| Iain Aitchison | Mr Spock as a toral tangram |
| We show by elementary means that every non-compact hyperbolic 3-manifold of finite volume admits a canonical CAT(0) spine and CAT(0) structure preserving cusp shapes, with a canonical deformation between the two geometric structures. | |
| Henning Andersen | $p$-filtrations for modular representations |
| By a $p$-filtration of a module $M$ for a reductive algebraic group $G$ over a field of prime characteristic $p$ we understand a sequence of $G$-submodules of $M$ whose quotients are tensor products of simple modules with the twists of Weyl modules. We shall prove some results about such modules with emphasis on a conjecture by Donkin. This conjecture relates the existence of $p$-filtrations of $M$ to properties of the tensor product of $M$ with the Steinberg module. | |
| Jon Carlson | Categorical equivalences and orthogonal subcategories over group algebras |
| Let $G$ be a finite group and let $k$ be a field of characteristic $p > 0$. We wish to consider the stable category ${\bf stmod}kG$ of finitely generated $kG$-modules modulo projectives. Let ${\cal K}$ denote the thick subcategory generated by the trivial $kG$-module $k$. Recent work with Raphael Rouquier has characterized all self stable equivalence of Morita type of ${\cal K}$. In the case that $G$ is a $p$-group this characterizes the self stable equivalences of ${\bf stmod}kG$. In other cases it gives the self stable equivalences of the principal block of $kG$. Old work of mine with Benson, Robinson determines when ${\cal K}$ is all of the stable category of the principal block. In a related paper Benson has proven a condition on varieties for the existence of thick subcategories of the principal block that are orthogonal to the category ${\cal K}$. Attempts to generalize the condition to other blocks have raised some very tricky questions. | |
| Jie Du | Finite dimensional algebras and cellular systems |
| We introduce the notion of a cellular system in order to deal with quasi-hereditary algebras. We obtain that a necessary and sufficient condition for an algebra to be quasi-hereditary is the existence of a full divisible cellular system. As a further application, we prove that the existence of a full local cellular system is a sufficient condition for a standardly stratified algebra. | |
| Swarup Gadde | Splittings of groups and intersection numbers |
| Intersection numbers are defined for almost invariant sets in groups. These correspond to immersed submanifolds. We show that there are splittings if the self intersection number is zero and if the intersection number of two splittings is zero, then both can be simultaneously realized in a graph of groups decomposition. Relationship with other results is also discussed. | |
| Jonathan Hillman | Applications of $L^2$-Betti numbers to low-dimensional topology |
| Atiyah defined $L^2$-Betti numbers for infinite coverings of manifolds and
cell complexes by means of the von Neumann dimension associated to the weak-*
completion of the group algebra (over $\Bbb{C}$) of the covering group.
These are in general non-negative real numbers, rather than integers, but their
alternating sum is the Euler characteristic of the base, and they satisfy
Poincar\'e duality. In addition, they are multiplicative in finite covers.
We shall outline the definition of these invariants and show how they may be applied to questions of combinatorial group theory and low dimensional topology. Among these are: (1) The Whitehead problem on subcomplexes of aspherical 2-complexes; (2) the Tits alternative for 3-manifold groups; (3) a criterion for a closed 4-manifold to be homotopy equivalent to a mapping torus; (4) a generalization and converse of Gottlieb's theorem on the Euler characteristic of aspherical finite complexes. These applications are essentially counting arguments, and rely on vanishing criteria for the first $L^2$-Betti number of a group. A more subtle application, using ideas of Gromov, gives a characterization of complex surfaces which fibre holomorphically over complex curves. (We shall not discuss all of these applications). The theory of $L^2$-(co)homology has been substantially recast and algebrized in recent years, by M.S.Farber and W.L\"uck; our main references for this technique are the papers of L\"uck (e.g., in Math. Ann. 309 (1997), 247-285, and a series of papers in Crelle). |
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| Bob Howlett | Automorphisms of Coxeter groups of rank 3 |
| Apart from the symmetric group on 6 letters, I don't know of any Coxeter group that has an interesting outer automorphism. In this talk I shall outline a proof of the fact that if W is an infinite rank 3 Coxeter group whose Coxeter graph contains no infinite bonds then the automorphism group of W is generated by graph automorphisms and inner automorphisms. | |
| Gus Lehrer | Action of a reflection group on the cohomology of its configuration space. |
| I shall describe 3 ways of computing the action of a unitary reflection group on the complement of its reflecting hyperplanes. Applications of all methods will be discussed and a new result concerning the connection between the group structure and trace formulae will be given. . | |
| Chi Kin Mak | Quasi-parabolic subgroups of generalised symmetric groups |
| Elements of symmetric group $S_r$ can be characterized by a sequence of distinct integers from $1$ to $r$ inclusive. Multiply each term of sequence by an $m$-th roots of unity, we obtain a set of $m^r r!$ elements. With suitably defined operation on this set we get a group that is isomorphic to a complex reflection group which is a wreath product of a cyclic group $C_m$ with the symmetric group $S_r$. These groups are called generalised symmetric groups. The theory of Young tableaux and distinguished coset representatives for Young (or parabolic) subgroups of symmetric groups is very important in the study of Hecke algebras and their representations. Du and Scott in their paper "The $q$-Schur$^2$ Algebra" defined quasi- parabolic Subgroups of Weyl group of type $B$. In this talk, I shall develop a similar theory for quasi-parabolic subgroups of generalised symmetric groups including the one-one correspondence between the set of double cosets of two given quasi-parabolic subgroups and a set of sequences of matrices with integer entries. | |
| Amnon Neeman | Automorphisms of the Yang--Baxter equations, for the chiral Potts model |
| In 1989, Au-Yang and Perk found a subgroup of order 4N^3 of the automorphism group of the chiral Potts curve. It has been an open question since, whether it is the full automorphism group. We will begin with background from mathematical physics to explain the relevance of the problem, then discuss the solution. | |
| Ruibin Zhang | Quantum superalgebras and Vassiliev invariants |
| Each finite dimensional irreducible representation of any Drinfeld-Jimbo quantum superalgebra gives rise to a link invariant, which can formally be regarded as a power series in the deformation parameter of the quantum superalgebra. Each coefficient of the power series is a Vassiliev invariant of links. We will develop a direct construction of such Vassiliev invariants by studying deformations of enveloping superalgebras more general than the Drinfeld-Jimbo quantum superalgebras. In particular, the Vassiliev invariants arising from the quantum $osp(1|2n)$ superalgebras will be analysed in detail. | |