Leeds

Let T be a (one-sorted first order) geometric theory (so T

has infinite models, T eliminates "there exist infinitely many" and

algebraic closure gives a pregeometry). I shall present some results

about T_P, the theory of lovely pairs of models of T as defined by

Berenstein and Vassiliev following earlier work of Ben-Yaacov, Pillay

and Vassiliev, of van den Dries and of Poizat. I shall present

results concerning superrosiness, the independence property and

imaginaries. As far as the independence property is concerned, I

shall discuss the relationship with recent work of Gunaydin and

Hieronymi and of Berenstein, Dolich and Onshuus. I shall also discuss

an application to Belegradek and Zilber's theory of the real field

with a subgroup of the unit circle. As far as imaginaries are

concerned, I shall discuss an application of one of the general

results to imaginaries in pairs of algebraically closed fields,

adding to Pillay's work on that subject.

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).

I will explain the connection between Shelah's recent notion of strongly dependent theories and finite weight in simple theories. The connecting notion of a strong theory is new, but implicit in Shelah's book. It is related to absence of the tree property of the second kind in a similar way as supersimplicity is related to simplicity and strong dependence to NIP.