Abstract: I will give an introduction to the isomorphism problem in ergodic theory. The classical successes for the two extreme kinds of ergodic transformations, the irational rotations of a circle (and their generalizations) on the one hand and the Bernoulli shifts on the other will be explained in some detail, as well as some recent results and open problems.
Abstract: Entropy of ergodic actions will be defined and I will explain the key role it plays in the isomorphism problem. A new characterization of the Shannon entropy of stationary processes will be described as well as some surprising connections with computable numbers.
Abstract: The notion of restricted orbit equivalence provides an umbrella under which reside a number of standard and nonstandard equivalence relations in ergodic theory. Two significant examples are conjugacy and even Kakutani equivalence. Less standard perhaps is equality of entropy. I will give a general overview of the notion of restricted orbit equivalence and then focus on the known examples. I will attempt to frame a number of both narrow and broad questions in the application of set theoretic methods to ergodic theory that arise in this context.
Abstract: In this talk, I will give an introduction to some recent work in set theory, whose goal is a theory of complexity of classification problems in mathematics, and discuss its connections with aspects of dynamical systems and in particular rigidity phenomena in the context of ergodic theory.
Abstract: I describe a technique for using the basic theorem of forcing to generalize results in ergodic theory.
Abstract: We consider the analytic equivalence relation of linear isomorphism on the class of separable Banach spaces. A question due to Godefroy asked for the determination of its complexity according the Borel reducibility ordering, and we solve this by showing that it has maximal complexity. Thus, at least theoretically, separable Banach spaces up to isomorphism provide complete invariants for virtually any class of separable structures encountered in analysis. Moreover, we shall indicate a surprising implication for the geometry of Banach space of our proof. [This is joint work with V. Ferenczi and A. Louveau]
Abstract: I'll discuss ways in which the Parametrized Diamond Principles scheme of Dzamonja, Hrusak and Moore can be adapted to the Covering Property Axiom.
Abstract: We prove that if ZFC is consistent so is ZFC + "for any σ-algebra A of sets of reals which is σ-generated by kappa-many sets, the Baire property can be extended to A", for suitably chosen cardinals kappa. We also discuss such an extension problem under the assumption of existence of measureable or weakly-compact cardinals.
This is a category analogue of the problem of extending Lebesgue measure to any collection of kappa-many sets. The results of this talk are a continuation of a work of Carlson, Zakrzewski and my Masters degree thesis. We study extensions of Baire Property and Lebesgue Measure in Cohen and Solovay models.
This is a joint work with Janusz Pawlikowski.
Abstract: We discuss the existence of a small compact space with special properties concerning the convergence of sequences and its connection to an open problem in the theory of Banach spaces. We show that in Hechler's model there is a compact space with slightly weaker properties.
Abstract: Fix a separable infinite dimensional complex Hilbert space H. Let B(H) be its algebra of bounded linear operators, K(H) its ideal of compact operators and C(H)=B(H)/K(H) the Calkin algebra. It can be considered as a quantized analogue of the quotient Boolean algebra P(N)/Fin. I will prove that OCA∞ and MA imply all automorphisms of the Calkin algebra are inner. Together with a recent result of Phillips and Weaver, this answers a 1977 question by Brown-Douglas-Fillmore.
Abstract: Consider the following problems:
given a Borel equivalence relation, how to recognize that it is Borel reducible to the orbit equivalence relation of a continuous Polish group action, and
given a Borel group, how to recognize from the orbit equivalence relations its continuous actions induce, that the group has a (unique) Polish group topology generating the original Borel structure on it.
In the talk, I will discuss the above problems and present a solution to the second one for Borel groups that are abelian.Abstract: Associated with any action of a countable group on a σ-complete Boolean algebra is the group of automorphisms which can be obtained from the action via countable decomposition. We will discuss several algebraic properties of these groups, and mention some connections to the study of countable Borel equivalence relations.
Abstract: AD is assumed. We consider specific types of unions of Borel sets, and whether or not a given pointset can be obtained by that type of union. For example, it is a classical theorem that a set is &Sigma12 iff it is the union of &omega1 Borel sets. There are two main theorems characterizing pointclasses via unions of Borel sets, in a manner analogous to the above result for &Sigma12: an old theorem for &Sigma12n and a new theorem for &Delta12n+1.
Abstract: A compact Hausdorff space is perfectly normal if every closed set is a countable intersection of open sets. Strong Baire Category Assumptions, such as Martin's Maximum, allow for a deep structural analysis within this class of spaces. Still, some fundamental problems remain open:
Question: (MM) Does every perfectly normal compactum admit an at most 2-to-1 continuous map into a metric space?
Question: Is every perfectly normal compact convex set in a locally convex topological vector space metrizable?
This talk will discuss how these and other related problems and conjectures developed. It will also present come recent advances in set theory and topology which may aid in their solution.
Abstract: Having exhausted the study of Borel hierarchies of countable ordinal length, the author proposes to take up the study of Borel hierarchies of length greater than &omega1. Doesn't this sound impossible?
Theorem. It is relatively consistent with ZF that the Borel hierarchy on the real line has &omega2 levels.
This partially answers a question raised by Peter Komjath. A similar argument produces models of ZF in which the Borel hierarchy on the real line has length any given limit ordinal less than &omega2, e.g., &omega or &omega1+&omega1.