November 2016

# Artur Bartoszewicz: On the sets of subsums of series

Tuesday, November 29, 2016 17:15

Room: D1-215

Speaker:
Artur Bartoszewicz

Title: On the sets of subsums of series

Abstract. The first observations connected with sets of subsums of series (s.c. achievement sets) belong to Kakeya and are over 100 years old. In my lecture I want to present the story of the studies of the problem and the results obtained by my cooperators and me quite recently. These results concern the series generating Cantorvals, connections between the achievement sets of series and the atractors of affine IFS's and achievement sets of conditionally convergent series in the plane.

# Daria Michalik: Degree of homogeneity of connes over locally connected curves

Tuesday, November 22, 2016 17:15

Room: D1-215

Speaker:
Daria Michalik

Title: Degree of homogeneity of connes over locally connected curves

Abstract. $$\mathcal{H}(X)$$ denotes the group of self-homeomorphisms of $$X$$. An orbit of a point $$x_0$$ in $$X$$is the set: $$\mathcal{O}_X(x_0) = \{h(x_0) : h\in\mathcal{H}(X)\}.$$

$$X$$ is $$1/n$$-homogeneous if $$X$$ has exactly $$n$$ orbits. In such a case we say that the degree of homogeneity of $$X$$ equals $$n$$. P. Pellicer Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases. We will generalize above result on connes over locally connected curves.

# Szymon Głąb: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Tuesday, November 15, 2016 17:15

Room: D1-215

Speaker:
Szymon Głąb

Title: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract. Let $$1 \le n \le\omega$$. Let $$A_n$$ be a set of natural numbers less than $$n$$. Define $$<$$ on $$A_n$$ so that for no $$x, y \in A_n$$ is $$x < y$$. Let $$B_n = A_n \times\mathbb{Q}$$ where $$\mathbb{Q}$$ is the set of rational numbers. Define $$<$$ on $$B_n$$ so that $$(k, p) < (m, q)$$ iff $$k = m$$ and $$p < q$$. Let $$C_n = B_n$$ and define $$<$$ on $$C_n$$ so that $$(k, p) < (m, q)$$ iff $$p < q$$. Finally, let $$(D, <)$$ be the universal countable homogeneous partially ordered set, that is a Fraisse limit of all finite partial orders.

A structure is called ultrahomogeneous, if every embedding of its finitely generated substructure can be extended to an automorphism. Schmerl showed that there are only countably many, up to isomorphism, ultrahomogeneous countable partially ordered sets. More precisely he proved the following characterization:

Let $$(H, <)$$ be a countable partially ordered set. Then $$(H, <)$$ is ultrahomogeneous iff it is isomorphic to one of the following:

1. $$(A_n, <)$$ for $$1 \le n \le\omega$$;

2. $$(B_n, <)$$ for $$1 \le n \le\omega$$;

3. $$(C_n, <)$$ for $$1 \le n \le\omega$$;

4. $$(D, <)$$.

Moreover, no two of the partially ordered sets listed above are isomorphic. Consider automorphisms groups $$Aut(A_\omega) = S_\infty$$, $$Aut(B_n)$$, $$Aut(C_n)$$ and $$Aut(D)$$. We prove that each of these groups contains two elements f, g such that the subgroup generated by f and g is free and dense. By Schmerl’s Theorem the automorphism group of a countable infinite partially ordered set is freely topologically 2-generated.

# Marcin Michalski: Universal sets for bases of $$\sigma$$-ideals

Tuesday, November 8, 2016 17:15

Room: D1-215

Speaker:
Marcin Michalski

Title: Universal sets for bases of $$\sigma$$-ideals

Abstract. We shall construct universal sets of possibly low Borel rank for classic $$\sigma$$-ideals of sets: $$\mathcal{N}$$-family of measure zero sets, $$\mathcal{M}$$-family of meager sets, $$\mathcal{M}\cap\mathcal{N}$$ and $$\mathcal{E}$$. We will also discuss briefly cases of other ideals.