Past Logic Seminar

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.
10 May 2012
17:00
Jamshid Derakhshan
Abstract
This is joint work with Raf Cluckers, Eva Leenknegt, and Angus Macintyre.<br /><br />We give a first-order definition, in the ring language, of the ring of p-adic integers inside the field&nbsp;p-adic numbers which works uniformly for all p and for valuation rings of all finite field extensions and of&nbsp;all local&nbsp;fields of positive characteristic p, and in many other Henselian valued fields as well. The formula canbe taken existential-universal in the ring language. Furthermore, we show the negative result that in the language&nbsp;of rings there does not exist a uniform definition by an existential formula and neither by a universal formula. For any fixed general p-adic field we give an existential formula in the ring language which defines the valuation ring.</p> <p>We also state some connections to some open problems.</p>

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