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.

Let $G=(V,E)$ be a finite connected graph and $D=[d(x,y)]_{x,y\in V}$ its distance matrix. The quadratic embedding constant (QEC) of a graph $G$ is defined by the conditional maximum of $\langle f, Df\rangle$, $f\in C(V)$, subject to two constraints $\langle f,f\rangle=1$ and $\langle \mathbf{1},f\rangle=0$. The QEC, introduced by Obata--Zakiyyah (2018), has recently been the focus of research as a new invariant for classifying graphs. In this talk, recalling the fundamental properties obtained so far, we discuss some new achievements on characterization of graphs $G$ with $\mathrm{QEC}(G)<-1/2$ and propose some challenges. This talk is partially based on the joint work with W. Młotkowski (Wroclaw) and E.T. Baskoro (Bandung).

Mechanics - 2024 Birkhäuser Distinguished Lecture

Lecturer:     Professor Camillo De Lellis, IAS, Princeton 

Title:            Flows of nonsmooth vector fields

Date:            Monday, 9th December, 2024 

Time:          08:30 New York, 14:30 Berlin, 13:30 London, 15:30 Cairo, 

                       16:30 Baghdad, 19:00 New Delhi, 21:30 Beijing, 22:30 Tokyo

Zoom link: https://springer.zoom.us/j/86946520694 (free access, no code)

Download: PDF Flyer (with Abstract and Bio)

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.

After a brief introduction to semigroups of ultrafilters, we shall discuss Schur ultrafilters on groups and with their help give a new description of Bohr compactifications of topological groups. Also, we show that Schur ultrafilters are crucial in distinguishing which chart group is a topological group. Namely, a chart group G is a (compact) topological group if and only if each Schur ultrafilter on G converges to the unit of G.