2006 Summer Meeting of the Association for Symbolic Logic: Logic Colloquium '06

s of invited tutorials RODNEY DOWNEY, Algorithmic randomness and computability. School of Mathematics, Statistics and Computer Science, Victoria University, PO Box 600, Wellington, New Zealand. E-mail: rod.downey@mcs.vuw.ac.nz. In these lectures, I will examine questions such as: what does it mean for a real to be random? what does it mean for one real to be more random than another? Does a real being random mean that it has high or low information content? There has been amazing progress in this area in recent years through that work of many authors. I plan to cover the basic ideas behind the theory of algorithmic randomness (randomness means (i) not having rare properties (ii) having initial segments that are hard to compress and (iii) random means hard to predict, for example). I will then look at recent work clarifying the relationship between randomness and computational power, as well as the classification of reals by initial segment complexity. Finally I plan to look at the deep work on triviality and lowness with its important implications in classical computability. Background material and details can be found in the references below. [1] R. Downey, Some computability-theoretical aspects of reals and randomness, The Notre Dame Lectures (P. Cholak, editor), Lecture Notes in Logic, vol. 18, pp. 97–148, Association for Symbolic Logic, 2005. [2] , Five lectures on algorithmic randomness, Proceedings of Computational Prospects of Infinity, World Scientific, to appear. [3] R. Downey and D. Hirschfeldt, Algorithmic randomness and complexity, Springer-Verlag Monographs in Computer Science, to appear. [4] R. Downey, D. Hirschfeldt, A. Nies, and S. Terwijn, Calibrating randomness, The Bulletin of Symbolic Logic, to appear. [5] A. Nies, Computability and randomness, to appear. LOGIC COLLOQUIM ’06 253 BOBAN VELICKOVIC, Forcing axioms. Équipe de Logique Mathématique, UFR de Mathématiques (case 7012), Université Paris 7, Denis-Diderot 2 place Jussieu, 75251 Paris Cedex 05, France. E-mail: boban@logique.jussieu.fr. Forcing axioms are natural combinatorial statements which decide many of the questions undecided by the usual axioms ZFC of set theory. The study of these axioms was initiated in the late 1960s by Martin and Solovay who introduced Martin’s Axiom, followed by the formulation of the Proper ForcingAxiom byBaumgartener and Shelah in the early 1980s and Martin’s Maximum by Foreman, Magidor and Shelah in the mid-1980s. In the mid 1990s Woodin’s work on Pmax extensions established deep connections between forcing axioms and the theory of large cardinals and determinacy. Nevertheless, some of the key problems remained open. In 2003 Moore formulated the Mapping Reflection Principle (MRP) which seems to be the missing ingredient needed in order to resolve many of the remaining open problems in the subject and a number of important developments in the subject followed. In this series of lectures we first survey the theory of forcing axioms and then present some recent results: Moore’s work on MRP, my work with A. Caicedo on definable well-orderings of the reals, Viale’s result that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis, etc. Abstracts of invited plenary talkss of invited plenary talks SAMSON ABRAMSKY, Categorical quantum logic. Oxford University Computing Lab., Wolfson Building, Parks Road, Oxford OX1 3QD, UK. E-mail: Samson.Abramsky@comlab.ox.ac.uk. We describe a novel approach to axiomatizing Quantum Mechanics in the setting of strongly compact closed categories. This axiomatization captures all the features which are significant for Quantum Information and Computation, and provides a basis for effective reasoning about quantum informatic processes. It can be presented in terms of a diagrammatic calculus which can be seen as a proof system for a logic, leading to a new perspective on what the right logical formulation for Quantum Mechanics should be. MARAT ARSLANOV, Problems of definability and structural differences among elementary theories of the Ershov difference hierarchy. Kazan State University, 420008, Kazan, Russia. E-mail: Marat.Arslanov@ksu.ru. In my talk I will give a survey on the current state of the research on differences among elementary theories of the Ershov difference hierarchy, including a recent negative solution of well-known Downey’s Conjecture on elementary equivalency of nand m-c.e. degree structures for n, m > 1 and n = m. Problems on definability of the relation “m-computably enumerable” in n-c.e. degree structures (n > m), and on m-c.e. degrees as Σ1-substructures of n-c.e. degree structures will be also considered, including recent solutions of a number of open questions in this area. New open questions will be raised, and trends of the research development in this area will be discussed. HARVEY FRIEDMAN, Search for consequences. Ohio State University, Room 754, Mathematics Building, 231 West 18th Avenue, Columbus, OH 43210, USA. E-mail: friedman@math.ohio-state.edu. We will discuss some new examples of necessary uses of set theory with large cardinals that fit squarely into finite graph theory. We will also discuss prospects for the future. 254 LOGIC COLLOQUIM ’06 MARTINGOLDSTERN,Applications of mathematical logic in algebra: the lattice of clones. DMG/Algebra, TUWien, Wiedner Haupstrasse 8-10/104, A-1040 Wien, Austria. E-mail: goldstern@tuwien.ac.at. A clone on a set X is a family of finitary operations on X which contains all projections and is closed under substitution (composition), such as the family of term functions on any universal algebra. The set of all clones on a fixed set X forms a complete lattice, whose structure is rather complicated already for finite sets X . I will discuss some old (1960s), intermediate and recent (2006) results about the structure (and more often: non-structure) of this lattice on infinite sets, as well as open questions. It turns out that methods and results from set theory (forcing, descriptive set theory, infinite combinatorics) are essential for this analysis. EHUD HRUSHOVSKI,Model theory of valued fields. Institute of Mathematics, Hebrew University (Giv’at Ram), Jerusalem 91904, Israel. E-mail: ehud@math.huji.ac.il. I will survey some of the developments in the subject, with emphasis on connections to abstract model theory and to geometry. JOCHEN KOENIGSMANN, Axiomatizing fields via Galois theory. Institut für mathematische Logik und Grundlagen der Mathematik, Universität Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany. E-mail: Jochen.Koenigsmann@unibas.ch. Grothendieck’s program of “anabelian geometry” is designed to recover a field K from its absolute Galois group GK := Gal(K/K). This has been achieved, e.g., for the class of finitely generated fields, but it doesn’t work in general. However, replacing GK by GK(x), the absolute Galois group of the rational function field K(x) over K , does the trick: almost all perfect fields K are determined up to isomorphism by GK(x). The model theoretic version of this theorem proves that K and GK(x) are biinterpretable where we consider the field K in the usual first order language for rings, and where GK(x) is understood as a structure in the language for profinite groups developed by Cherlin, van den Dries, Macintyre and Chatzidakis. We will also discuss the impact on decidability questions for fields like C(t) or Fp((t)). ANDREW LEWIS, On constructing strong minimal covers. Dipartimento di ScienzeMatematiche ed Informatiche RobertoMagari, University of Siena, Pian dei Mantellini 44, 53100 Siena, Italy. E-mail: andy@aemlewis.co.uk. After giving an introduction to the splitting tree technique of minimal degree construction we shall discuss the generality of this approach and applications to the construction of strong minimal covers. ANTONIO MONTALBÁN, Embeddability and decidability in the Turing Degrees. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chacago, IL 60637, USA. E-mail: antonio@math.uchicago.edu. One of themain goals of computability theory is to understand the shape of the structure of the Turing degrees. Two common ways of studying this is by looking at structures that can be embedded into theTuringDegrees andby looking a the decidability of the theory of this structure varying the language or taking substructures. Wedescribe the author’s results in this area. ERIK PALMGREN, Intuitionism, Bishop constructivism and point-free thinking. Department of Mathematics, Uppsala University, PO Box 480, S-571 06 Uppsala, Sweden. E-mail: palmgren@math.uu.se. LOGIC COLLOQUIM ’06 255 A consequence of the Heine-Borel theorem is that a point-wise continuous function on the closed unit interval is uniformly continuous. Brouwer incorporated this theorem into intuitionism by developing new mathematical axioms. Topology could then be developed using essentially the classical point-based theory. These axioms, incompatible with the interpretation of logic in terms of computable functions, were given up in the later development of constructive mathematics by the followers of Errett Bishop. Instead various techniques to provide necessary information about (local) uniformity were developed. A parallel development was that within locale theory, where a point is a derived concept of the topology, while the notion of cover is basic. This made it possible to naturally extend the notion of being locally uniformly continuous to general maps between topological spaces. We show how the theory of Bishop and the point-free theory can be reconciled. The foundational problems of point-free topology are interesting from a predicativistic point of view, and have been addressed by many researchers since the mid 1980s. We discuss some recent developments in this area. We also discuss examples of point-free methods in measure theory. WOLFRAM POHLERS, Iterations of ordinal operators. Institut fürmathematischeLogik undGrundlagenforschung, Einsteinstr. 62, 48149, Münster, G