Past Logic Seminar

13 November 2008
17:00
Bob Coecke
Abstract
[This is a joint seminar with OASIS] A formulation of quantum mechanics in terms of symmetric monoidal categories provides a logical foundation as well as a purely diagrammatic calculus for it. This approach was initiated in 2004 in a joint paper with Samson Abramsky (Ox). An important role is played by certain Frobenius comonoids, abstract bases in short, which provide an abstract account both on classical data and on quantum superposition. Dusko Pavlovic (Ox), Jamie Vicary (Ox) and I showed that these abstract bases are indeed in 1-1 correspondence with bases in the category of Hilbert spaces, linear maps, and the tensor product. There is a close relation between these abstract bases and linear logic. Joint work with Ross Duncan (Ox) shows how incompatible abstract basis interact; the resulting structures provide a both logical and diagrammatic account which is sufficiently expressive to describe any state and operation of "standard" quantum theory, and solve standard problems in a non-standard manner, either by diagrammatic rewrite or by automation. But are there interesting non-standard models too, and what do these teach us? In this talk we will survey the above discussed approach, present some non-standard models, and discuss in how they provide new insights in quantum non-locality, which arguably caused the most striking paradigm shift of any discovery in physics during the previous century. The latter is joint work with Bill Edwards (Ox) and Rob Spekkens (Perimeter Institute).
30 October 2008
17:00
Jochen Koenigsmann
Abstract
I will present a universal definition of the integers in the field of rational numbers, building on work discussed by Bjorn Poonen in his seminar last term. I will also give, via model theory, a geometric criterion for the non-diophantineness of Z in Q.
16 October 2008
17:00
Kobi Peterzil
Abstract
<p> (joint work with E. Hrushovski and A. Pillay)<br /> <br /> If G is a definably compact, connected group definable in an o-minimal structure then, as is known, G/Z(G) is semisimple (no infinite normal abelian subgroup).<br /> <br /> We show, that in every o-minimal expansion of an ordered group: </p> <p> If G is a definably connected central extension of a semisimple group then it is bi-intepretable, over parameters, with the two-sorted structure (G/Z(G), Z(G)). Many corollaries follow for definably connected, definably compact G.<br /> Here are two: </p> <p> 1. (G,.) is elementarily equivalent to a compact, connected real Lie group of the same dimension. </p> <p> 2. G can be written as an almost direct product of Z(G) and [G,G], and this last group is definable as well (note that in general [G,G] is a countable union of definable sets, thus not necessarily definable).<br /> <br /> </p>
9 October 2008
17:00
Amador Martin-Pizarro
Abstract
In 2006, a bad field was constructed (together with Baudisch, Hils and Wagner) collapsing Poizat's green fields. In this talk, we will not concentrate on the general methodology for collapsing specific structures, but more on a specific result in algebraic geometry, a weaker version of the Conjecture on Intersection with Tori (CIT). We will present a model theoretical proof of this result as well as discuss the possible generalizations to positive characteristic. We will try to make the talk  self-contained and aimed for an audience with a basic acquaintance with Model Theory.<br /> <br />
23 July 2008
14:30
Bart Kastermans
Abstract
Cofinitary groups are subgroups of the symmetric group on the natural numbers (elements are bijections from the natural numbers to the natural numbers, and the operation is composition) in which all elements other than the identity have at most finitely many fixed points. We will give a motivation for the question of which isomorphism types are possible for maximal cofinitary groups. And explain some of the results we achieved so far.
13 June 2008
15:15
Alex Prestel
Abstract
We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on $S(h)$, admits a representation $f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$ with each $\sigma_i$ being a sum of squares of real polynomials. If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean. There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however, interested in an effective procedure to decide whether h is archimedean or not. In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision problem. Recently, S. Wagner has proved decidability for all dimensions using among others model theoretic tools like the Ax-Kochen-Ershov Theorem.

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