This thesis aims to provide a classification of low-dimensional Lie algebras. We make emphasis on several structural properties, such as nilpotency, solvability and (semi) simpli- city. The first two properties relate to two fundamental theorems by Lie and Engels which classification results will be presented in a table for ease of access. / <p>I presented my thesis on 1st of October 2021.</p>

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In this licentiate thesis we present a proof of the initiality conjecture for Martin-Löf’s type theory with 0, 1, N, A+B, ∏AB, ∑AB, IdA(u,v), countable hierarchy of universes (Ui)iєN closed under these type constructors and with type of elements (ELi(a))iєN. We employ the categorical semantics of contextual categories. The proof is based on a formalization in the proof assistant Agda done by Guillaume Brunerie and the author. This work was part of a joint project with Peter LeFanu Lumsdaine and Anders Mörtberg, who are developing a separate formalization of this conjecture with respect to categories with attributes and using the proof assistant Coq over the UniMath library instead. Results from this project are planned to be published in the future. We start by carefully setting up the syntax and rules for the dependent type theory in question followed by an introduction to contextual categories. We then define the partial interpretation of raw syntax into a contextual category and we prove that this interpretation is total on well-formed input. By doing so, we define a functor from the term model, which is built out of the syntax, into any contextual category and we show that any two such functors are equal. This establishes that the term model is initial among contextual categories. At the end we discuss details of the formalization and future directions for research. In particular, we discuss a memory issue that arose in type checking the formalization and how it was resolved. / <p>Licentiate defense over Zoom.</p>

Swedish preschool has undergone some changes over the last decades. The preschool curriculum, LpFö98/2010, underlines the teacher’s responsibilities to work against the reproduction of traditional gender patterns. The focus of this study is to identify preschool teachers’ perceptions of what is male and female and how these may affect the dynamics amongst colleagues. In order to do so we have conducted eight interviews with preschool teachers of both genders and analyzed the material with the help of gender theories from Hirdman and Connell. The analysis shows that traditional gender patterns frequently occurs and that both genders tend to preserve the differences between men and women. The call for male preschool teachers affects the dynamics within the institutions. Male colleagues often benefit due to their gender, even though the perceptions of masculinity, to a certain extent, presents a broader variety of roles.

This licentiate thesis consists of one paper about unitary representationtheory of ample groupoids and semigroups together with generalizationsto étale and non-Hausdorff groupoids. In the paper we study algebraically the type I and CCR properties forample Hausdorff groupoids. Clarke and Van Wyk proved that both ofthese properties admit a topological characterization for Hausdorff second countable groupoids in terms of separation properties of their orbitspace and the isotropy groups. Using a Stone type duality between ample groupoids and Boolean inverse semigroups with meets, we exploit thischaracterization to get a purely algebraic statement. We also apply thoseresults to get characterizations of the type I and CCR properties for inverse semigroups using their Boolean inverse completions. The generalization is about characterizing the same properties for both étale and ample non-necessarily Hausdorff groupoids which nonethelesshave Hausdorff unit spaces. In this setup, we first give a direct proofof the topological characterization for the CCR property which doesn't rely on the disintegration theory. The argument cannot be adapted toget an easier proof in the type I case, but we rather explain how to geta proof following the original ideas of Clark and Van Wyk in that case.Finally, we state for both étale and ample groupoids algebraic conditionsequivalent to the CCR and GCR properties on their pseudogroup of openand compact open bisections respectively.

We construct a category of short hammocks and show that it has the weak homotopy type of mapping spaces. In doing so we tackle the problem of applying the nerve to large categories without the use of multiple universes. We also explore what the mapping space is. The main tool in showing the connection between hammocks and mapping spaces will be the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding bar constructions. / <p>QC 20141208</p>

Aus der Einleitung: „The Guarded Fragment of first-order logic, introduced by Andréka, van Benthem, and Németi, has been a succesful attempt to transfer many good properties of modal, temporal, and description logics to a larger fragment of predicate logic. Among these are decidability, the finite modal property, invariance under an appropriate variant of bisimulation, and other nice modal theoretic properties. ...”

This essay focusses on the relation between timbre and musical logic. I try to draw a connection between Schoenberg’s remarks on »Klangfarbenmelodie« in his Harmonielehre (1911) and the concept of »harmonie-timbre« represented by the spectral composers Gérard Grisey and Tristan Murail in the 1970s. At first glance, this relation between free atonality and spectral music seems far-fetched as Grisey and Murail rarely commented on Schoenberg’s theories. Nevertheless, a detailed investigation shows a parallelism of both analytical and aesthetic problems. Both the free atonal Schoenberg and the spectralists argued for developing sound sequences out of the innermost quality of timbre. Besides, the forming of free atonal and spectral sound sequences cannot resort to a structural preformation of material as in dodecaphonic or serial music. Therefore qualities of listening gain major importance. In order to realize free atonal or spectral sound sequences, it is necessary to apply a wide notion of perception: on the one hand, the composer enters the »inner« world of timbre by eaves- dropping on timbral nuances carefully. This perceptual attitude (»Lauschen«) brings to mind the transient presence and alterity of sound. On the other hand, this fascination with an »inner« quality of timbre should not lead to the assumption that free atonal or spectral sound sequences are unfolding in an »auto-generative« manner. Although composers can surrender themselves to a »passive« perception of sonic nuances, they are also compelled to make decisions about the direction of the music – decisions within the area of timbre, which seems to be resistant against decision-making. These decisions eventually constitute what might be perceived as musical logic. As a result, listeners of free atonal and spectral music are confronted with a fundamental tension between »inner« and »outer« compositional aspects: they perceive an illusory perfect and organic growth of sound (cf. Adorno: »Schein des Organischen«). The idea of a subject which articulates itself within a »pure« sound unleashes a tension between sense and its subversion, between active and passive listening. This tension seems to be one of the reasons why Schoenberg hesitated between sound and musical »sense«, and why he was fascinated by the »futuristic phantasy« to reconcile the apparent opposites of timbre and musical logic.

This thesis looks at characterising countably infinitely categorical theories. That is theories for which every countably infinite model is isomorphic to every other countably infinite model. The thesis looks at the Lindenbaum-Tarski algebra, Henkin theories, types and then ends with the Ryll-Nardzewski theorem which provides several equivalences to a theory being countably infinitely categorical.