Seminaria

Streszczenie. W latach pięćdziesiątych Gagliardo wykazał, że dla obszaru $\Omega$ z regularnym brzegiem operator śladu z przestrzeni Sobolewa $W^1_1(\Omega)$ do przestrzeni $L^1(\partial \Omega)$ jest surjekcją. Zatem naturalne jest pytanie o istnienie prawego odwrotnego operatora do operatora śladu. Petree udowodnił, że w przypadku półpłaszczyzny $\mathbb{R}x\mathbb{R}_{+}$ nie istnieje prawy odwrotny operator do operatora śladu. Podczas referatu przedstawię prosty dowód twierdzenia Petree, który wykorzystuje tylko pokrycie Whitney'a danego obszaru oraz klasyczne własności przestrzeni Banacha. Następnie zdefiniujemy operator śladu z przestrzeni Sobolewa $W^1_1(K)$, gdzie $K$ jest płatkiem Kocha. Przez pozostałą część mojego referatu skonstruujemy prawy odwrotny do operatora śladu na płatku Kocha. W tym celu scharakteryzujemy przestrzeń śladów jako przestrzeń Arensa-Eelsa z odpowiednią metryką oraz skorzystamy z twierdzenia Ciesielskiego o przestrzeniach funkcji hölderowskich.

Cuntz algebras $O_n$, $n>1$, are celebrated examples of a separable infinite simple C*-algebra with a number of fascinating properties. Their K-theory allows an embedding of $O_m$ in $O_n$ whenever $n-1$ divides $m-1$. In 2009, Kawamura provided a simple and explicit formula for all such embeddings. It turns out that his formulas can be easily deduced by viewing Cuntz algebras as graph C*-algebras, known as operator algebras that one can see. Better still, playing the game of graphs and using both the covariant and contravariant functoriality of assigning graph C*-algebras to directed graphs, we can show how to embed Cuntz algebras into matrices over Cuntz algebras via straightforward polynomial formulas. Based on joint work with Yang Liu.

Cucker-Smale model, introduced in 2007, describes the evolution of particles aligning their velocities according to nonlocal interaction protocol. We will focus on the strongly singular case and see that, under some mild assumptions, measure valued solutions to kinetic Cucker-Smale equations are in fact monokinetic, which reduces them further to the hydrodynamic scale, known as Euler-alignment system. Consequently, we are able to perform a direct micro- to macroscopic mean field limit, proving the existence of weak solutions to the Euler-alignment system with almost any initial data, including the possibility of a vacuum. The talk is based on joint work with Jan Peszek (University of Warsaw) and Young-Pil Choi (Yonsei University).

For a model M and a topological space C we say that a map f: M -> C is definable if for any two disjoint closed sets in C their preimages by f are separable by a definable set. For a definable structure N in a model M we say that f is a definable compactification of N if f is a compactification of N and f is a definable map. We say that a definable compactification of N is universal if every definable compactification of N factors by f via a continuous map. It turns out that if N is a group (Gismatullin, Penazzi & Pillay) or a ring (Gismatullin, Jagiella & Krupiński) then N/N^{00}_M is the universal definable compactification of N, where N^{00}_M is the type-definable connected component of N over M. A module can be represented as N in two ways. The first one is a definable abelian group and a ring that is a part of the language. The second one is a definable abelian group and a definable ring. This talk: In this talk we will prove that for a definable module N (in both senses) N/N^00_M is a universal definable compactification of N (the definition of N^00_M for a module will be given). We will also analyze how N^00_M depends on the type-definable connected component of the abelian group. To do this we will prove a theorem that shows how connected components help in creating sets that are closed under commutative addition (generalization of the proof by Krzysztof Krupiński for approximate rings). Using the theorem, we will describe the type-definable connected component of modules, rings and semidirect products of groups. We will also show that in many cases of structures analyzed during this talk adding homomorphism/monomorphism/automorphism/differentiation to the structure N does not change the type-definable connected component of N. During the talk we will come across a few open questions. This is a joint work with Krzysztof Krupiński and is a part of a bigger project with Grzegorz Jagiella.

In this talk, we will discuss the similarities and differences between ultrafilters and finite additive measures on the natural numbers, with a particular emphasis on the Rudin-Keisler and Rudin-Blass orderings and their generalization to measures.