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, 349–368.
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.
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