All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!

**Seminars for the autumn term 2010.**

- 28th September:
**Jack Button**(Cambridge): Finitely Generated versus Finitely Presented Groups - 5th October:
**Jahan Zahid**(Bristol): Forms in many variables and recent progress on Artin's conjecture (abstract) - 12th October:
**Jan Hladký**(Warwick): Hamilton cycles in dense vertex-transitive graphs (abstract) - 19th October:
**Mark Wildon**(RHUL): Commuting conjugacy classes: an overview (abstract) - 26th October:
**Graeme Taylor**(Bristol): Cyclotomic Matrices and Graphs (abstract) - 2nd November:
**Graham Niblo**(Southampton): Topological superrigidity (abstract) - 9th November,
**3pm**:**Jozef Skokan**(LSE): Ramsey-goodness (abstract) - 16th November:
**Nick Gill**(Bristol) Growth in solvable subgroups of GL_{r}(Z/pZ) - 23rd November:
**Demetres Christofedes**(Warwick): Winning lines in generalised Tic-Tac-Toe - 30th November:
**Keith Martin**(RHUL): The rise and fall and rise of combinatorial key predistribution (abstract)

**Abstracts:**

Jahan Zahid: Forms in many variables and recent progress on Artin's conjecture

After a brief survey of the charted and uncharted landscape of large dimensional projective varieties, we shall focus our attention to such objects defined over local fields. In particular we shall discuss recent progress on a conjecture due to Emil Artin.

Jan Hladký: Hamilton cycles in dense vertex-transitive graphs

We prove that every large dense connected vertex-transitive graph G contains a Hamilton cycle, that is a cycle through all the vertices of G. This answers partially a question of Babai and Lovasz. The proof is based on the Regularity Lemma. This is a joint work with Demetres Christofides and Andras Mathe (both Warwick).

Mark Wildon: Commuting conjugacy classes: an overview

Let us say that two conjugacy classes of a group commute if they contain representatives that commute. When G is a finite group with a normal subgroup N such that G/N is cyclic, one can use this definition, together with Hall's Marriage Theorem, to describe the distribution of the conjugacy classes of G across the cosets of N. I will give an overview of this result, and then talk about some more recent work on commuting conjugacy classes in symmetric and general linear groups. This talk is on joint work with John Britnell.

Graeme Taylor: Cyclotomic Matrices and Graphs

Lehmer's problem on the Mahler measure of integer polynomials motivates the study of matrices satsifying certain eigenvalue constraints. For integer symmetric matrices, a complete classification of cyclotomic matrices - those with all eigenvalues in [-2,2] - and minimal noncyclotomic matrices was obtained by McKee and Smyth. These results confirm Lehmer's conjecture for a broad class of polynomials, but the general problem remains open. I'll give an overview of the rational integer case, as well as some recent work generalising their approach - based on charged signed graphs - to matrices / graphs over the rings of integers of some imaginary quadratic fields, where Lehmer's conjecture also holds.

Graham Niblo: Topological superrigidity

There is a recurring theme in topology of starting with a map satisfying some control condition, and deriving the existence of a map with much more control. Examples include Whitney's embedding theorem, the sphere theorem, Waldhausen's torus theorem and the geometric superrigidity theorem. I will outline a new result in this spirit concerning the existence of codimension-1 embeddings in aspherical manifolds of high dimension and suggest some applications and possible generalisations. The results use a mixture of ideas from geometric group theory, surgery theory, Poincaré duality and rigidity. This is joint work with Aditi Kar.

Given two graphs G and H, the Ramsey number R(G,H) is the smallest N such that, however the edges of the complete graph K_{N }are coloured with red and blue, there exists either a red copy of G or a blue copy of H. Burr gave a simple general lower bound on the Ramsey number R(G,H), valid for all connected graphs G: defining sigma(H) to be the smallest size of any colour class in any colouring of H with chi(H) colours, we have R(G,H) ≥ (chi(H)-1)(|G|-1) + sigma(H). For a given graph H, it is natural to ask which connected graphs G attain this bound. A class of graphs is called Ramsey-good if, for each fixed H, Burr's bound is attained for all sufficiently large graphs in the class.

In this talk we will give an overview of some known results about Ramsey-goodness, and offer some new results. In particular, we shall explore connections between Ramsey-goodness and the bandwidth. Joint work with Peter Allen and Graham Brightwell.

Keith Martin:The rise and fall and rise of combinatorial key predistribution

There are many applications of symmetric cryptography where the only realistic option is to predistribute key material in advance of deployment, rather than provide online key distribution. The problem of how most effectively to predistribute keys is inherently combinatorial. We revisit some early combinatorial key predistribution shemes and discuss their limitations. We then explain why this problem is back "in fashion" after a period of limited attention by the research community. We consider the appropriateness of combinatorial techniques for key distribution and identify potential areas of further research.

**Useful links**:

Campus map (The McCrea Building is number 17)