

Titles and abstracts for invited talks at the 2005 BCC Speaker: Ben Green Title: Finite field models in additive combinatorics
Speaker: Oliver King Title: The
subgroup structure of finite classical groups in terms of geometric
configurations L.E.Dickson's approach to the subgroups of PSL_{2}(q)
(the Linear Fractional Group) gives rise to a description of subgroups as fixing
one of: a real point; a pair of real points; a pair of imaginary points; a
subline; and so on. H.H. Mitchell took a similar approach in describing
subgroups of PSL_{3}(q) and PSp_{4}(q)
(for odd q). In the 1980s, Aschbacher gave a description of subgroups of
classical groups as either lying in one of eight classes or being almost simple;
the eight classes can largely be described geometrically. The remaining
subgroups have not yet been completely determined but a certain amount of
geometric structure can be identified. Our paper gives a survey of progress
towards a geometric description of subgroups of the classical groups. Speaker: Patric Östergĺrd Title: Constructing
combinatorial objects via cliques Speaker: Tim Penttila Title: Flocks of circle planes Flocks of
finite circle planes  inversive, Minkowski and Laguerre planes  are surveyed,
including their connections with projective planes, generalised quadrangles and
ovals. Speaker: Alex Scott Title: Judicious
partitions and related problems Speaker: Oriol Serra Title: An isoperimetric method for the small sumset problem We survey
applications of an isoperimetric method to the small sumset problem in Additive
Theory. The small sumset problem asks for lower bounds of the cardinality of the
sum of two sets in a group. Sample proofs are presented to illustrate the
application of the method, which is based on connectivity properties of graphs.
In the final part we describe some applications to several problems in number
theory, group theory and combinatorics. Speaker: Paul Seymour (Coauthor: Maria Chudnovsky) Title: The structure of clawfree graphs A graph is clawfree if no vertex has three pairwise nonadjacent neighbours. At first sight, there seem to be a great variety of types of clawfree graphs. For instance, there are line graphs, the graph of the icosahedron, complements of trianglefree graphs, and the Schläfli graph (an amazingly highlysymmetric graph with 27 vertices), and more; for instance, if we arrange vertices in a circle, choose some intervals from the circle, and make the vertices in each interval adjacent to each other, the graph we produce is clawfree. There are several other such examples, which we regard as ``basic'' clawfree graphs. Nevertheless, it is possible to prove a complete structure theorem for clawfree graphs. We have shown that every connected clawfree graph can be obtained from one of the basic clawfree graphs by simple expansion operations. In our paper we explain the precise statement of the theorem, sketch the proof, and give a few applications. Speaker: Alan Sokal Title: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
Speaker: Angelika Steger Title: The sparse regularity lemma and its applications
Szemerédi's
regularity lemma is one of the most celebrated results in modern graph theory.
However, in its original setting it is only helpful for studying large dense
graphs, that is, graphs with n vertices and Θ(n^{2})
edges. The main reason for this is that the underlying concept of εregularity
is not meaningful when dealing with sparse graphs, since for large enough n
every graph with o(n^{2}) edges is εregular. In
1997 Kohayakawa and Rödl independently introduced a modified definition of εregularity
which is also useful for sparse graphs, and used it to prove an analogue of
Szemerédi's regularity lemma for sparse graphs. However, some of the key tools
for the application of the regularity lemma in the dense setting, the socalled
embedding lemmas or, in their stronger forms, counting lemmas,
are not known to be true in the sparse setting. In fact, counterexamples show
that these lemmas do not always hold. However, Kohayakawa, Łuczak, and Rödl
formulated a probabilistic embedding lemma that, if true, would solve several
longstanding open problems in random graph theory. In this survey we give an
introduction to Szemerédi's regularity lemma and its generalisation to the
sparse setting, describe embedding lemmas and their applications, and discuss
recent progress towards a proof of the probabilistic embedding lemma. In
particular, we present various properties of εregular graphs in the
sparse setting. We also show how to use these results to prove a weak version of
the conjectured probabilistic embedding lemma.

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