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.

In this talk we present closed formulas for the conditional expectation of resolvents of arbitrary non-commutative polynomials in free random variables onto the subalgebras generated by subsets of the variables. More precisely, given a linearization of the resolvent we compute a linearization of its conditional expectation. The coefficients of the expressions obtained in this process involve certain Boolean cumulant functionals, which can be computed by solving a system of matrix equations.

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.

The Maximality Principle (introduced by Hamkins) asserts that any forceably necessary statement is true. The Recurrence Axiom (introduced by Fuchino and Usuba) asserts that any forceable statement is true in some ground, where an inner model W is said to be a ground if VV is a set-sized forcing extension of W. In this talk, we will explore natural variants of these principles by restricting the complexity of statements, allowing parameters, and varying the class of forcing posets. For example, we discuss the (in)compatibility of these variants with the Ground Axiom (introduced by Hamkins and Reitz), which asserts that there are no non-trivial grounds. This talk is based on joint work with Sakaé Fuchino and Francesco Parente.