October 2023

# Aleksander Cieślak: Cofinalities of tree ideals and Shrinking Property

Tuesday, October 31, 2023 17:00

Location: room 601, Mathematical Institute, University of Wroclaw

Speaker:
Aleksander Cieślak

Title: Cofinalities of tree ideals and Shrinking Property

Abstract: If $$\mathcal{T}$$ is a collection of trees on $$\omega^\omega$$, then we define the tree ideal $$t_0$$ as a collection of these $$X\subset \omega^\omega$$ such that each $$T\in \mathcal{T}$$ has a subtree $$S\in \mathcal{T}$$ which shares no branches with $$X$$. We will be interested in the cofinalities of the tree ideals. In particular, we will focus on the condition, called "Incompatibility Shrinking Property", which implies that $$cof(t_0)>\mathfrak c$$. We will consider under what assumptions this property is satisfied for the two types of trees, which are Laver and Miller trees which split positively according to some fixed ideal on $$\omega$$.

# Maciej Korpalski: Straightening almost chains into barely altenating ones

Tuesday, October 24, 2023 17:00

Location: room 601, Mathematical Institute, University of Wroclaw

Speaker:
Maciej Korpalski

Title: Straightening almost chains into barely altenating ones

Abstract: Consider an almost chain $$\mathcal{A} = \{A_x \subset \omega: x \in X\}$$ for some separable linearly ordered set $$X$$. Such a chain is barely alternating if for all $$n \in \omega$$ we cannot find elements $$x_1 < x_2 < x_3 < x_4$$ in $$X$$ satisfying $$n \in A_{x_1}, A_{x_3}$$, $$n \notin A_{x_2}, A_{x_4}$$. We will show that under $$MA(\kappa)$$, if $$|X| \leq \kappa$$, then we can straighten our almost chain $$\mathcal{A}$$ into a barely alternating one by changing at most finitely many elements in each set $$A_x$$.

# Viktoriia Brydun: Monad on FMS(•) (Fuzzy Metric Spaces Category)

Tuesday, October 17, 2023 17:00

Location: room 601, Mathematical Institute, University of Wroclaw

Speaker:
Viktoriia Brydun (Ivan Franko Lviv National University)

Title: Monad on FMS(•) (Fuzzy Metric Spaces Category)

Abstract: Monad on FMS(•) (Fuzzy Metric Spaces Category)

# Arturo Martinez: Cardinal invariants related to free sets

Tuesday, October 10, 2023 17:00

Location: room 601, Mathematical Institute, University of Wroclaw

Speaker:
Arturo Martinez

Title: Cardinal invariants related to free sets

Abstract: (Joint work with T. Żuchowski) Given a function $$f$$ without fixed points, an infinite set $$A$$ is called free for $$f$$ if $$f[A] \cap A = \emptyset$$. In this talk we will discuss some cardinal invariants related to families of free sets and we will discuss their relation between some cardinal invariants related to category and measure.