May 2022
Rafał Filipów: Does there exist a Hindman space which is not a van der Waerden space?
30/05/22 09:19
Tuesday, May 31, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Rafał Filipów (University of Gdańsk)
Title: Does there exist a Hindman space which is not a van der Waerden space?
Abstract: Both Hindman spaces and van der Waerden spaces were defined by M. Kojman (Proc. AMS 130(2002), no. 3 and no. 6) with the aid of Hindman's finite sum theorem and van der Waerden's theorem on arithmetic progressions, respectively. Then M. Kojman and S. Shelah (Proc. AMS 131(2003), no. 5) proved that there exists a van der Waerden space which is not a Hindman space. The question whether there exists a Hindman space which is not a van der Waerden space is still open. In my talk I will show how this question about topological spaces can be reduced to a question only about Katetov order of two ideals of subsets of \(\mathbb{N}\). This result is from our joint paper with K. Kowitz, A. Kwela nad J. Tryba (Proc. AMS 150(2022), no. 2).
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Rafał Filipów (University of Gdańsk)
Title: Does there exist a Hindman space which is not a van der Waerden space?
Abstract: Both Hindman spaces and van der Waerden spaces were defined by M. Kojman (Proc. AMS 130(2002), no. 3 and no. 6) with the aid of Hindman's finite sum theorem and van der Waerden's theorem on arithmetic progressions, respectively. Then M. Kojman and S. Shelah (Proc. AMS 131(2003), no. 5) proved that there exists a van der Waerden space which is not a Hindman space. The question whether there exists a Hindman space which is not a van der Waerden space is still open. In my talk I will show how this question about topological spaces can be reduced to a question only about Katetov order of two ideals of subsets of \(\mathbb{N}\). This result is from our joint paper with K. Kowitz, A. Kwela nad J. Tryba (Proc. AMS 150(2022), no. 2).
Krzysztof Leśniak: Enriching IFS fractals with symmetry
23/05/22 08:00
Thursday, May 26, 2022 17:15
Location: room P.01 C-11
Speaker: Krzysztof Leśniak (Nicolaus Copernicus University in Toruń)
Title: Enriching IFS fractals with symmetry
Abstract: Let \(\mathcal{F}=(X; f_i:i\in I)\) be an iterated function system (IFS) consisting of a finite number of Banach contractions \(f_i\) acting on a complete metric space \(X\). According to the seminal result of Hutchinson (1981), \(F\) admits an attractor, denoted by \(A_{\mathcal{F}}\). Let \(g:X\to X\) be a \(p\)-periodic isometry, \(p>1\), which admits a (not necessarily unique) fixed point.
Proposition: The IFS \(\widetilde{\mathcal{F}} =F \cup \{g\}\) admits a semiattractor \(A^{\flat}\) (in the Lasota—Myjak sense) which is compact and \(g\)-symmetric.
The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.
Theorem (L & Snigireva): \(A^{\flat}\) is an attractor of any of the following contractive IFSs
\begin{eqnarray*}
\mathcal{G}
= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),
%\label{eq:IFS-Gconj}
\\
GF
= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),
%\label{eq:IFS-GF}
\\
\widehat{\mathcal{G}} =
(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).
%\label{eq:IFS-Gdoubletilde}
\end{eqnarray*}
Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).
Remark: \(\widehat{\mathcal{G}}\) appears in Symmetry in Chaos by Field & Golubitsky, cf. http://larryriddle.agnesscott.org/ifskit/IFShelp/howtoCreateSymmetricFractal.html by L.R. Riddle
The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).
Location: room P.01 C-11
Speaker: Krzysztof Leśniak (Nicolaus Copernicus University in Toruń)
Title: Enriching IFS fractals with symmetry
Abstract: Let \(\mathcal{F}=(X; f_i:i\in I)\) be an iterated function system (IFS) consisting of a finite number of Banach contractions \(f_i\) acting on a complete metric space \(X\). According to the seminal result of Hutchinson (1981), \(F\) admits an attractor, denoted by \(A_{\mathcal{F}}\). Let \(g:X\to X\) be a \(p\)-periodic isometry, \(p>1\), which admits a (not necessarily unique) fixed point.
Proposition: The IFS \(\widetilde{\mathcal{F}} =F \cup \{g\}\) admits a semiattractor \(A^{\flat}\) (in the Lasota—Myjak sense) which is compact and \(g\)-symmetric.
The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.
Theorem (L & Snigireva): \(A^{\flat}\) is an attractor of any of the following contractive IFSs
\begin{eqnarray*}
\mathcal{G}
= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),
%\label{eq:IFS-Gconj}
\\
GF
= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),
%\label{eq:IFS-GF}
\\
\widehat{\mathcal{G}} =
(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).
%\label{eq:IFS-Gdoubletilde}
\end{eqnarray*}
Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).
Remark: \(\widehat{\mathcal{G}}\) appears in Symmetry in Chaos by Field & Golubitsky, cf. http://larryriddle.agnesscott.org/ifskit/IFShelp/howtoCreateSymmetricFractal.html by L.R. Riddle
The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).
David Chodounsky: Sacks indestructible ultrafilters and reaping families
19/05/22 14:44
Tuesday, May 24, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: David Chodounsky (Czech Academy of Sciences)
Title: Sacks indestructible ultrafilters and reaping families
Abstract: Preservation of reaping families and especially ultrafilters on countable sets is a well studied theme in set theory of the reals. A. Miller proved that if an ultrafilter remains a reaping family in some forcing extension, then it has to be also Sacks indestructible. The existence of Sacks indestructible ultrafilters in ZFC is an open question. A related problem is Sacks indestructibility of reaping families which are complements of ideals. We prove that complements of most classical ideals are indestructible with one notable exception, the ideal of sets asymptotic density zero.
The presented results are from an upcoming paper with O. Guzman and M. Hrusak.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: David Chodounsky (Czech Academy of Sciences)
Title: Sacks indestructible ultrafilters and reaping families
Abstract: Preservation of reaping families and especially ultrafilters on countable sets is a well studied theme in set theory of the reals. A. Miller proved that if an ultrafilter remains a reaping family in some forcing extension, then it has to be also Sacks indestructible. The existence of Sacks indestructible ultrafilters in ZFC is an open question. A related problem is Sacks indestructibility of reaping families which are complements of ideals. We prove that complements of most classical ideals are indestructible with one notable exception, the ideal of sets asymptotic density zero.
The presented results are from an upcoming paper with O. Guzman and M. Hrusak.
Mirna Dzamonja: Reasonable structures of size \(\aleph_1\)
13/05/22 08:27
Tuesday, May 17, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Mirna Dzamonja (Université deParis-Cité)
Title: Reasonable structures of size \(\aleph_1\)
Abstract: We are interested to develop a theory of structures of size \(\aleph_1\) which are ’tame’ in the sense that they in some sense or other preserve the nice properties that we are used to seeing on the countable structures.
We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Mirna Dzamonja (Université deParis-Cité)
Title: Reasonable structures of size \(\aleph_1\)
Abstract: We are interested to develop a theory of structures of size \(\aleph_1\) which are ’tame’ in the sense that they in some sense or other preserve the nice properties that we are used to seeing on the countable structures.
We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.
Damina Sobota: On sequences of finitely supported measures on products of compact spaces
08/05/22 20:10
Tuesday, May 10, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Damina Sobota (KGRC, Vienna)
Title: On sequences of finitely supported measures on products of compact spaces
Abstract: Cembranos, Freniche, and Khurana (all independently) proved that for every two infinite compact spaces \(K\) and \(L\) the Banach space \(C(K\times L)\) contains a complemented copy of the space \(c_0\). To obtain this copy all the three proofs utilize in some way the Josefson-Nissenzweig theorem which more or less asserts that there is a sequence \((mu_n)\) of normalized signed Radon measures on \(K\times L\) such that \(mu_n(f)\) converges to \(0\) for every \(f\) from \(C(K\times L)\). Since most (if not all) of the known proofs of the J-N theorem are non-constructive, it follows that the (known to me) proofs of Cembranos et al. are also non-constructive. During my talk I'll show a generalization of the theorem of Cembranos et al. whose proof uses a direct construction of a sequence \((mu_n)\) of finitely supported measures on \(K\times L\) as above. I'll also discuss the case of pseudocompact spaces \(K\) and \(L\) and pose some questions.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Damina Sobota (KGRC, Vienna)
Title: On sequences of finitely supported measures on products of compact spaces
Abstract: Cembranos, Freniche, and Khurana (all independently) proved that for every two infinite compact spaces \(K\) and \(L\) the Banach space \(C(K\times L)\) contains a complemented copy of the space \(c_0\). To obtain this copy all the three proofs utilize in some way the Josefson-Nissenzweig theorem which more or less asserts that there is a sequence \((mu_n)\) of normalized signed Radon measures on \(K\times L\) such that \(mu_n(f)\) converges to \(0\) for every \(f\) from \(C(K\times L)\). Since most (if not all) of the known proofs of the J-N theorem are non-constructive, it follows that the (known to me) proofs of Cembranos et al. are also non-constructive. During my talk I'll show a generalization of the theorem of Cembranos et al. whose proof uses a direct construction of a sequence \((mu_n)\) of finitely supported measures on \(K\times L\) as above. I'll also discuss the case of pseudocompact spaces \(K\) and \(L\) and pose some questions.
Ondrej Zindulka: Microscopic sets, Hausdorff measures and their cardinal invariants
04/05/22 19:27
Monday, May 9, 2022 15:15
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Ondrej Zindulka (Czech Technical University, Prague)
Title: Microscopic sets, Hausdorff measures and their cardinal invariants
Abstract: A set in a metric space is microscopic it admits, for every \(\varepsilon>0\), a cover \(\{E_n\}\) such that the diameter of each \(E_n\) is at most \(\varepsilon^n\). The notion was introduced 21 years ago and since then a number of people contributed to the theory. I will provide a brief account of the state of art and present new results and in particular the so far overlooked relation to Hausdorff measures. Attention will be paid to cardinal invariants of the ideal of microscopic sets and sets of Hausdorff measure zero in metric spaces and Polish groups.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Ondrej Zindulka (Czech Technical University, Prague)
Title: Microscopic sets, Hausdorff measures and their cardinal invariants
Abstract: A set in a metric space is microscopic it admits, for every \(\varepsilon>0\), a cover \(\{E_n\}\) such that the diameter of each \(E_n\) is at most \(\varepsilon^n\). The notion was introduced 21 years ago and since then a number of people contributed to the theory. I will provide a brief account of the state of art and present new results and in particular the so far overlooked relation to Hausdorff measures. Attention will be paid to cardinal invariants of the ideal of microscopic sets and sets of Hausdorff measure zero in metric spaces and Polish groups.