Department of Pure Mathematics & Mathematical Statistics

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# Abstracts



## 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 2-subgroups.

Let G be a finite group, let {\cal J} be the set of involutions j of G such that m2(CG(j)) = m2(G), and write {\cal L} for the set of 2-components of CG(j) for j ∈ {\cal J}.

Define G to be J-locally 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 J-locally Goldschmidt. Most of the program is to be carried out in the category of saturated 2-fusion systems, to avoid a proof of the B-conjecture. I'll discuss results that show simple J-locally Goldschmidt groups resemble the GLS groups of even type. 1

## Inna Capdeboscq

### Some subgroups of topological Kac-Moody 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 Kac-Moody groups. In particular, we show topological finite generation for the pro-p Sylow subgroups in many topological Kac-Moody groups and discuss the implications of this result on the subgroup structure of the ambient Kac-Moody group.

## George Glauberman

### A partial analogue of Borel's Fixed Point Theorem for finite p-groups

Borel's Fixed Point Theorem states that a solvable connected algebraic group G on a non-empty 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 p-groups, 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.

### Finiteness results for Hochschild cohomology of p-blocks of finite groups

A dominant theme of the modular representation theory of finite groups is the relationship between the structure of a p-block 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 p-blocks 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 p-blocks of positive defect are determined by their values on p-singular 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 quasi-simple groups.

## Peter Sin

### Integral invariants of skew lines in projective space over a finite field

Two distinct lines in projective 3-space 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 non-adjacency matrices for singular points on a quadric in 5-dimensional 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 E8(p), p ≥7, over the rationals

We show that the finite simple groups E8(p), p≥5, occur as Galois groups of regular extensions of Q(t). The proof involves Deligne-Lusztig 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

## Pham Tiep

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 2p-2.

## Richard Weiss

### Involutions acting on Bruhat-Tits buildings

Bruhat-Tits 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 Bruhat-Tits building is unramified if it acts non-trivially 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 Bruhat-Tits building. This is joint work with Bernhard Mühlherr and Holger Petersson.

Last updated: 6 September 2013