impan seminar:

working group in applications of set theory


Previous talks this semester:

24. 05. 2012.

Maciej Malicki (Łazarski) and Marcin Sabok (IMPAN and UWr) on the work I. Ben Yaacov, A. Berenstein, J. Melleray: Polish topometric groups (Continuation)

17. 05. 2012.

Maciej Malicki (Łazarski) on the work I. Ben Yaacov, A. Berenstein, J. Melleray Polish topometric groups

10. 05. 2012.

David Guerrero Sanchez (Universidad de Murcia and UNAM Mexico) talked on some covering properties of weak topologes based on A. Dow, H. Junnila, J. Pelant, Weak covering properties of weak topologies. Proc. London Math. Soc. (3) 75 (1997), no. 2, 349368.

19. 04. 2012.

Mirna Dzamonja (University of East Anglia, Norwich)

Isomorphic embeddings of Banach spaces and universality questions

Universality questions abound in mathematics. In the most general form they are formulated as follows: given a class C of objects and a notion of quasi-order <= between them, find a subclass D of smallest cardinality which has the property that every element of C is <= an element of D. An example of this in the theory of Banach spaces is the search for an isomorphically or an isometrically universal element in the class of Banach spaces in a given density. There are interesting independence results in the theory of isomorphically universal Banach spaces (see the work of Brech and Koszmider) and on the other hand, model theoretic results (Shelah and Usvyatsov) in the theory of isometrically universal spaces. The latter have the advantage that in a certain sense they are absolute, specifically, they depend only on cardinal arithmetic and not on the set-theoretic universe, but a disadvantage that they cannot talk about isomorphism, only about isometry. We would like to have means to obtain similar "semi-absolute" results in the isomorphic theory, and we present some initial results in this direction.

12. 04. 2012.

Tomasz Weiss (UPHS):

On meager additive and null additive sets in the Cantor space 2w and in R.

We will say that a subset X of 2w (R) is meager additive (respectively, null additive) iff for every meager (respectively, every null) set A, X+A is meager (respectively, null).
The following question was asked by several set theorits (among them T. Bartoszynski): Suppose that there exists an uncountable meager (respectively, null) additve set in 2w. Is it true that there is an uncountable meager (respecyively, null) additive set in R? And how about the converse implication?
We will sketch the startegy that solves 3/4 of the above problem and we will present the main arguments that give (hopefully) a complete solution.

29. 03. 2012.

Wiesław Kubiś (Czech Academy of Sciences and UJK Kielce):

On a class of Banach spaces generated by pushout iterations.

We describe a general way of constructing Banach spaces by using the operation of a pushout with finite-dimensional spaces. This method leads in particular to non-separable variants of the Gurarii space. We shall show that for every cardinal kappa not smaller than the continuum, there exists a unique Banach space of density kappa in this class, containing isometric copies of all finite-dimensional spaces and homogeneous for finite-dimensional subspaces. Our main arguments are of category-theoretic nature and therefore can be adapted to Boolean algebras and other structures. In paritcular, we are able to significantly improve the results of a recent work of A. Aviles and C. Brech (Topology and its Applications 158 (2011) 1534-1550). Preprint.

15. 03. 2012.

Piotr Koszmider: On the operator algebra of the Banach space of continuous functions on the first uncountable ordinal w1

Continuation from the previous week.

08. 03. 2012.

Piotr Koszmider: On the operator algebra of the Banach space of continuous functions on the first uncountable ordinal w1

We will show a dichotomy for weak* compact subsets in the space of measures on w1. This will allow to get new results on the internal structure of the operators in the above mentioned Banach space. And this in turn has applications in the structure of the operator algebra. We will focus on the combinatorial aspects of the results.

Based on joint work with T. Kania and N. Laustsen from Lancaster University

01. 03. 2012.

Marcin Sabok: Automatic continuity for U(l2) - Continuation from January.

We will discuss a recent paper of Tsankov, which proves that the group of unitary operators on the infinite dimensional separable Hilbert space has the automatic continuity property. This means that arbitrary algebraic homomorphism from this group into a separable group is continuous. Automatic continuity property has many interesting consequences, for example, it implies uniqueness of a Polish group topology on the group if such topology exists.

Based on T. Tsankov; Automatic continuity for the unitary group

Talks in the first semester of 2011-12.

Time and place: Thursdays 10.00- 13.00, room 408

The scope of the seminar: Set-theoretic methods in diverse fields of mathematics, in particular in Banach spaces, C*-algebras, topology, the structure of the real line, measure theory, Boolean algebras, topological groups. Here we include both the developing of such methods as forcing, descriptive set theory, Ramsey theory as well as their concrete applications in the fields mentioned above.

Working group style: We will make efforts so that this seminar has more a working character rather than the presentation style. This means that we encourage long digressions, discussions, background preparations and participation of everyone. To achive this, the duration of the seminar will not be rigidly fixed and may reach even 3 hours with the breaks. We would like to immerse ourselves into the details of the mathematical arguments studied. We hope that such a seminar may complement presentations at the Wednesday set theory seminar at the University of Warsaw.

Participants this semester:

  • Zofia Adamowicz (IMPAN)
  • Joanna Jureczko (UKSW)
  • Piotr Koszmider (IMPAN)
  • Adam Kwela (Ph. D. student IMPAN)
  • Michał Korch (Ph. D. student UW)
  • Adam Krawczyk (UW)
  • Mikołaj Krupski (Ph. D. student IMPAN)
  • Maciej Malicki (Łazarski)
  • Henryk Michalewski (IMPAN and UW)
  • Claudia Corrêa de Andrade Oliveira (Ph. D. student USP)
  • Marcin Sabok (IMPAN and UWr)
  • Adam Skalski (IMPAN)
  • Tomasz Weiss (UPHS)
  • Rafał Zduńczyk (UŁ)
Forthcoming talks:

No more talks this semester