Thursday, 8.06. 2017, 10¹⁵, room 105, Hugh Wark (Aviva UK Life) "A non separable uniformly convex Banach space on which there are few operators"

Abstract: "It will be shown using Shelah’s extension of an “anti-Ramsey” theorem of Todorcevic that there exists a uniformly convex Banach space of density character ω₁ on which every bounded linear operator is the sum of a scalar multiple of the identity and an operator of separable range".

Thursday, 1.06. 2017, 10¹⁵, room 105, Piotr Koszmider (IMPAN) "Massive quotients of algebras in various models" continuation from 18.05

Thursday, 25.05. 2017, 10¹⁵, room 105, Maciej Malicki (Warsaw School of Economics) "Generic representations of countable groups"

Abstract: Suppose that X is a mathematical object, and let Aut(X) denote the group of all automorphisms of X. A representation of a group G on X is a homomorphism of G into Aut(X), and the space of all such representations is denoted by Rep(G,X). Typically, X is a Hilbert space H (then Aut(X) is the unitary group U(H)) but one can consider other situations as well, e.g. X being the Urysohn space U or a countable structure such as the natural numbers with equality (then Aut(X) is the symmetric group of all permutations of natural numbers) or the random graph.

If G is countable and discrete, and Aut(X) is endowed with the topological structure of a Polish group, Rep(G,X) can be endowed with a Polish topology as well. Then, for a given G and X, it is natural to ask whether there exists a generic representation, that is, whether there exists a comeager orbit under the conjugation action of Aut(X) on Rep(G,X). After an introduction to this topic, I will discuss the case of X equal to U or H. For U, there never exists a generic representation, and similarly for H, if G is a group with the Haagerup property and with densely many finite dimensional representations in the unitary dual. The general situation for H is not clear but I will present some results that may either lead to examples of groups with a generic unitary representation or a proof that groups with the Kazhdan property, and with densely many finite dimensional representations in the unitary dual, do not exist. The latter would answer a question of Lubotzky and Shalom.

This is joint work in progress with Michal Doucha.

Thursday, 18.05. 2017, 10¹⁵, room 105, Piotr Koszmider (IMPAN) "Massive quotients of algebras in various models"

Abstract: We will discuss some questions, results and proofs concerning the dependence of properties of massive quotients of algebras (P(N)/Fin, l_∞/c₀, B(l₂)/K(l₂)) on additional set-theoretic assumptions. In particular we are interested in the question what is the corona algebra M(A∩A*)/A∩A* where A is a maximal left-ideal of the Calkin algebra or any SAW* C*-algebra if A∩A* is nonunital (cf. it is one dimensional if A is II₁-factor (Sakai) or if A is C(K) for K extremally disconnected with no isolated points).
We will spend the main part of the talk on the proofs of the corresponding commutative results showing that the Cech-Stone reminder of N*-{x} may have just one point (Gillman) or may be a nonmetrizable compact space (Malykhin), depending on consistent set-theoretic assumptions, where N* is the Gelfand space of the Banach algebra l_∞/c₀ or the Stone space of the Boolean algebra P(N)/Fin, i.e., N*=βN-N. To follow the talk no foundational knowledge is necessary as we will use combinatorial characterizations of P(N)/Fin in various models due to Parovichenko (CH) and Dow-Hart (Cohen model).

Thursday, 11.05. 2017, 10¹⁵, room 105, Witold Marciszewski (MIMUW) "Twisted sums of c₀ and C(K) spaces"

Abstract: "A twisted sum of Banach spaces Z and Y is a short exact sequence

We investigate the following problem posed by Cabello Sanchez, Castillo, Kalton, and Yost:

Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c₀ and C(K)?

Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2^ω₁ or K is a separable scattered compact space of height 3 and weight ω₁, then every twisted sum of c₀ and C(K) is trivial.

This is a joint research with Grzegorz Plebanek."

Thursday, 04.05. 2017, 10¹⁵, room 105, Saeed Ghasemi (IMPAN) "SAW* and corona algebras"

Abstract: "SAW*-algebras were introduced by G. K. Pedersen as noncommutative analogous of sub-Stonean spaces (F-spaces) i.e., spaces for which two disjoint open, σ-compact sets have disjoint closures. Many properties of sub-Stonean spaces were generalized to general SAW*-algebras. For example Pedersen showed that the corona algebras of sigma-unital C*-algebras are SAW*, which generalizes the fact that Cech-Stone remainders of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces. I will show that SAW*-algebras can not be isomorphic to C*-tensor products of two infinite-dimensional C*-algebras. This in particular answers a question of Simon Wassermann who conjectured that the same is true for the Calkin algebra. It also generalizes the fact that the product of two sub-Stonean spaces is not sub-Stonean."

Thursday, 27.04. 2017, 10¹⁵, room 105, Tristan Bice (IMPAN) "Domains and Distances"

Abstract: "We aim to give a gentle introduction to the theory of domains - directed complete continuous posets. The emphasis will be on definitions and examples. We then discuss generalisations where order relations are replaced by non-symmetric distances, with examples from C*-algebras."

Friday, 21.04. 2017, 9³⁰, room 105, Damian Sobota (Kurt Godel RC, Vienna) "On ultrafilters related to Rosenthal's lemma"

Note unusual time!

Abstract: "An infinite matrix (m_{k, n}) with real entries is called Rosenthal if m_{k, n} ≥ 0 for every k, n ∈ N and ∑_{n∈ N}m_{k, n} ≤ 1 for every k ∈ N. A non-empty family F of infinite subsets of natural numbers is called Rosenthal if for every Rosenthal matrix and ε>0 there exists A in F such that for every k in A the following inequality holds: ∑_{n∈ A, n≠k}m_{k, n}<ε. Well-known and useful Rosenthal's lemma states that the family of all infnite subsets of natural numbers is Rosenthal. In my PhD thesis I proved that every selective ultrafilter is Rosenthal (assuming such ultrafilters exist). During the talk I will show a construction of an ultrafilter which is simultaneously: 1) a Rosenthal family and 2) a P-point but 3) not a Q-point (hence it is not selective). I will also make some comments suggesting that probably every ultrafilter is a Rosenthal family."

Thursday, 13.04. 2017, 10¹⁵, room 105, Eva Pernecká (IMPAN) "Weak sequential completeness of Lipschitz-free spaces (part two)" (joint work with T. Kochanek)

Abstract: "Adapting the proof of a classical Bourgain's result, we have extended an earlier work due to Cúth, Doucha and Wojtaszczyk, and we have proved that for every compact subset M of a superreflexive Banach space the Lipschitz-free (Arens-Eels) space Æ(M) is weakly sequentially complete. So, in particular, it does not contain a copy of c₀. In the second part of the talk, we shall present the proof of the announced result."

Thursday, 6.04. 2017, 10¹⁵, room 105, Tomasz Kochanek (IMPAN) "Weak sequential completeness of Lipschitz-free spaces (part one)" (joint work with E. Pernecká)

Abstract: Adapting the proof of a classical Bourgain's result, we have extended an earlier work due to Cúth, Doucha and Wojtaszczyk, and we have proved that for every compact subset M of a superreflexive Banach space the Lipschitz-free (Arens-Eels) space Æ(M) is weakly sequentially complete. So, in particular, it does not contain a copy of c₀. In the first part of the talk, we shall explain the role of the superreflexivity assumption and connections between our result and a still open question of Dutrieux and Ferenczi about structural properties of Arens-Eels spaces. We will prove a certain combinatorial lemma about geometry of superreflexive spaces which strengthens one of classical James' characterizations. We will also provide several nontrivial examples of metric spaces M to which our result applies, and which extend the list of hitherto known examples of M with Æ(M) being weakly sequentially complete like, e.g., uniformly discrete spaces, snowflakings and finite-dimensional cubes. Finally, we will present some ideas for continuation of this research.

Talks in the first semester of 2016-17.

Talks in the second semester of 2015-16.

Talks in the first semester of 2015-16.

Talks in the second semester of 2014-15.

Talks in the first semester of 2014-15.

Talks in the second semester of 2013-14.

Talks in the first semester of 2013-14.

Talks in the second semester of 2012-13.

Talks in the first semester of 2012-13.

Talks in the second semester of 2011-12.

Talks in the first semester of 2011-12.