Everyone is welcome.

"Elliptic curves and Hilbert's tenth problem"
Graham Everest (UEA)
October 1st 2007

In 1970 Matiyasevic resolved Hilbert's Tenth Problem for the ring of integers. Currently the problem is unresolved for the field of rational numbers. I will report on recent work of Poonen which took a giant leap in that direction by resolving HTP for some large subrings of the rationals. Another interesting feature of his work is his use of elliptic curves. I will report too of my own modest efforts to give qualitative results of the same kind.

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"Commuting elements in conjugacy classes: an application of Hall's Marriage Theorem"
Mark Wildon (Swansea)
October 8th 2007

This talk is on joint work with John Britnell (Newcastle). If G is a finite group and H is a subgroup of G of index 2 then there is a well-known relationship between the conjugacy classes of G and H. In my talk I will present two related generalisations. The first has a combinatorial flavour, and includes an interesting application of Hall's Marriage Theorem to group theory. The second has connections with character theory.

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"Dimensions of compact invariant sets of some expanding maps"
Yuki Yamama (UEA)
October 15th 2007

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by considering a general Sierpinski carpet represented by a shift of finite type. Applying results of Ledrappier-Young and Shin, we study the Hausdorff dimension of such a general Sierpinski carpet for the case when there is a saturated compensation function. We givesome conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

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"Tilting modules and some 3 part decomposition numbers"
Alison Parker (Leeds)
October 22th 2007

This talk will look at what is known about tilting modules for SL_3 over an algebraically closed field of prime characteristic. I will first explain what a tilting module is and how finding the characters of the tilting modules then tells us what the 3 part decomposition numbers for the symmetric group are. I will then look at specific calculations of tilting modules and what is thought to be true in general (and why this is such a hard problem!)

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"On the Howe correspondence"
Alberto Minguez (UEA)
October 29th 2007

The aim of this talk is to introduce the audience to the theory of local Howe correspondence. For the dual pair of type (Gl(n), Gl(m)) we will show a new proof which allows us to describe the correspondence in terms of Langlands parameters. At the end, we will discus about the possibility of having such a correspondence for $l$-modular representations.

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"Relative cohomology theories for group algebras"
Matt Grime (Bristol)
November 5th 2007

There are many triangulated categories that arise in the study of group cohomology: the derived, stable or homotopy categories, for example. In this talk I shall describe the relative cohomological versions and the relationship with ordinary cohomology. I will explain what we know (and what we would like to know) about these categories, and how the representation type of certain subgroups makes a fundamental difference.

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"On a Conjecture of Foulkes"
Johannes Siemons (UEA)
November 12th 2007

For the integers a and b let P(a^b) be all partitions of the set N= {1,..., ab} into parts of size a. Further, let CP (a^b) be the corresponding permutation module for the symmetric group on N. A conjecture of Foulkes says that CP(a^b) is isomorphic to a submodule of CP(b^a) for all a not larger than b. The conjecture goes back to the 1950's but has remained open. Nevertheless, for small values of b there has been some progress. I will discuss some proofs and further conjectures. There is a close correspondence between the representations of the symmetric groups and those of the general linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore has implications for GL- representations, a fact first observed by Roger Howe at Yale in the 1980's. I hope to have the time to mention some of these interesting connections to classical invariant theory.

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"Almost counterexamples"
Mirna Džamonja (UEA)
November 19th 2007

Say that a family F of subsets of a set A is 1/2-dense if every finite F\subseteq A has a subset F_0 in F satisfying |F_0| >= 1/2 |F|. The notorious problem DU on Fremlin's list is if every 1/2-dense family on \omega_1 which is closed under subsets admits an infinite set whose all finite subsets are in F. Under CH the answer is known to be negative. In the hope of getting closer to the solution of DU we try to understand the relevance of the two requirements on the family. We show that there is a 1/2-dense family F of subsets of the continuum c with the property that every infinite subset of c has arbitrarily large finite subsets in F but there is no infinite set whose all finite subsets are in F. Michalewski found an easy example example which closed under subsets but where 1/2-density is weakened. By a result of Fremlin, modulo a measurable cardinal it is consistent that for every 1/2-dense closed under subsets family on c there is an infinite set whose all finite subsets are in the family, therefore the above examples are closest that one can get to a counterexample in ZFC.

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"Some results on harmonic analysis on p-adic symmetric spaces"
Patrick Delorme (Marseille)
November 23th 2007

I will describe basic facts on p-adic symmetric spaces. In particular σ-split tori, σ-split parabolic subgroups related to results of A.Helminck with S.P.Wang and G.Helminck. I will describe the Cartan decomposition for p-adic symmetric spaces (Benoist-Oh, Delorme-Sécherre). I will describe results of Lagier, Kato-Takano and myself on H-fixed linear forms on smooth-modules and Jacquet modules. This leads to the definition of the constant term of smooth functions on a p-adic symmetric space and of cuspidal functions. The construction of Eisenstein integrals will be described (Blanc-Delorme).

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"Hochschild cohomology and support varieties"
Nicole Snashall (Leicester)
November 26th 2007

Hochschild cohomology is an important invariant in the representation theory of algebras. In joint work with Solberg, we used the Hochschild cohomology ring to construct a support variety for any module over a finite-dimensional algebra. This was motivated by the work of Carlson, who introduced the support variety of a module over a group algebra; this is now a powerful invariant in the modular representation theory of finite groups. I will describe the construction and properties of support varieties constructed using Hochschild cohomology, with specific reference to certain finiteness conditions, under which we have analogues of many of the properties of the group ring situation. This work motivated our conjecture that the Hochschild cohomology ring modulo nilpotence of a finite-dimensional algebra is always finitely generated as an algebra. The second part of my talk will focus on the current position of this conjecture.

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"rincipal series types for p-adic general linear groups"
Peter Campbell (Bristol)
December 3rd 2007

This is a report on joint work with Monica Nevins (University of Ottawa). Types are an important tool for understanding the representation theory of p-adic groups and they have been extensively studied in the case of supercuspidal representations. We consider the principal series representations of GL(n) and, in particular, reexamine earlier work of Howe.

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"The plan geometry of Norwich cathedral"
Nigel Hiscock (Oxford Brooks)
December 10th 2007

The evidence for the use of geometry in the planning of medieval cathedrals and in their constructional design is conclusive. Accordingly, this paper will consider what evidence there is for the presence of geometry in the planning of Norwich Cathedral.

Geometry was fundamental to an architect's training and to the working practices of the masons' lodge. This resulted in plans sometimes being based on grids of squares, known as designing ad quadratum, also in architectural details being produced from squares in the technique known as quadrature.

The modernist critique of medieval architecture argued that these techniques were the purely practical procedures of masons which signified nothing beyond themselves. On the other hand, geometry, as one of the disciplines of the liberal arts, was regarded as a tool of reason for observing the universe. God was repeatedly portrayed as the Divine Architect, compass in hand, creating order from chaos, and geometric figures commonly carried symbolic meaning.

Norwich Cathedral was chosen for the investigation because its Norman layout stands virtually intact and a digitised laser survey has recently been completed.

The prominence of the square in medieval architectural practice has led numerous studies to propose the ratio of the side to the diagonal of a square, as $1: \sqrt(2)$, as the basis of much medieval plan design. Although the system carries with it certain problems, including its application to Norwich Cathedral, such is the weight of circumstantial evidence, it represents a case that needs to be answered.

Another system that has been applied to Norwich involves the figures of Platonic geometry. As the plane figures of the regular polyhedra, with which Plato associated the elements and the universe, much importance was given to them in medieval design. When the geometry of these figures was applied to the plan of Norwich Cathedral, and tested mathematically against the survey, a correlation was found between the geometric system and the building to within 1.5%, a degree of accuracy which would have appeared exact to anyone restricted to medieval draughting and surveying techniques.

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