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Logic Seminar (past)

Thu, 16/05
17:00 		Tom Leinster (Edinburgh) Logic Seminar Add to calendar L3
Ultraproducts, categorically
It has long been a challenge to synthesize the complementary insights offered by model theory and category theory. A small fragment of that challenge is to understand ultraproducts categorically. I will show that, granted some general categorical machinery, the notions of ultrafilter and ultraproduct follow inexorably from the notion of finiteness of a set. The machine in question, known as the codensity monad, has existed in an underexploited state for nearly fifty years. To emphasize that it was not constructed specifically for this purpose, I will mention some of its other applications. This talk represents joint work with an anonymous referee. Little knowledge of category theory will be assumed.
Thu, 09/05
17:00 		Dan Isaacson (Oxford) Logic Seminar Add to calendar L3
POSTPONED
Thu, 07/03
17:00 		Jeff Paris (Manchester) Logic Seminar Add to calendar L3
Pure Inductive Logic
I shall give a non-technical survey of Pure Inductive Logic, a branch of Carnap's Inductive Logic which was anticipated early on in that subject but has only recently begun to be developed as an area of Mathematical Logic. My intention is to cover its origins and aims, and to pick out some of the key concepts which have emerged in the last decade or so.
Thu, 28/02
17:00 		Gareth Jones (Manchester) Logic Seminar Add to calendar L3
Rational values of certain analytic functions
Masser recently proved a bound on the number of rational points of bounded height on the graph of the zeta function restricted to the interval [2,3]. Masser's bound substantially improves on bounds obtained by Bombieri-Pila-Wilkie. I'll discuss some results obtained in joint work with Gareth Boxall in which we prove bounds only slightly weaker than Masser's for several more natural analytic functions.
Thu, 21/02
17:00 		Ivan Tomasic (QMUL) Logic Seminar Add to calendar L3
Multiplicity in difference geometry
The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to count the number of solutions to difference polynomial equations over fields with powers of Frobenius. We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.
Thu, 14/02
17:00 		Jonathan Kirby (UEA) Logic Seminar Add to calendar L3
Quasiminimal structures and categoricity
Thu, 07/02
17:00 		Peter Holy (Bristol) Logic Seminar Add to calendar L3
The Outer Model Programme
The Outer Model Programme investigates L-like forcing  extensions of the universe, where we say that a model of Set Theory  is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk  about its history, motivations, recent results and applications. I  will be presenting joint work with Sy Friedman and Philipp Luecke.
Thu, 24/01
17:00 		Bahareh Afshari (Oxford) Logic Seminar Add to calendar L3
Closure ordinals for fixpoints in modal mu-calculus
Thu, 17/01
17:00 		Jonathan Kirby (UEA) Logic Seminar Add to calendar L3
***POSTPONED***
Thu, 06/12/2012
17:00 		George Metcalfe (Bern) Logic Seminar Add to calendar L3
An application of proof theory to lattice-ordered groups
(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.
Thu, 29/11/2012
17:00 		Martin Hils (Paris) Logic Seminar Add to calendar L3
Valued difference fields and NTP2
(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).
Thu, 22/11/2012
17:00 		Boris Zilber (Oxford) Logic Seminar Add to calendar L3
A non-desarguesian projective plane of analytic origin
(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, avoiding a direct use of Hrushovski's construction. Instead we make use of the field of complex numbers with a holomorphic function  (Liouville function) which is an omega-stable structure by results of A.Wilkie and P.Koiran.  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. 
Thu, 15/11/2012
17:00 		Jonathan Pila (Oxford) Logic Seminar Add to calendar L3
Some special point problems
Thu, 08/11/2012
17:00 		Davide Penazzi (Leeds) Logic Seminar Add to calendar L3
Topological dynamics and model theory of SL(2,R)
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.
Thu, 25/10/2012
17:00 		Alexandre Borovik (Manchester) Logic Seminar Add to calendar L3
Oracles and involutions: interpretability in black box groups
Thu, 18/10/2012
17:00 		Mirna Dzamonja (UEA) Logic Seminar Add to calendar L3
Embeddings of the spaces of the form C(K)
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.
Thu, 11/10/2012
17:00 		Frank Wagner (Lyon) Logic Seminar Add to calendar L3
Plus ultra
I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.
Thu, 14/06/2012
17:00 		Özlem Beyarslan (Bogazici) Logic Seminar Add to calendar L3
Algebraic closure in pseudofinite fields
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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