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# Program for Sunday

Ludomir Newelski, Uniwersytet Wrocławski.

** Hour: ** 9:00 - 10:00

** Title: ** Topological methods in model theory.

** Abstract: ** I will survey the development of topological methods in model theory. I
will discuss Morley rank and forking in stable theories and also various
ways to generalize it to the unstable theories. I will focus on
topological dynamics. Topological dynamics yields general counterparts of
the notion of a generic type in a stable group.

Krzysztof Krupiński, Uniwersytet Wrocławski.

** Hour: ** 10:20 - 11:20

** Title: ** Polish structures.

** Abstract: ** Abstract: I will define the notion of Polish structure introduced by myself a few years ago, and discuss counterparts of some fundamental notions from stability theory in the context of Polish structures. Then I will focus on structural results about groups and rings in the context of Polish structures, some of which were obtained in collaboration with F. Wagner or J. Dobrowolski.

Anand Pillay, University of Leeds.

** Hour: ** 11:30 - 12:30

** Title: ** Transcendence, differential equations, and model theory

** Abstract: ** I give an application of model theory, specifically the theory DCF_0 of differentially closed fields, to the problem of understanding algebraic relations between solutions of ordinary differential equations belonging to the Painleve family. I will give the required background as well as describing elementary aspects of the proof (joint with my student J. Nagloo).

Yiannis Moschovakis, UCLA.

** Hour: ** 14:30 - 15:30

** Title: ** Effective Descriptive Set Theory.

** Abstract: ** Andrzej Mostowski first noticed the similarities between
the classical Borel and projective sets of real numbers and
Kleene's arithmetical and analytical sets of natural numbers and
used them to define the hyperarithmetical sets. These “analogies”
between the two subjects, as they were first called, eventually
developed into a common “effective” theory of Borel, analytic and
projective sets on Polish spaces, a common extension of the
classical and the recursive hierarchies which illuminates and
enriches them both.

My aim in this talk is to give a very brief introduction to Effective Descriptive Set Theory, its aims, its methods and some of its applications. I will describe a development of the subject which is more general and easier to apply than the standard presentations from the 1970s, based on a set of notes that I am writing together with Vassilis Gregoriades.

Leszek Kołodziejczyk, University of Warsaw.

** Hour: ** 15:50 - 16:50

** Title: ** The importance of approximate counting in bounded arithmetic.

** Abstract: ** Bounded arithmetic is a collective name for a family of relatively
weak theories of first-order arithmetic in which the exponential
function is not total and the induction scheme can only be applied to
bounded formulas. Such theories, which were given their modern form by
Sam Buss in the 1980's, are studied largely because of their
connections to computational complexity and propositional proof
complexity. However, they are also of interest from a more traditional
foundational point of view, as a way of understanding how much of
finite mathematics can be developed without exponentiation or other
“infeasibly fast-growing” functions.

The fundamental problem in bounded arithmetic is whether the induction scheme for the class of all bounded formulas can be finitely axiomatized. This problem and related questions appear to be beyond the reach of current methods, so much effort has gone into the study of relativized theories, for which unprovability and separation results are more accessible. I will discuss the two main open problems on the frontier of research on relativized bounded arithmetic, and explain recent work suggesting that a single theory plays a major role in the context of both problems. The theory in question, axiomatized by a fragment of the induction scheme and a version of the so-called weak pigeonhole principle, was introduced by Emil Jerabek in order to formalize the notion of approximate cardinality of definable sets.

The talk will be based on joint work with Sam Buss, Neil Thapen and Konrad Zdanowski.

Jouko Väänänen, University of Helsinki and University of Amsterdam.

** Hour: ** 17:00 - 18:00

** Title: ** The many lives of generalized quantifiers.

** Abstract: ** Generalized quantifiers have thrived in logic, linguistics and computer science in ways perhaps unanticipated by Mostowski who introduced them in 1957. They have come to manifest a perfect symbiosis between model theory and set theory. Their model theoretic properties reflect a rich array of set-theoretical concepts such as large cardinals and combinatorial principles. I will end with a few remarks on dependence logic as a new way of thinking about generalized quantifiers.