Past Logic Seminar

6 December 2012
17:00
George Metcalfe
Abstract
(Joint work with Nikolaos Galatos.) Proof-theoretic methods provide useful tools for tackling problems for many classes of algebras. In particular, Gentzen systems admitting cut-elimination may be used to establish decidability, complexity, amalgamation, admissibility, and generation results for classes of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing a family resemblance to groups, such methods have so far met only with limited success. The main aim of this talk will be to explain how proof-theoretic methods can be used to obtain new syntactic proofs of two core theorems for the class of lattice-ordered groups: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.
29 November 2012
17:00
Martin Hils
Abstract
(Joint work with Artem Chernikov.) In the talk, we will first recall some basic results on valued difference fields, both from an algebraic and from a model-theoretic point of view. In particular, we will give a description, due to Hrushovski, of the theory VFA of the non-standard Frobenius acting on an algebraically closed valued field of residue characteristic 0, as well as an Ax-Kochen-Ershov type result for certain valued difference fields which was proved by Durhan. We will then present a recent work where it is shown that VFA does not have the tree property of the second kind (i.e., is NTP2); more generally, in the context of the Ax-Kochen-Ershov principle mentioned above, the valued difference field is NTP2 iff both the residue difference field and the value difference group are NTP2. The property NTP2 had already been introduced by Shelah in 1980, but only recently it has been shown to provide a fruitful ‘tameness’ assumption, e.g. when dealing with independence notions in unstable NIP theories (work of Chernikov-Kaplan).
22 November 2012
17:00
Boris Zilber
Abstract
(This is a joint result with Katrin Tent.) We construct a series of new omega-stable non-desarguesian projective planes, including ones of Morley rank 2,&nbsp;<br />avoiding a direct use of Hrushovski's construction. Instead we make use of the field of complex numbers with a holomorphic function &nbsp;(Liouville function) which is an omega-stable structure by results of A.Wilkie and P.Koiran. &nbsp;We first find a pseudo-plane interpretable in the above analytic structure and then "collapse" the pseudo-plane to a projective plane applying a modification of Hrushovski's mu-function.&nbsp;</p>
8 November 2012
17:00
Davide Penazzi
Abstract
Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.
18 October 2012
17:00
Mirna Dzamonja (UEA)
Abstract
<p>We discuss the question of the existence of the smallest size of a family of Banach spaces of a given density which embeds all Banach spaces of that same density. We shall consider two kinds of embeddings, isometric and isomorphic. This type of question is well studied in the context of separable spaces, for example a classical result by Banach states that C([0,1]) embeds all separable Banach spaces. However, the nonseparable case involves a lot of set theory and the answer is independent of ZFC.</p>
11 October 2012
17:00
Frank Wagner (Lyon)
Abstract
<p>I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.</p>
14 June 2012
17:00
Özlem Beyarslan (Bogazici)
Abstract
A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can be realised as ultraproducts of finite fields. A group $G$ is geometrically represented in a theory $T$ if there are modles $M_0\prec M$ of $T$, substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$ and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of pseudofinite fields. We show that, geometric representation of a group whose order is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity in models of $T$. As a consequence of this, we show that, for almost all completions of the theory of pseudofinite fields, over a substructure $A$, algebraic closure agrees with definable closure, if $A$ contains the relative algebraic closure of the prime field. This is joint work with Ehud Hrushovski.

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