This talk is designed as an introduction to the cohomology of groups, and particularly of finite groups. I shall describe the algebraic, topological and number theoretic views of the subject and the connections between them. If I have time, I shall say something about the derived commutative algebra of cohomology rings.

This is a talk about the interplay between finite group theory and homotopy theory. We review the concept of fusion system and linking system and how they relate to classifying spaces and self maps. As an application, we show some equivalences between fusion systems of finite groups of Lie type and the relation between self maps of p-completed classifying spaces and fusion preserving automorphisms of Sylow p-subgroups.

We will report on recent progress towards a characterisation of rank one groups over local fields among all locally compact groups, in the spirit of Hilbert's fifth problem. We will explain how trees whose set of ends is a Moufang set appear in that context, and present a classification of those Moufang sets in characteristic zero.

Localities are "objective partial groups" which are at the same time "transporter systems" in the sense of homotopical group theory developed by Broto, Levi, Oliver, and their many co-workers. They have provided the basis for a constructive proof of the result that to any saturated fusion system over a finite p-group there corresponds a unique centric linking system. I'll discuss some more recent developments, with a view towards obtaining a suitable notion of "homomorphism" of linking systems, and of "normal subsystem" of a linking system.

The Hopf-square map on a graded bialgebra often has a natural combinatorial/probabilistic interpretation. On the free associative algebra, it leads to the Gilbert-Shannon-Reeds model of riffle shuffling. Applied to symmetric functions, it leads to a rock-breaking model of Kolmogorov. Using this, the combinatorics of the free algebra and an idea of Drinfeld leads to a description of all the eigen-values and eigen-vectors. These are often natural, useful, and interpretable. This is joint work with Amy Pang and Arun Ram.

In geometry special properties of collineations have been studied
since the 19^{th} century. In particular involutions were used
as subsets of groups to describe geometries. In my work I study
conjugacy classes D of involutions generating a finite group G: Let ω
be a set of natural numbers; then D is a class of ω-transpositions of
G if different elements of D have products of order in ω. I plan to
talk on recent results classifying {3,4}-transposition-groups.

Let A be an elementary abelian r-group of rank at least 3. We will define the notion of an A-signalizer amalgam. A proof of the classification of the simple A-signalizer amalgams will be described. It is a corollary that every A-signalizer amalgam arises from the action of A on an r'-group. This gives a new proof of McBride's Nonsolvable signalizer Functor Theorem. Consequently, this work will be a contribution to the Gorenstein-Lyons-Solomon proof of the Classification of the Finite Simple Groups.

The proof of the classification of A-signalizer amalgams has many features in common with that of the CFSG, however it is very much simpler.

30 years ago McKay noticed

Saturated fusion systems model the p-local structure of finite groups. They were introduced by Puig in the early 1990's under the name of full Frobenius categories to unify phenomena occurring both in block theory and in finite groups. Later, Broto, Levi and Oliver introduced the now standard terminology and extended Puig's theory for the purposes of homotopy theory. Recent work of Aschbacher and others has enriched the theory of saturated fusion systems by concepts and theorems which have analogues of fundamental importance in finite group theory. In this talk we will focus on classification theorems for fusion systems. In particular, I will outline the current progress of a large project proposed by Michael Aschbacher for the classification of all simple 2-fusion systems leading to a new proof of the classification of finite simple groups. Moreover, I will report about my work towards a classification of minimal non-solvable fusion systems.

In 1968, John Thompson proved the following solvability criterion for
finite groups: *A finite group is solvable if and only if every pair of its
elements generates a solvable subgroup.* Several other *Thompson-like*
results have appeared recently in the literature, where the solvability of
all 2-generator subgroups is replaced by a weaker condition, restricting
the required set of 2-generator subgroups. We proved the following new
criterion for solvability: *A finite group G is solvable if and only if for
all elements x and y of G of prime power order, there exists a conjugate
z of y such that x and z generate a solvable subgroup of G.* This is a joint
paper with Silvio Dolfi, Bob Guralnick and Cheryl Praeger.

(joint work with I. Biringer, K Bou-Rabee, and F. Matucci)

A group is called residually finite if the intersection of all finite index subgroups is trivial. This can be quantified in two ways

- what is the minimal index of a normal subgroup, which does not contain a given element; and
- what is the index of the intersection of all normal subgroups of small index.

Many questions in the modular representation theory of finite groups concern the connection between global representation-theoretic invariants (e.g. character degrees, cartan numbers, decomposition numbers...) and local structure (e.g. defect groups, fusion systems...). I will speak about the recent resolution of the forward direction of the following problem, posed by Richard Brauer, and known as the height zero conjecture:

We give a description of definable subsets in a free non-abelian group F that follows from our work on the Tarski problems. As a corollary we show that proper non-abelian subgroups of F are not definable (solution of Malcev's problem) and prove Bestvina and Feighn's result that definable subsets in a free group are either negligible or co-negligible. This is joint work with A. Myasnikov.

I shall define what I mean by a platonic polygonal complex, and state a recent classification result for platonic polygonal complexes with certain vertex links. I shall also explain the connection with incidence geometries and with some earlier work. The original parts of the talk concern joint work with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler.

Group theoretic questions can often be reduced to the study of finite simple groups. We show how Lusztig's character theory, which ultimately rests on the Weil conjectures, can be used to obtain strong results on the structure constants of simple groups of Lie type. These in turn lead to the solution of a conjecture of Peter Neumann on fixed spaces in representations, and of a conjecture of Bauer, Catanese and Grunewald on the existence of Beauville surfaces. This is joint work with Robert Guralnick.

Around 1989 Ronan and Tits introduced twin buildings which are motivated by the theory of Kac-Moody groups. Twin buildings are pairs of buildings endowed with a codistance function. Twin trees provide a most interesting special case of this theory. In the 1990's Serre pointed out that there is a natural generalisation of twin trees to multiple trees based on the theory of line bundles over rational function fields. The examples arising in this context are at the origin of Ronan's work on the Moufang property of multiple trees.

In joint work with M. Grüninger we proved that multiple Moufang trees involving at least three factors are of algebraic origin (i.e. related to S-arithmetic groups where |S| is the number of factors). By work of Rémy and Ronan one knows that one cannot expect such a result if there are only two factors. In my talk I will present this result and discuss some related results for twin trees and in the higher rank case.

The notion of weak commensurability was introduced in the ongoing joint work with Gopal Prasad on length-commensurable and isospectral locally symmetric spaces. We have been able to determine when two arithmetic subgroups are weakly commensurable. This leads to various geometric results, some of which are related to the famous question "Can one hear the shape of a drum?"

Non-commutative analogues of Waring problem in number theory were studied extensively in recent years, where the goal is to express group elements as short products of special elements; these may be powers, commutators, values of a general word w, or elements of a given conjugacy class, or of certain subgroups or subsets. Such problems arise naturally in profinite groups, finite groups, and finite simple groups in particular.

I will describe background, recent results (with various coauthors), and relations to representations, geometry, and growth. I will conclude with some applications and conjectures. The talk will be accessible for a wide audience.

(or How I Learned to Inhale Without Smoking)

The talk is mainly about a group of young students from Frankfurt. How they got influenced by Fischer's ideas and came to work with him in Bielefeld. and what some of them made out of it later.

The talk is meant for a general audience of mathematicians, and maybe also for people who got used to the mysteries of life, like pieces of incomprehensible or unexplained terminology.

Anabelian geometry describes arithmetic and geometry of algebraic curves over number fields in terms of their etale fundamental groups. The section conjecture of Grothendieck in particular, suggests to describe rational points in terms of conjugacy classes of splittings of the fundamental extension. After introducing this anabelian circle of ideas we will show how an abelian approximation to the section conjecture, replacing rational points by 0-cycles of degree 1, relates to an old question of Cassels and Bashmakov. The question is about divisibility properties of elements of the Tate-Shafarevich group inside the Weil-Chatelet group of an abelian variety. The reported results are joint work with Mirela Ciperiani.

In this talk we will report on some relations between ^{2}E_{6}(2)
and F_{2} with supporting actor F_{4}(2). This starts with the
centralizer of a {3,4}-transposition in F_{2} and the identification by
this centralizer. Then it continues with the geometry given by these
transpositions, which generalizes to c-extended buildings of F_{4}-type
and their classification. Finally it ends with the study of groups
with a large subgroup initiated by U. Meierfrankenfeld, B. Stellmacher
and G. Stroth, in which context ^{2}E_{6}(2) and F_{2}
show up as honorary groups of Lie type over 𝔽_{3}.
[PDF]

I shall lecture on SL_{2} and Dirichlet series, and discuss a possible
connection with the Fischer-Griess group.

Parapolar spaces are point line geometries with main axiom: if one takes two points of distance two, than there exists either a unique common neighbour or the set of common neighbours is a polar space. In the talk parapolar spaces with 3 points on each line will be considered, for which the exceptional Lie-type groups over GF(2) and certain large sporadic groups (for example monster and baby monster) with point set the set of 2-central involutions provide examples.

The group of outer automorphisms of a free group acts on a space of
finite graphs known as Outer space, and a classical theorem of Hurwicz
implies that the homology of the quotient by this action is an
invariant of the group. A more recent theorem of Kontsevich relates
the homology of this quotient to the Lie algebra cohomology of a
certain infinite-dimensional symplectic Lie algebra. Using this
connection, S. Morita discovered a series of new homology classes for
Out(F_{n}). In joint work with J. Conant and M. Kassabov, we
reinterpret Morita's classes in terms of hairy graphs, and show how
this graphical picture then leads to the construction of large numbers
of new classes, including some based on classical modular forms for
SL(2,ℤ).

Like the three Fischer groups Fi_{22}, Fi_{23} and
Fi_{24}, the Moufang quadrangles "of type E_{6},
E_{7} and E_{8}" arise inductively from one special
case in the solution to a larger classification problem. We will
describe how this happens and try to indicate why this induction, like
Fischer's, stops after just three steps. We will also discuss various
properties of these remarkable geometries and their automorphism
groups. For example, if G is the group of linear automorphisms of one
of these quadrangles, then F*(G) is simple (but not finite); these
simple groups are the exceptional groups of our title.

While Lie theory has been very successful at unifying the theory of finite simple groups, its size and complexity present a barrier to beginning students. Moreover, it begins from the adjoint representation, which in general is not the smallest representation, and is not a representation of the generic cover.

Alternative approaches can give a more direct construction, overcoming all of these obstacles, though at the cost of uniformity. This is well-known in the case of classical groups: we present some new approaches to exceptional groups.

last modified on 6 March 2012