skip to content
 

Seminars (GRA)

Videos and presentation materials from other INI events are also available.

Search seminar archive

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 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.
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 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.
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 systems-II
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 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.
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 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.
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 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.
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 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.
GRAW01 9th January 2020
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.
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 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.

GRAW01 9th January 2020
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.
GRAW01 9th January 2020
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/.

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 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.
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 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.
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 non-permutation 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 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.
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 1-spaces.

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 simple-minded 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 AT-groups. 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 AT-group. 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 cross-characteristic 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 Leedham-Green) 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 production-grade 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 non-associative 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 5-modular Character Table of the Lyons Group
We will talk about the determination of the 5-modular 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 111-dimensional 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 double-checked 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 Borho-Kraft 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 x86-64 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.
This talk will explain some of the changes in technique that are needed to achieve this - both algorithmic and technological - that seem quite radical at the moment, but which I expect to become more mainstream in future.

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 open-ended (and impossible-to-answer) 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 minimal-degree 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 S-subalgebras). 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 S-subalgebras.
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 quasi-affine. In this talk, I will present a Gröbner-basis-based 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.]
Quotient algorithms have been a principal tool for the computational
investigation of finitely presented groups as well as for constructing groups.
We describe a method for a nonsolvable quotient algorithm, that extends a
known finite quotient with a module.
Generalizing ideas of the $p$-quotient algorithm, and building on results of
Gaschuetz on the representation module, we construct, for a finite group
$H$, an irreducible module $V$ in characteristic $p$, and a given number of
generators $e$ a covering group of $H$, such that every $e$-generator
extension of $H$ with $V$ must be a quotient thereof. This construction uses
a mix of cohomology (building on rewriting systems) and wreath product methods.
Evaluating relators of a finitely presented group in such a cover of a known
quotient then yields a maximal quotient associated to the cover.
I will describe theory and implementation of such an approach and discuss
the scope of the method.

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 Calogero-Moser 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 self-contained 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 non-solvable 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 polynomial-time 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 non-abelian 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 p-groups 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 by-product, 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 on-going 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 non-degenerate 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 non-degenerate 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), 227-255, by Heiko Dietrich, C.R.Leedham-Green, 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'Nan-Scott 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 vertex-stabiliser 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 vertex-transitive 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 3/2-Generation
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 3/2-generation. This answers a 1962 question of Steinberg. In this talk I will report on recent progress towards classifying the finite 3/2-generated groups, and I will discuss joint work with Casey Donoven in which we found the first nontrivial examples of infinite 3/2-generated groups.




GRA 18th February 2020
11:00 to 12:00
Lucas Ruhstorfer Jordan decomposition for the Alperin-McKay conjecture
In recent years, many of the famous global-local conjectures in the representation theory of finite groups have been reduced to the verification of certain stronger conditions on the characters of finite quasi-simple 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 Bonnafé-Dat-Rouquier 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 Alperin-McKay conjecture.





GRA 20th February 2020
16:00 to 17:00
Pierre-Emmanuel Caprace Hyperbolic generalized triangle groups, property (T) and finite simple quotients
It is a long-standing 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 p-subgroups 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 p-subgroups 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 p-groups 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 non-projective 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 Jacobson-Morozov-Kostant. 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 Benson-Solomon 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 2-fusion systems, namely the Benson-Solomon systems. The Benson-Solomon fusion systems are closely related to 7-dimensional 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 Benson-Solomon systems in the context of Aschbacher's program for the classification of simple 2-fusion systems of odd type.  The Benson-Solomon 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 quasi-isolated 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 non-defining characteristic. I will present some recent developments which allow us to consider tackling this problem for the quasi-isolated blocks of finite groups of Lie type in non-defining 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 X-homogeneous 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 X-homogeneity is vertex-transitivity. 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 Breuillard-Green-Tao and Pyber-Szabo 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 pro-p 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 so-called "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 Leedham-Green 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.




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