impan seminar:

working group in applications of set theory



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".

Previous talks this semester

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

0→Z→X→Y→0,
where X is a Banach space and the maps are bounded linear operators. Such a twisted sum is called trivial if the exact sequence splits, i.e. if the map Z → X admits a left inverse (in other words, if the map X → Y admits a right inverse). This is equivalent to saying that the range of the map Z → X is complemented in X.

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 2ω1 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 (mk, n) with real entries is called Rosenthal if mk, n ≥ 0 for every k, n ∈ N and ∑n∈ Nmk, 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≠kmk, 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.

Time and place: Thursdays 10.15-12.00 am, room 105, Sniadeckich 8


The scope of the seminar: Set-theoretic combinatorial and topological methods in diverse fields of mathematics, with a special emphasis on abstract analysis like Banach spaces, Banach algebras, C*-algebras, 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. We would like to immerse ourselves into the details of the mathematical arguments studied. Also the talks are usualy devoted to research in progress or fascinating results leading to some project not yet resolved. While ready final results could be presented at other seminars at IM PAN or UW.

Participants this semester so far:

  • Tristan Bice (IM PAN/WCNM)
  • Tadeusz Dobrowolski (Pittsburg State U.)
  • Saeed Ghasemi (IM PAN/WCNM)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Eva Pernecká (IM PAN)
  • Tatiana Shulman (IM PAN)
  • Karen Strung (IM PAN)
  • Jarosław Swaczyna (Lodz UT)
Forthcoming talks :

  • 15.06.2017 Corpus Christi. No seminar.
  • 22.06.2017 No seminar due to the Scientific Council of IMPAN
  • No more seminars this semester