes of CRAGs are in bijective correspondence with the isomorphism classes of root data. This begs the question, when are two root data isomorphic? In this talk we will describe an algorithmic solution to this problem. Part of this is joint work with Jean Michel. 

GRAW02 
31st January 2020 11:30 to 12:20 
Christopher Jefferson 
Backtrack Search in Permutation Groups
While there are many problems can be solved in polynomial
time, some important fundamental problems can only be solved by backtrack
searches, which are often exponential time. These include many important
permutation group problems including group and coset intersection, stabilizer,
normaliser, and canonical image problems.
This talk will give an overview of backtracking algorithms in permutation groups, explaining both the fundamental ideas, and the most improvements. In particular this will cover Leon's Partition Backtrack algorithm and the more recent Graph Backtracking algorithm. [This talk includes joint work with Rebecca Waldecker, Wilf Wilson and others] 

GRAW02 
31st January 2020 13:45 to 14:35 
Cheryl Praeger 
Classical groups, and generating small classical subgroups
I will report on ongoing work with Alice Niemeyer and Stephen Glasby. In trying to develop for finite classical groups, some ideas Akos Seress had told us about special linear groups, we were faced with the question: "Given two nondegenerate subspaces U and W, of dimensions e and f respectively, in a formed space of dimension at least e+f, how likely is it that U+W is a nondegenerate subspace of dimension e+f?" Something akin to this question, in a similar context is addressed in Section 5 of "Constructive recognition of classical groups in even characteristic" (J. Algebra 391 (2013), 227255, by Heiko Dietrich, C.R.LeedhamGreen, Frank Lubeck, and E. A. O’Brien). We wanted explicit bounds for this probability, and then to apply it to generate small classical subgroups. 

GRAW02 
31st January 2020 14:45 to 15:35 
Martin Liebeck 
Computing with conjugacy classes in classical groups
I will discuss some theory and algorithms for performing the following tasks with unipotent elements of finite classical groups: (1) writing down conjugacy class representatives (2) computing centralizers (3) solving the conjugacy problem, and finding conjugating elements. 

GRA 
4th February 2020 11:00 to 12:00 
Joshua Maglione 
Isomorphism, derivations, and Lie representations
By bringing in tools from multilinear algebra,
we introduce a general method to aid in the computation of isomorphism for
groups. Of particular interest are nilpotent groups where the only classically
known proper nontrivial characteristic subgroup is the derived subgroup. This
family of groups poses the biggest challenge to all modern approaches. Through
structural analysis of the biadditive commutator map, we leverage the
representation theory of Lie algebras to prove efficiency for families of
nilpotent groups. We report on joint work with Peter A. Brooksbank, Uriya
First, and James B. Wilson. 

GRA 
5th February 2020 16:00 to 17:00 
Csaba Schneider 
A new look at permutation groups of simple diagonal type
Permutation groups of simple diagonal
type form one of the classes of
(quasi)primitive permutation groups identified by the O'NanScott
Theorem. They also occur among the maximal
subgroups of alternating and symmetric groups. Until now, they were not
considered as a geometric class in the sense that they were not viewed as stabilizers
of geometric or combinatorial objects. In this talk I will report on some new research, carried out in collaboration with
Cheryl Praeger, Peter Cameron and Rosemary Bailey, whose results show that
these groups can also be viewed as full stabilizers of certain combinatorial
structures. I will also show that a permutation
group of simple diagonal type is the automorphism group of a graph
which is constructed as the edge union of
Hamming graphs. The results hold also for
infinite permutation groups. 

GRA 
11th February 2020 16:00 to 17:00 
Luke Morgan 
Graphs with lots of symmetry  a local perspective
In this talk we will focus on groups acting on graphs
with a good amount of symmetry, such as vertex transitivity. Several
conjectures have connected global and local properties of the graphs in this
class. In particular, “global" can refer to the number of automorphisms of
a graph, and local then refers to certain conditions placed on the local
action, that is, the action induced by a vertexstabiliser on the neighbours of
the vertex it fixes. A conjecture of Weiss from 1978 asserts that under mild
conditions on this local action the number of automorphisms of a connected
vertextransitive graph should be bounded by a function of the valency. This
conjecture is still very much open. I will report on recent progress on an
expanded version of the conjecture which uses tools from group theory developed
for the classification of the finite simple groups.


GRA 
13th February 2020 16:00 to 17:00 
Scott Harper 
3/2Generation
Many interesting and surprising
results have arisen from studying generating sets for groups, especially simple
groups. For example, every finite simple group can be generated by just two
elements. In fact, Guralnick and Kantor, in 2000, proved that in
a finite simple group every nontrivial element is contained in a generating
pair, a property known as 3/2generation. This answers
a 1962 question of Steinberg. In this talk I will report on
recent progress towards classifying the finite 3/2generated groups, and I will
discuss joint work with Casey Donoven in which we found the first nontrivial
examples of infinite 3/2generated groups.


GRA 
18th February 2020 11:00 to 12:00 
Lucas Ruhstorfer 
Jordan decomposition for the AlperinMcKay conjecture
In recent years, many of the famous globallocal
conjectures in the representation theory of finite groups have been reduced to
the verification of certain stronger conditions on the characters of finite
quasisimple groups. It became apparent that checking these conditions requires
a deep understanding of the action of group automorphisms on the characters of
a finite simple group of Lie type. On the other hand, the Morita equivalence by BonnaféDatRouquier has become an indispensable tool to study the representation theory of groups of Lie type. In this talk, we will discuss the interplay of this Morita equivalence with group automorphisms. We will then show how this can be applied in the context of the AlperinMcKay conjecture. 

GRA 
20th February 2020 16:00 to 17:00 
PierreEmmanuel Caprace 
Hyperbolic generalized triangle groups, property (T) and finite simple quotients
It is a longstanding open problem in Geometric
Group Theory to determine whether all Gromov hyperbolic groups are residually finite. Contributions of Olshanskii imply that, in order to answer this question in the negative, it suffices to find a hyperbolic group that does not admit finite simple quotients of arbitrarily large rank. In this talk, I will report on efforts in identifying explicit candidates of such a hyperbolic group, and explain a connection with Kazhdan's property (T). This is partly based on an experimental case study on generalized triangle groups, conducted jointly with Marston Conder, Marek Kaluba and Stefan Witzel. 

GRA 
25th February 2020 11:00 to 12:00 
Valentina Grazian 
Exotic fusion systems
Fusion
systems are structures that encode the properties of conjugation between
psubgroups of a group, for p any prime number. Given a finite group G, it is
always possible to define the saturated fusion system realized by G on one of
its Sylow psubgroups S: this is the category where the objects are the
subgroups of S and the morphisms are the restrictions of conjugation maps
induced by the elements of G. However, not all saturated fusion systems can be
realized in this way. When this is the case, we say that the fusion system is
exotic. The understanding of the behavior of exotic fusion systems (in
particular at odd primes) is still an important open problem. In this talk we
will offer a new approach to the study of exotic fusion systems at odd primes.
We will show that an important role is played by the fusion systems defined on
pgroups of maximal nilpotency class and we will present new results concerning
the classification of such fusion systems.


GRA 
27th February 2020 16:00 to 17:00 
David Benson 
Summands of tensor powers of modules for a finite group
In modular representation theory of finite
groups, one of the big mysteries is the structure of tensor products of modules, with the diagonal group action. In particular, given a module $M$, we can look at the tensor powers of $M$ and ask about the asymptotics of how they decompose. For this purpose, we introduce an new invariant $\gamma(M)$ and investigate some of its properties. Namely, we write $c_n(M)$ for the dimension of the nonprojective part of $M^{\otimes n}$, and $\gamma_G(M)$ for $\frac{1}{r}$", where $r$ is the radius of convergence of the generating function $\sum z^n c_n(M)$. The properties of the invariant $\gamma(M)$ are controlled by a certain infinite dimensional commutative Banach algebra associated to $kG$. This is joint work with Peter Symonds. We end with a number of conjectures and directions for further research. 

GRA 
3rd March 2020 11:00 to 12:00 
Mikko Korhonen 
Unipotent elements in irreducible representations of simple algebraic groups
Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p ≥ 0. In this talk, I will discuss the following question and some related problems.
Let f:G → I(V) be a rational irreducible representation, where I(V) = SL(V), I(V) = Sp(V), or I(V) = SO(V). For each unipotent element u ∈ G, what is the conjugacy class of f(u) in I(V)?
Solutions to this question in specific cases have found many applications, one basic motivation being in the problem of determining the conjugacy classes of unipotent elements contained in maximal subgroups of simple algebraic groups. In characteristic zero, there is a fairly good answer by results of JacobsonMorozovKostant. I will focus on the case of positive characteristic p > 0, where much less is known and few general results are available. When G is simple of exceptional type, computations due to Lawther describe the conjugacy class of f(u) in SL(V) in the case where V is of minimal dimension (adjoint and minimal modules). I will discuss some recent results in the case where G is simple of classical type.


GRA 
5th March 2020 16:00 to 17:00 
Justin Lynd 
Fusion systems with BensonSolomon components
In the 1960s and 70s, involution centralizer
problems gave rise to several new sporadic simple groups. With the benefit of about 30 years of
hindsight, one such problem considered by Solomon also gave rise to an infinite
family of exotic 2fusion systems, namely the BensonSolomon systems. The
BensonSolomon fusion systems are closely related to 7dimensional orthogonal
groups over fields of odd order, and they currently comprise the known simple
exotic systems at the prime 2. In this talk, I'll discuss the solution to the
involution centralizer problem for the BensonSolomon systems in the context of
Aschbacher's program for the classification of simple 2fusion systems of odd
type. The BensonSolomon problem is
intertwined with the solution to Walter's Theorem for fusion systems, one of
the four main steps of the program. This
is joint work with E. Henke. 

GRA 
10th March 2020 11:00 to 12:00 
Niamh Farrell 
Decomposition matrices for quasiisolated blocks of exceptional groups
Decomposition numbers encapsulate the relationship
between ordinary and modular representations of finite groups. Determining
decomposition matrices is a difficult problem, particularly for the finite
groups of Lie type in nondefining characteristic. I will present some recent
developments which allow us to consider tackling this problem for the
quasiisolated blocks of finite groups of Lie type in nondefining
characteristic, and discuss some of the strategies and challenges involved in
this project.


GRA 
12th March 2020 16:00 to 17:00 
Joanna Fawcett 
Homogeneity in graphs
Let X be a class of graphs. A graph G is Xhomogeneous if every graph isomorphism f:H>K between finite induced subgraphs H and K of G with H in X extends to an automorphism of G. For example, if X consists of the graph with one vertex, then Xhomogeneity is vertextransitivity. In this talk, we will discuss various interesting choices for X.


GRAW03 
16th March 2020 10:00 to 11:00 
William Timothy Gowers 
Partial associativity and rough approximate groups
Let X be a finite set and let o be a binary operation on X that is injective in each variable separately and has the property that x o (y o z) = (x o y) o z for a positive proportion of triples (x,y,z) with x,y,z in X. What can we say about this operation? In particular, must there be some underlying group structure that causes the partial associativity? The answer turns out to be yes ? up to a point. I shall explain what that point is and give some indication of the ideas that go into the proof, which is joint work with Jason Long. I shall also report on a natural strengthening that one might hope for. We identified a likely counterexample, which was recently proved to be a counterexample by Ben Green, so in a certain sense our result cannot be improved. (However, there are still some interesting questions one can ask.) 

GRAW03 
16th March 2020 11:30 to 12:30 
Michael Giudici  Simple groups and graph symmetry  
GRAW03 
16th March 2020 14:30 to 15:30 
Emmanuel Breuillard 
Expanders and word maps.
An understanding of word maps on semisimple Lie
groups helps for establishing spectral gap bounds for finite simple groups of
Lie type. In this talk I will discuss how this can also be reversed. Joint work
with P. Varj\'u and O. Becker 

GRAW03 
17th March 2020 10:00 to 11:00 
Tim Burness  The length and depth of a group  
GRAW03 
17th March 2020 11:30 to 12:30 
Aner Shalev 
Subset products and applications
In the past two decades there has been intense interest in products of subsets in finite groups. Two important examples are Gowers' theory of Quasi Random Groups and its applications by Nikolov, Pyber, Babai and others, and the theory of Approximate Subgroups and the Product Theorem of BreuillardGreenTao and PyberSzabo on growth in finite simple groups of Lie type of bounded rank, extending Helfgott's work. These deep theories yield strong results on products of three subsets (covering, growth). What can be said about products of two subsets? I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem, focusing on products of two normal subsets of finite simple groups, and deriving some applications. The proofs involve algebraic geometry, representation theory and combinatorics. 

GRAW03 
17th March 2020 14:30 to 15:30 
Nikolay Nikolov 
Rank gradient in profinite groups
In this talk I will focus on the growth of generators in open subgroups of profinite groups. In the case of abstract groups this topic has received a lot of attention but in the profinite situation there are not many results and they tend to be different from those for abstract groups. For example a prop group of positive rank gradient must have a dense free subgroup. As far as I know the analogous question for profinite groups is still open. 

GRAW03 
17th March 2020 16:00 to 17:00 
Cheryl Praeger 
Kirk Lecture: The mathematics of Shuffling
The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two ``perfect shuffles'' was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a socalled "many handed dealer'' shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k is not 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, including all (k, n) with k > n. We initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles. This is joint work with Carmen Amarra and Luke Morgan. 

GRAW03 
18th March 2020 10:00 to 11:00 
Martin Kassabov  Presentation of simple groups of Ree type  
GRAW03 
18th March 2020 11:30 to 12:30 
Martin Liebeck  Binary groups  
GRA 
24th March 2020 11:00 to 12:00 
Pablo Spiga 
CANCELLED On the asymptotic > enumeration of Cayley digraphs and their friends
In this seminar, we give an overview to the
recent solution to a conjecture of Babai and Godsil on the asymptotic
enumeration of finite Cayley digraphs. We also discuss some related results,
concerning the asymptotic enumeration of other classes of graphs admitting a
rich group of automorphisms. Finally, we briefly talk about some open problems. 

GRA 
26th March 2020 16:00 to 17:00 
Derek Holt 
CANCELLED Polynomial time computation in > matrix groups and applications
This talk will be in two parts. In the first part, we
discuss a recent joint result with Charles LeedhamGreen and Eamonn O'Brien
that there is a version of the CompositionTree program for finite matrix groups
that runs in polynomial time subject to the availability of oracles for
discrete logarithm and integer factorisation. The existence of such an
algorithm was established by Babai, Beals and Seress in 2009 but, unlike our
new version, their algorithm was not intended for or suitable for practical
implementation.
In the second part we discuss possible applications of
CompositionTree and the related functionality that has been implemented in
Magma and GAP.
In particular, in his recent PhD thesis at Manchester
(supervised by Peter Rowley), Alexander McGaw used the Magma implementations to
carry out structural computations in the group E_8(2), with a view to settling
remaining uncertainties about its maximal subgroups.
