SEEMOD is the South and East of England Model Theory Network, which has had meetings at UEA, in London, and in Oxford. This is its first meeting in Cambridge. It is supported by a Scheme 3 grant from the London Mathematical Society. The co-ordinators are Charlotte Kestner and Jonathan Kirby.

All talks will be in the room MR3 (level -1).

12-1 arrival and lunch. Some people will meet at 12noon at Churchill College cafeteria. For those who arrive later, the CMS central core cafe should have some food (perhaps sandwiches).

1pm - 1:50pm Emmanuel Breuillard

1:50pm - 2:40pm Ivan Tomasic

2:40pm - 3:30pm tea and discussions

3:30pm - 4pm Yibei Li

4pm - 4:50pm Laura Capuano

followed by drinks/dinner in Cambridge city centre

Emmanuel Breuillard

Title: Approximate groups and projective geometries.

Abstract: The structure of finite subsets A of an ambient algebraic
group G, which do not grow much under multiplication, say
|AA|<|A|^{1+\epsilon}, is well understood after the works of
Hrushovski, Pyber-Szabo and Breuillard-Green-Tao on approximate
subgroups of algebraic groups. A more general question, tackled by
Elekes and Szabo, asks for the structure of Cartesian products A_1
\times ... \times A_n of finite subsets of size N of an arbitrary
d-dimensional algebraic variety W, with large (i.e. >N^{\dim V/d})
intersection with a given subvariety V \leq W^n (the case n=3, W=G, A_i=A, V={(x,y,xy)} corresponds to the above mentioned
approximate group problem). In joint work with Martin Bays, we
completely characterize the algebraic varieties V that can admit a
(general position) family of such finite Cartesian products with large
intersection. We show that they are in algebraic correspondence with a
subgroup of a commutative algebraic group endowed with an extra
structure arising from a certain division ring of group endomorphisms.
The proof makes use of the Veblen-Young theorem on abstract projective
geometries, generalized Szemeredi-Trotter bounds and Hrushovski's
formalism of pseudo-finite dimensions.

Ivan Tomasic

Title: Graphons, Tao's regularity and difference polynomials

In joint work with Mirna Dzamonja, we formulate Tao's spectral proof of the algebraic regularity lemma for certain classes of finite structures in the context of graphons. We apply these techniques to study expander difference polynomials over fields with powers of Frobenius.

Yibei Li

Title: Automorphism groups of homogeneous structures with stationary weak independence relations

Abstract: Tent and Ziegler showed that if a homogeneous structure admits a stationary independence relation, then its automorphism group can be generated by some special automorphism. We will discuss a generalisation of this statement and look at some examples.

Laura Capuano

Title: An effective criterion for periodicity of l-adic continued fractions

It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a l-adic continued fraction, and the definition depends on the chosen system of residues mod l. It turns out that the theory of l-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the l-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the "real" value of the l-adic continued fraction.