Department of Pure Mathematics & Mathematical Statistics 

Michael Aschbacher
Groups of component type
A Goldschmidt group is a nonabelian finite simple group that is either of Lie type of
even characteristic and Lie rank 1, or has abelian Sylow 2subgroups.
Let G be a finite group, let {\cal J} be the set of involutions j of G such that m_{2}(C_{G}(j)) =
m_{2}(G), and write {\cal L} for the set of 2components of C_{G}(j) for j ∈ {\cal J}.
Define G to be Jlocally Goldschmidt if for each K ∈ L, K=O(K) is a Goldschmidt
group.
There is a program to classify the members of a large subclass of the class of simple
groups of component type, which would reduce the classification of the finite simple
groups to a class of groups that are Jlocally Goldschmidt. Most of the program is to
be carried out in the category of saturated 2fusion systems, to avoid a proof of the Bconjecture. I'll discuss results that show simple Jlocally Goldschmidt groups resemble
the GLS groups of even type.
1
Inna Capdeboscq
Some subgroups of topological KacMoody groups
In this talk we discuss a recent joint work with B. Remy (Lyon) in which we initiate a study
of some subgroups of topological KacMoody groups.
In particular, we show topological finite generation for the prop Sylow subgroups in many topological KacMoody groups
and discuss the implications of this result on the subgroup structure of the ambient KacMoody group.
George Glauberman
A partial analogue of Borel's
Fixed Point Theorem for finite pgroups
Borel's Fixed Point Theorem states that a solvable connected algebraic group G on a nonempty complete variety V must have a fixed point. Thus, if V consists of subgroups of G, and G acts on V by conjugation, then some subgroup in V isnormal in G.
Although G is infinite or trivial here, we can use the method of proof to obtain applications to finite pgroups, such as extensions of Thompson's Replacement Theorem. We plan to discuss some applications and some open problems. No previous knowledge of algebraic groups is needed.
Radha Kessar
Finiteness results for Hochschild cohomology of pblocks of finite groups
A dominant theme of the modular representation theory of finite groups is the relationship between the structure of a pblock of a finite group and the local structure of the block. The Hochschild cohomology of a finite dimensional algebra is an invariant of the underlying module category. I will present some results which show that Hochschild cohomology of pblocks is strongly controlled by their defects. This is joint work with Markus Linckelmann.
Geoff Robinson
Virtual projectives of norm 2
A general problem of Brauer was to determine the extent to which the values of irreducible characters
in pblocks of positive defect are determined by their values on psingular elements. One oobvious obstacle to such control is virtual projective characters of weight 2. In this talk, we show that the only virtual projectives of weight 2 which can't be described in terms of similar virtual characters of smaller groups are virtual characters of quasisimple groups.
Peter Sin
Integral invariants of skew lines in projective space over a finite field
Two distinct lines in projective 3space either
intersect or are skew. Over a finite field one can consider the
incidence matrix for skewness (with respect to some fixed but arbitrary ordering of
the lines). The integral invariants (invariant factors or elementary
divisors) of such a matrix are independent of the ordering.
The incidence matrices can also be interpreted as the nonadjacency
matrices for singular points on a quadric in 5dimensional
projective space, via the Klein correspondence.
We present the computation of these invariants, in joint work with
Andries Brouwer and Josh Ducey.
Ron Solomon
The house that John built
I will describe some of my favorite gems from
John's work, discuss their significance to the simple group
classification, and present some recent variations and extensions on
these themes.
Stephen Smith
Failure of Thompson Factorization and its descendants
The talk will be a historical sampling of some of the ideas arising from the Thompson subgroup, including e.g. failure of factorization, pushing up, weak closure methods, and Oliver's conjecture.
Gunter Malle
Galois realization of E_{8}(p), p ≥7, over the rationals
We show that the finite simple groups E_{8}(p), p≥5,
occur as Galois groups of regular extensions of Q(t). The proof
involves DeligneLusztig theory to evaluate a certain structure
constant, as well as detailed knowledge on subgroups containing regular
unipotent elements. As a byproduct, we note a remarkable symmetry
between the character table of a finite reductive group and that
of its dual group.
This is joint work with R. Guralnick
JeanPierre Serre
Unitary groups and Galois extensions in characteristic 2
Pham Tiep
Adequate subgroups
The notion of adequate subgroups was introduced by Thorne.
It is a weakening of the notion of big subgroups used by Wiles and Taylor
in proving automorphy lifting theorems for certain Galois representations.
Using this idea, Thorne was able to prove some new lifting theorems.
It was shown recently by Guralnick, Herzig, Taylor, and Thorne that if the degree is small compared to the characteristic then all absolutely irreducible representations are adequate. We will discuss extensions of
this result obtained recently in joint work with R. M. Guralnick and
F. Herzig. In particular, we show that almost all absolutely irreducible
representations in characteristic p of degree less than p are adequate.
We will also address a question of Serre about indecomposable modules in
characteristic p of dimension less than 2p2.
Richard Weiss
Involutions acting on BruhatTits buildings
BruhatTits buildings are, roughly speaking, those affine buildings that are classified by spherical buildings whose field of definition F is complete with respect to a discrete valuation; their residues are spherical buildings defined over the residue field $\bar F$.
An involution acting on a BruhatTits building is unramified if it acts nontrivially on the field of definition of its residues. In this talk we will describe recent results about the fixed points of an unramified involution acting on a BruhatTits building. This is joint work with Bernhard
Mühlherr and Holger Petersson.
Last updated: 6 September 2013