   Here you may download some of my recent talks given while attending conferences, visiting other universities, or taking part in some special events like the $$\pi$$ Days. Some of results in the conference talks available here are still in progress and being prepared for submission. If you have any questions concerning these materials, please feel free to contact me via e-mail.

ISFE (2012) The talk A variant of the Gleason-Kahane-Żelazko theorem given during the last International Symposium on Functional Equations in Hajdúszoboszló (Hungary), June 2012.

Lancaster (2012) The lecture Stability of vector measures and twisted sums of Banach spaces given during my visit in Lancaster University (UK), in June 2012, at the Pure Mathematics Seminar.

KDWS (2012) The talk Non-trivial extensions of $$c_0$$ by reflexive spaces given during the last Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities in Hajdúszoboszló (Hungary), January 2012.

At the end of this talk you may find the following problem which, I believe, is quite interesting and has an elementary formulation. Here it goes:

Let $$\mathscr{F}$$ be the set algebra given by $$\mathscr{F}=\{A\subset\mathbb{N}\colon\,\vert A\vert<\omega\mbox{ or }\vert\mathbb{N}\setminus A\vert<\omega\}.$$ Is it true that for arbitrarily large number $$M>0$$ there exists a function $$\nu\colon\mathscr{F}\to\ell_2$$ enjoying the following three properties:
• $$\nu$$ is $$1$$-additive, which means that is satifies the inequality $$\|\nu(A\cup B)-\nu(A)-\nu(B)\|\leq 1\quad\mbox{for }A,B\in\mathscr{F},\, A\cap B=\varnothing;$$
• for each $$n\in\mathbb{N}$$ we have $$\nu\{n\}=Me_n$$, where $$e_n$$ stands for the $$n$$th canonical unit vector in $$\ell_2$$;
• $$\nu$$ is a bounded function?

I arrived at this problem when trying to show that every infinite-dimensional Banach space $$X$$ which has the $$\mathsf{SVM}$$ property (see the introduction to this paper) must contain an isomorphic copy of $$c_0$$. The positive answer to the question above would imply that it is the case (the details may be found in the presentation). Moreover, it would have also some nice (and quite strong) consequences in the structural theory of Banach spaces. For instance, the positive answer would, in a sense automatically, generate non-trivial twisted sums of every reflexive space and the space $$c_0$$, that is, it would imply that $$\mathrm{Ext}(c_0,X)\not=0$$ for every reflexive Banach space $$X$$ (for $$X=\ell_2$$ it was shown by F. Cabello Sánchez and J.M.F. Castillo, and it is by no means an easy result).

Several people had looked at this problem but there is no serious progress at the moment. I suspect it may be solved by using some more advanced techniques (like the Kalton and Peck non-trivial twisted sum of $$\ell_2$$ with itself), but maybe it is reachable by elementary methods. Anyway, any ideas would be much appreciated!

$$\pi$$ Days (2012) The lecture Projekt matematyczny (in Polish) given during the last $$\pi$$ Days in the Institute of Mathematics, University of Silesia, March 2012.

Twierdzenie spektralne (2011) The talk Twierdzenie spektralne (in Polish) given during the 31st Students' Math. Society session Motivations, Intuitions, Constructions in Szczyrk, November 2011.

Problem Schrödera-Bernsteina (2011) The talk Problem Schrödera-Bernsteina dla przestrzeni Banacha (in Polish) given during the 30th Students' Math. Society session Pathologies and Paradoxes in Mathematics in Szczyrk, May 2011.