Previous talks this semester:
Thursday, 07.06. 2018, 10:15, room 105, Michał Tomasz Godziszewski
(Ph. D. student IMPAN) "The rearrangement number"
Abstract: "The Riemann Series Theorem says that for any conditionally convergnet
series Σ_{n ∈ N}a_{n} of real numbers and for any number
S in the extended real line there exists a permutation σ of N such that the series
Σ_{n ∈ N} a_{σ(n)}
converges to S or diverges by oscillation.
A natural settheoretic question is: what is the minimal cardinality of a family C
of permuations of natural numbers such that for any conditionally convergent
series Σ_{n ∈ N}a_{n} of real numbers there is a permutation σ
in C such that the series Σ_{n ∈ N} a_{σ(n)} converges to a different real number, diverges to infinity or diverges by oscillation? Such a minimal cardinality is denoted by the rearrangement number rr.
The question has been investigated in a recent paper by A. Blass, J. Brendle, W. Brian,
J. Hamkins, M. Hardy, P. Larson on "The Rearrangement Number".
During the talk, we will demonstrate some results from the paper, in particular comparing (in ZFC) the number rr and its variants to some wellknown cardinal characteristics of the continuum, as e.g. the bounding number b, the dominating number d, as well as the covering and uniformity for measure and category, i.e. cov(N), cov(M), non(N), non(M).
If the time permits, we will also give some consistency results concerning the rearrangement numbers and show open questions in the subject. "
Thursday, 17.05. 2018, 10:15, room 105, Piotr Koszmider (IMPAN) "Uncountable irredundant sets in large algebras"
Abstract:
"A subset X of a structure A is irredundant if no x from X belongs to the substructure
generated by the remaining elements of X. For example irredundant subsets of Banach spaces correspond to biorthogonal systems,
and it is known that the existence of uncountable biorthogonal systems in nonseparable Banach spaces is undecidable.
Similar results hold for Boolean algebras. In my talk I will focus on irredundant sets in C*algebras. Even in the commutative
case of C(K) it is not known if there is an absolute example of a nonmetrizable compact K such that C(K) has
no uncountable irredundant set. But the totally disconnected case behaves like the Banach space case.
I will start with the review of the commutative case where most of the results are due to Todorcevic and will touch
the noncommutative case where partial results were obtained together with my Ph.d. student from São Paulo, Clayton Suguio Hida"
Thursday, 10.05. 2018, 10:15, room 105, Tomasz Kochanek (IMPAN/UW) "On the property (X) of Banach spaces"
Abstract: "The property (X) was introduced by Godefroy and Talagrand in 1981 in order to study the problem of uniqueness of preduals. It is a relatively simply formulated condition for veryfing whether a given element of the bidual space actually belongs to the original space, and which implies that the underlying Banach space is the strongly unique predual of its dual. We will discuss some reformulations of this property (for example in terms of a certain Edgar's ordering between Banach spaces) and its connections with the better known Pelczynski's properties (V) and (V*). We shall focus, in particular, on the still open problem posed by Godefroy and Lerner whether the Lipschitzfree spaces over finite dimensional cubes have property (X). Some possible approaches will be discussed, based on a joint paper with E. Pernecka, where it was shown that these Lipschitzfree spaces have a somewhat weaker property (V*)."
Thursday, 12.04. 2018, 10:15, room 105, Piotr Koszmider (IMPAN) "Applications of Todorcevic's construction schemes"
Abstract: "Recently Todorcevic (S. Todorcevic, A construction scheme for nonseparable structures. Adv. Math. 313 (2017), 564589.)
proposed combinatorial structures whose existence follows from the diamond principle. These ideas can be traced back to the previous attempts of Magidor
and Malitz (1977), Shelah (1985) and Velleman (1984)), but construction schemes seems to a more elegant framework providing quick constructions of complicated objects
ranging from combinatorics to functional analysis. I will show at least a construction of a Souslin tree from such a scheme."
Thursday, 05.04. 2018, 10:15, room 105, Damian Sobota (TU Wien) "Grothendieck C(K)spaces of small density"
Abstract: "A Banach space X is Grothendieck if every weak* convergent
sequence of bounded functionals on X is weakly convergent. A typical
example of a Grothendieck space is the space l_{∞} or more
generally every Banach space C(K) of realvalued continuous
functions on the Stone space K of a σcomplete Boolean
algebra. During my talk I will investigate relations between densities
of the Grothendieck spaces of the form C(K) for K being the Stone
space of a nonσcomplete Boolean algebra and classical
cardinal characteristics of the continuum."
Thursday, 22.03. 2018, 10:15, room 105, Tomasz Kochanek (IMPAN/UW) "A short proof of a theorem of Pfitzner"
Abstract: "In 1953 Grothendieck proved a classical result on characterization of weakly compact subsets of duals of C(K)spaces. His criterion of (non)weak compactness defines the socalled Pelczynski property (V). A noncommutative analogue was provided by Pfitzner (Math. Ann. 1994) who showed that all C*algebras have that property (V), which can be also expressed by saying that weak compactness in duals of C*algebras is determined commutatively. During the talk, I shall present a more elementary proof of this result due to FernándezPolo and Peralta (Quart. J. Math. 2009).
Thursday, 01.03. 2018, 10:15, room 105, Wieslaw Kubis (Czech Academy and UKSW)
"On the weak amalgamation property "
Abstract: "The somewhat technical weak amalgamation property (weak AP) of
models with embeddings was identified first by Ivanov in 1999 during
his study of generic automorphisms, later rediscovered by Kechris and
Rosendal (2007), again studying generic automorphisms. Recently, we have
discovered, jointly with Adam Krawczyk, that the weak amalgamation
property plays essential role in characterizing the existence of a
winning strategy in the abstract BanachMazur game, played with models
and embeddings. We shall describe the main results relating this game
with the weak AP. Next, we shall present some new examples of hereditary
classes of finite structures with the weak AP, failing its stronger
version called the cofinal AP. The material comes from joint works with
A. Krawczyk, A. Kruckman, and A. Panagiotopoulos."
Talks in the first semester of 201718.
Talks in the second semester of 201617.
Talks in the first semester of 201617.
Talks in the second semester of 201516.
Talks in the first semester of 201516.
Talks in the second semester of 201415.
Talks in the first semester of 201415.
Talks in the second semester of 201314.
Talks in the first semester of 201314.
Talks in the second semester of 201213.
Talks in the first semester of 201213.
Talks in the second semester of 201112.
Talks in the first semester of 201112.
