## SEEMOD Workshop 9

### 17th May 2019, Centre for Mathematical Sciences, Cambridge

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.

### Speakers

Emmanuel Breuillard (Cambridge), Laura Capuano (Oxford), Yibei Li (Imperial College London), Ivan Tomasic (Queen Mary, University of London)

### Directions

Directions to the Centre for Mathematical Sciences can be found on the CMS website. The U bus service runs from the railway station to the CMS. It runs every 15 minutes and the journey time is also about 15 minutes.

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

### Registration and funding

There is no registration for the meeting. Some money is available, particularly for PhD students, for travel expenses and to cover additional caring costs (e.g. childcare). Please contact Jonathan Kirby if you would like to claim expenses.

### Provisional schedule

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

### Abstracts

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.

### Thanks

The workshop is supported by an LMS Scheme 3 grant and by the ERC grant GeTeMo no 617129.