2009-01-08Jindřich Zapletal (University of Florida and Czech Academy of Sciences)Quantitative Infinite Ramsey Theorems
I will introduce the concept of parametrizing Ramsey theorems with a probability measure. I will state several new theorems--parametrizations of old and well-known results--and I will sketch a proof of one of them. A large number of open problems remains.

2009-01-15Hans Adler (University of Leeds)Two aspects of stability theory
Stability theory as practised by Shelah has a set theoretic flavour. It uses methods of infinite combinatorics to get results that often vary with the model of ZFC. Mainstream stability theory is much more geometric in character, and Hrushovski's group construction is analogous to a famous lattice theoretic theorem of von Neumann. The distinction becomes more important now that we are moving to more general contexts. Dependent (NIP) theories and rosy theories both generalise stable theories. The former have most of the combinatorial properties of stable theories, while the latter preserve most of the geometric properties.

2009-01-22Douglas S. Bridges (University of Canterbury, New Zealand)Constructive Reverse Mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Constructive reverse mathematics is reverse mathematics carried out in Bishop-style constructive math (BISH)---that is, using intuitionistic logic and, where necessary, constructive ZF set theory. There are two primary foci of constructive reverse mathematics:

* first, investigating which constructive principles are necessary to prove a given constructive theorem; * secondly, investigating what non-constructive principles are necessary additions to BISH in order to prove a given non-constructive theorem.

I will introduce basic ideas of constructive mathematics and constructive reverse mathematics, paying particular attention to connections between various continuity properties, versions of the fan theorem, and a simple principle due to Ishihara.

2009-01-29Katie Thompson (KGRC)A universality spectrum for graphs without GCH
A universal model for structures of size K is one which embeds all other such structures of size K. For first-order theories of small size, like graphs, there are universals in all infinite cardinals under GCH. For unstable theories, the question: in which cardinals do universals exist in models of failures of GCH, requires more set-theoretic techniques to answer.

The main results are joint work with Sy Friedman.

2009-03-05Sergei S. Goncharov (Novosibirsk State University)Strongly minimal and categorical theories
2009-03-19Hiroaki Minami (KGRC)Katetov order, cardinal invariants of ideals on omega and Mathias forcing
2009-03-26Ekaterina Fokina (KGRC)Computable categoricity of structures
2009-04-02Lyubomyr Zdomskyy (KGRC)A new upper bound on the cofinality of the infinite symmetric group
We shall prove that if an ultrafilter F is not coherent to a Q-point, then cof(sym(w)) ≤ b(F).

This improves some results of J. Sharp and S. Thomas.

2009-04-23Prerna Juhlin (KGRC)Geometric Aspects of Semiminimality
I will give a brief overview of some notions in geometric stability theory, then discuss how they help us understand dependence in a superstable theory of finite rank, via semiminimal types.

2009-04-30Philip Welch (University of Bristol)Weak deteminacy, inductive operators and subsystems of analysis
It has been long known that the theory of Π11 and Σ11 monotone operators have links with Σ01 and Σ02 Determinacy respectively, and constructible ranks for strategies for such games have been found. In analysis the remaining Σ03 case (since by H Friedman Σ04 Determinacy is not provable in analysis) seems little investigated. We attempt to remedy this. We have in one version:

Δ13-CA0 << Σ03 Determinacy << Δ13-CA0 + a Σ13-absoluteness'' principle.

The above theorem is deliberately given in the spirit of weak subsystems of analysis; however we shall really be investigating low levels of the constructible hierarchy, and a concomitant notion of quasi-inductive operator.

2009-05-07Sy-David Friedman (KGRC)The effective theory of Borel equivalence relations (joint work with Fokina and Törnquist)
2009-05-14Riccardo Camerlo (Politecnico di Torino)Standard universal dendrites as small Polish structures
2009-05-28Alberto Marcone (Udine)The complexity of colored linear orders
Let C be a countable partial order. A C-colored linear order (X,f) is a countable linear order X with a map f:X → C. If (X,f) and (X',f') are two such C-colored linear orders, an embedding is an order-preserving φ:X → X' such that f(x) ≤C f'(φ(x)) for all x in X. The quasi-order of C-colored linear orders under embeddability is analytic. Its complexity in the reduction hierarchy depends on the combinatorial properties of C. We will present some results and some open problems.

2009-07-28Kentaro Sato (Japan Society for the Promotion of Science)An Impact of Infinity
Both in the absence and in the presence of infinity, the logical strengths of several principles are investigated. Surprisingly, the structure of the logical strengths turns out to be completely changed by infinity. This also shows the very strong utility of the framework of set theory.

2009-10-06Victor Selivanov (Insitute of Informatics Systems, Siberian Division of Russian Academy of Science)Some reducibilities on k-partitions
We discuss some extensions of the classical Wadge reducibility on Borel subsets of the Baire space. Main emphasis is made on the extension from the case of sets (i.e., 2-partitions) to the case of partitions of a space to k parts, for any k>1. We show, in particular, that for k>2 the structure of Wadge degreed becomes much more complicated (but still manageable) than for the case of sets. We also settle model-theoretic properties (like characterisation of the definable predicates) of some initial segments of the structures Wadge degrees.

2009-10-08Pavel Semukhin (National University of Singapore)A survey of automatic structures
Automatic structures are the algebraic structures whose basic relations and functions are recognizable by finite automata. In this survey talk, I will first give main definitions and discuss general properties of automatic structures. In the second part, we will consider the problem of characterizing the automatic structures in some particular classes of models, like linear orders, groups, Boolean algebras, etc. As an example, we will prove non-automaticity of the ordinal ωω and (N,*), the natural numbers with multiplication.

2009-10-15Thomas Johnstone (KGRC and City Tech, New York)The Resurrection Axioms
I will discuss a new class of forcing axioms, the Resurrection Axioms RA), and the Weak Resurrection Axioms (wRA). While Cohen's method of forcing has been designed to change truths about the set-theoretic universe you live in, the point of Resurrection is that some truths that have been changed by forcing can in fact be resurrected, i.e. forced to hold again. In this talk, I plan to illustrate how RA is tied to forcing axioms such as MA and BPFA, and how it affects the size of the continuum. The main theorem will show that RA and many instances of wRA are equiconsistent with the existence of an uplifting cardinal, a large cardinal notion consistent with V=L. This is joint work with Joel Hamkins.

2009-10-19Sy-David Friedman (KGRC)Some remarks about PFA and BPFA
It is consistent that PFA holds and some inner model with the correct ω2 does not contain all reals. This contrasts with results of Velickovic and Caicedo-Velickovic, which say that this cannot hold for SPFA nor for BPFA if one requires that the inner model also satisfy BPFA. I'll also discuss the compatibility of BPFA with projective wellorders.

2009-10-29Vera Fischer (KGRC)Cardinal characteristics and projective wellorders
Using countable support iteration of S-proper posets, we show that the existence of a Δ13 definable wellorder of the reals is consistent with each of the following: '''d''' < '''c''', '''b''' < '''a''' = '''s''', '''b''' < '''g'''. This is a joint work with Sy Friedman.

2009-11-05Miguel Angel Mota (KGRC)Forcing anti-diamond principles together with the continuum being large
As an example of our technique of ''rigid iterations with side conditions'', we will force some strengthenings of the negation of weak club guessing together with the continuum being larger than ω2. This is joint work with David Asperó.

2009-11-12Ekaterina Fokina (KGRC)Computable categoricity vs. relative computable categoricity
The notion of computable categoricity of structures is one of the fundamental in computable model theory. A computable structure A is computably categorical if for every computable structure B isomorphic to A, there exists a computable isomorphism from A to B. We will study the question of syntactical characterization of the notion of computable categoricity. The answer exists only under some additional effectiveness condition. Examination of this characterization without the effectiveness condition will lead us to the notion of relative computable categoricity.

2009-11-26Hans Adler (KGRC)First-order theories without the independence property
Roughly speaking, a first-order formula has the independence property (relative to a complete theory T) if it cannot be used to represent arbitrary subsets of an infinite set in a model of T. T itself is said to have the independence property if one of its formulas has it. The independence property defines a very significant dividing line in the sense of Shelah. In the last years theories without the independence theory (also known as NIP theories) have received wider attention as a potentially fruitful generalisation of stable theories.

I will give an elementary introduction into this field, with a special focus on some of Shelah's remarkable results related to indiscernible sequences.

2009-12-17Sy-David Friedman (KGRC)Abstract Elementary Classes and Axioms for Set Theory