June 16, 2016, 11¹⁵-13, room 105, Tomasz Kochanek (IM PAN/UW)

Title: Structural properties of Banach spaces in terms of certain classes od Baire-1 functions"
Abstract: "We will discuss some connections between structural properties of separable Banach spaces and certain classes of functions acting on the unit dual ball with the weak* topology. One of the most classical results in this spirit, due to Odell and Rosenthal, says that a separable Banach space X contains an isomorphic copy of l₁ if and only if the unit ball of X** contains a non-Baire-1 element. We will describe another class of functions which is responsible for containing a copy of c₀ (via Rosenthal's c₀-theorem) and leads quite naturally to other more exotic classes which are responsible for producing spreading models equivalent to the canonical bases of l₁ or c₀. The talk will be largely based on the paper by R. Haydon, E. Odell and H. Rosenthal, "On certain classes of Baire-1 functions with applications to Banach space theory", Lecture Notes in Math. 1470 (1991), 1-35."

June 9, 2016, 11¹⁵-13, room 105, Marcin Sabok (McGill/IM PAN)

Title: Classification of operator systems
Abstract: I will discuss the problem of classification of separable operator systems up to complete order isomorphism. While the relation for arbitrary separable operator systems is a complete orbit equivalence relation, the restriction to the class of finite-dimensional ones gives a smooth equivalence relation. This is joint work with M. Argerami, S. Coskey, M. Kennedy, M. Kalantar and M. Lupini."

June 2, 2016, 11¹⁵-13, room 321, Damian Sobota (Ph.D. student IM PAN) NOTE THE UNUSUAL ROOM!

Title: Phillips's lemma, Schur's theorem and cardinal invariants of the continuum.
Abstract: Phillips's lemma asserts that for every sequence of measures (μ_n) on P(N) satisfying lim_n μ_n(A)=0 for every subset A of N, we have: lim_n Σ_j |μ_n({j})|=0. The lemma has many consequences in vector measure theory and Banach space theory. E.g. Schur's theorem stating that every weakly convergent sequence in l₁ is norm convergent is an easy application of Phillips's lemma. During my talk I will show in ZFC that there exists a family F of subsets of N such that |F|=cof(N) (the cofinality of the Lebesgue measure zero ideal) and satisfying the following Phillips-like condition: for every sequence of measures (μ_n) on P(N) satisfying lim_n μ_n(A)=0 for every A in F, we have: lim_n Σ_j |μ_n({j})|=0. On the other hand, I will also show that no family F of subsets of N such that |F|<p (the pseudo-intersection number) can be used."

May 19, 2016, 11¹⁵-13, room 105, Maciej Malicki (The Warsaw School of Economics)

Title: Amenable groups.
Abstract: I will continue an overview of some aspects of classical and contemporary studies of amenable groups. This time I will focus on properties of universal minimal flows of non-archimedean groups (i.e. automorphism groups of countable models), and the notion of extreme amenability. Recall that a flow is a continuous action of a topological group on a compact space. A flow is called minimal if all its orbits are dense. For every topological group there exists a universal such flow, and, it turns out that there exist many "large" groups - called extremely amenable - for which it is trivial. I will explain how to construct the universal minimal flow for a non-archimedean group of the form Aut(M) using Stone-Čech compactifications, and how to relate extreme amenability to certain properties of ultrafilters naturally associated with the model M. This connection is based on Ramsey theory."

May 12, 2016, 11¹⁵-13, room 105, Gonzalo Martínez Cervantes (Ph. D, student, Murcia)

Title: Weakly Radon-Nikodým compact spaces.
Abstract: A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of l₁. In this talk I will show some topological properties of this class of compact spaces and its relation with other classes of compact spaces such as Radon-Nikodým or Corson compacta. Most of the results of this talk are contained in the paper 'On weakly Radon-Nikodým spaces' which is available on arxiv.org.

May 5, 2016, 11¹⁵-13, room 105, Przemysław Ohrysko (Ph. D. student, IM PAN)

Title: The Gelfand space of the measure algebra.
Abstract: In this talk I would like to present some recent results concerning the Banach algebra M(T) of Borel regular measures on the circle group with the convolution product. Since it is well-known that the spectrum of a measure can be much bigger then the closure of the values of its Fourier-Stieltjes transform (the Wiener-Pitt phenomenon) it is natural to ask what kind of topological properties of the Gelfand space Ge(M(T)) are responsible for this unusual spectral behaviour. It follows immediately from the existence of the Wiener-Pitt phenomenon that the set Z identified with Fourier-Stieltjes coefficients is not dense in Ge(M(T)). However, it is not clear if any other countable dense subset of this space exists. During my talk, I will disprove this fact - i.e. I will show the non-separability of the Gelfand space of the measure algebra on the circle group. This result is contained in a paper 'On topological properties of the measure algebra on the circle group' written in a collaboration with Michał Wojciechowski which has not been published yet but is available on arxiv.org with identifier: 1603.05864.

April 28, 2016, 11¹⁵-13, room 105, Tomasz Kochanek (IMPAN/UW)

Title: The Szlenk power type of injective tensor products of Banach spaces.
Abstract: We shall discuss the notion of the Szlenk power type (strictly related to the Szlenk index) and its relationships with asymptotic geometry of Banach spaces. In particular, we will describe how the so-called tree maps and subsequential tree estimates can be used as tools to understand the dynamics of the Szlenk derivations. Next, we will prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with the Szlenk index at most ω (a joint result with S. Draga). This allows us, for example, to determine the moduli of asymptotic smoothness of the spaces of compact operators between l_p-spaces.

April 21, 2016, 11¹⁵-13, room 105, Gonzalo Martínez Cervantes (Ph. D, student, Murcia)

Title: Riemann integrability versus weak continuity.
Abstract: We introduce some properties of Banach spaces related with Riemann integrability and we study the relation between weak-continuity and Riemann integrability. In particular, a Banach space is said to have the weak Lebesgue property if every Riemann integrable function from the unit interval into it is weakly continuous almost everywhere. We present several results concerning the weak Lebesgue property.

April 7, 2016, 11¹⁵-13, room 105, Tomasz Żuchowski (Ph. D. student, UWr)

Title: Nonseparable growth of N supporting a strictly positive measure.
Abstract: We will construct in ZFC a compactification γN of N such that its remainder γN\N is not separable and carries a strictly positive measure, i.e. measure positive on all nonempty open subsets. Moreover, the measure on our space is defined by the asymptotic density of subsets of N. Our remainder is the Stone space of some Boolean subalgebra of the algebra Bor(2^N) of all Borel subsets of 2^N containing all clopen sets. This line of research is motivated by the problem of characterizing the Banach spaces c₀⊆ X⊆ l_∞ such that the space c₀ is complemented in X.

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.