Previous talks this semester
Thursday, 1.06. 2017, 10^{15}, room 105, Piotr Koszmider (IMPAN)
"Massive quotients of algebras in various models" continuation from 18.05
Thursday, 25.05. 2017, 10^{15}, 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, 10^{15}, 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_{∞}/c_{0}, B(l_{2})/K(l_{2}))
on additional settheoretic assumptions. In particular we are interested in the question
what is the corona algebra M(A∩A*)/A∩A* where A is a maximal leftideal of the Calkin algebra or
any SAW* C*algebra if A∩A* is nonunital (cf. it is one dimensional if A is II_{1}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 CechStone reminder of N*{x} may have just one point (Gillman) or may be a nonmetrizable compact space (Malykhin),
depending on consistent settheoretic assumptions, where N* is the Gelfand space of the Banach algebra
l_{∞}/c_{0} or the Stone space of the Boolean algebra P(N)/Fin, i.e., N*=βNN. 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 DowHart (Cohen model).
Thursday, 11.05. 2017, 10^{15}, room 105, Witold Marciszewski (MIMUW)
"Twisted sums of c_{0} 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 c_{0} and C(K)?
Using additional settheoretic 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 ω_{1},
then every twisted sum of c_{0} and C(K) is trivial.
This is a joint research with Grzegorz Plebanek."
Thursday, 04.05. 2017, 10^{15}, room 105, Saeed Ghasemi (IMPAN)
"SAW* and corona algebras"
Abstract:
"SAW*algebras were introduced by G. K. Pedersen as noncommutative
analogous of subStonean spaces (Fspaces) i.e., spaces for which two disjoint open,
σcompact sets have disjoint closures.
Many properties of subStonean spaces were generalized to general SAW*algebras.
For example Pedersen showed that the corona algebras of sigmaunital C*algebras are SAW*,
which generalizes the fact that CechStone remainders of locally
compact σcompact Hausdorff spaces are subStonean spaces.
I will show that SAW*algebras can not be isomorphic to C*tensor products of two infinitedimensional 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 subStonean
spaces is not subStonean."
Thursday, 27.04. 2017, 10^{15}, 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 nonsymmetric distances, with examples from C*algebras."
Friday, 21.04. 2017, 9^{30}, 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 nonempty 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. Wellknown 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
Ppoint but 3) not a Qpoint (hence it is not selective). I will also
make some comments suggesting that probably every ultrafilter is a
Rosenthal family."
Thursday, 13.04. 2017, 10^{15}, room 105, Eva Pernecká (IMPAN)
"Weak sequential completeness of Lipschitzfree 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 Lipschitzfree (ArensEels) space Æ(M) is weakly sequentially complete.
So, in particular, it does not contain a copy of c_{0}.
In the second part of the talk, we shall present the proof of the announced result."
Thursday, 6.04. 2017, 10^{15}, room 105, Tomasz Kochanek (IMPAN)
"Weak sequential completeness of Lipschitzfree 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 Lipschitzfree (ArensEels) space Æ(M) is weakly sequentially complete.
So, in particular, it does not contain a copy of c_{0}. 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 ArensEels 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 finitedimensional cubes.
Finally, we will present some ideas for continuation of this research.
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.
