April 2015

Wojciech Bielas: An example of a rigid \(\kappa\)-superuniversal metric space

Tuesday, May 5, 2015 17:15

Room: D1-215

Speaker:
Wojciech Bielas

Title: An example of a rigid \(\kappa\)-superuniversal metric space

Abstract. For an uncountable cardinal \(\kappa\) a metric space \(X\) is called to be \(\kappa\)-superuniversal if for every metric space \(Y\) with \(|Y | < \kappa\) every partial isometry from a subset of \(Y\) into \(X\) can be extended over the whole space \(Y\). It is easy to prove that if a \(\kappa\)-superuniversal metric space is of cardinality \(\kappa\), then it is also \(\kappa\)-homogeneous, i.e. every isometry of a subspace \(Y\) of the space with \(|Y | < \kappa\) can be extended to an isometry of the whole space. I will discuss an example of a \(\kappa\)-superuniversal metric space which has exactly one isometry.

Filip Strobin: Spaceability of particular family of continuous functions with nowhere continuous inverses

Tuesday, April 28, 2015 17:15

Room: D1-215

Speaker:
Filip Strobin

Title: Spaceability of particular family of continuous functions with nowhere continuous inverses

Abstract. We will show that the family of continuous injections \(T:l_p\to l_p\) (where \(p>0\)) with nowhere continuous inverses (together with zero function) contains isometric copy of \(l_p\). In particular, it means that the set of such functions is spaceable.

Mirna Džamonja: WQOs, FACs and their width

Tuesday, April 21, 2015 17:15

Room: D1-215

Speaker:
Mirna Džamonja

Title: WQOs, FACs and their width

Abstract. A quasi-order is WQO if it has no infinite antichains or infinite decreasing sequences. A partial order is FAC if it has no infinite antichains. These restrictions on the orders mean that there are several naturally defined ordinal valued ranks that can be used to study them, for example, the rank of the tree of antichains, called the width. These ranks have been studied from the point of view of order theory, Ramsey theory, and also the theory of algorithms, since it turns out that a large class of « well structured systems « of algorithms can be modeled using the wqo. We shall present certain structural results connecting FAC and WQO orders and then some calculations of the ranks. The new results presented in the talk come from a collaborative work with Schnoebelen and Schmitz.

Katarzyna Chrząszcz: On some properties of microscopic sets

Tuesday, April 14, 2015 18:45

Room: D1-215

Speaker:
Katarzyna Chrząszcz

Title: On some properties of microscopic sets

Abstract: The notion of microscopic set appeared for the first time in paper 'Insiemi ed operatori “piccoli” in analisi funzionale' (APPELL, J., Rend. Istit. Mat. Univ. Trieste 33 (2001), 127–199).
 
Def. A set  \(A\subseteq\mathbb{R}\) is called microscopic if for every  \(\varepsilon>0\) there exists a sequence of segments \((I_n)_{n\in\mathbb{N}}\) such that \(A\subseteq\bigcup\limits_{n\in\mathbb{N}} I_n\) and \(|I_n|\leq\varepsilon^n\) for \(n\in\mathbb{N}\).

We will give generalizations of given notion to the case of arbitrary metric space. We will analyze algebraic and set-theoretic properties of the family of microscopic sets.

Marek Bienias: General methods in algebrability

Tuesday, April 14, 2015 17:15

Room: D1-215

Speaker:
Marek Bienias

Title: General methods in algebrability

Abstract: During last 15 years new idea of measuring sets appeared and become popular.

Def.
Let \(\kappa\) be a cardinal number and let \(\mathcal{L}\) be a commutative algebra. Assume that \(A\subseteq\mathcal{L}\). We say that \(A\) is:
  • \(\kappa\)-algegrable if \(A\cup \{0\}\) contains \(\kappa\)-generated algebra \(B\);
  • strongly \(\kappa\)-algegrable if \(A\cup \{0\}\) contains \(\kappa\)-generated free algebra \(B\).

In many recent articles authors studied algebrability of sets naturally appering in mathematical analysis. It seems that required results are the general methods of algebrability which can cover known methods and give new constructions.

We will describe two methods:  independent Bernstein sets and exponential like. They let us prove many results concerning algebrability and strong algebrability of subsets of algebras \(\mathbb{R}^\mathbb{R}\), \(\mathbb{C}^\mathbb{C}\), \(\mathbb{R}^\mathbb{N}\), \(C[0,1]\), \(\mathcal{l}_\infty\). Most of presented applications give the best possible result in terms of complication of built algebraic structure and cardinality of set of generators of this structure (in most cases \(\mathfrak c\) or \(2^{\mathfrak c}\)).