<p>Författaren har bytt namn och heter nu: Kevin Chauwinoir</p>

Let G ≤ SLn+1(C) act on R = C[X1, ..., Xn+1] by change of variables. Then, the skew-group algebra R*G is bimodule (n+1)-Calabi-Yau. In certain circumstances, this algebra admits a locally finite-dimensional grading of Gorenstein parameter 1, in which case it is the (n+1)-preprojective algebra of its n-representation infinite degree 0 piece. If the group G is abelian, the (n+1)-preprojective algebra is said to be of type Ã. For a given group G, it is not obvious whether R*G admits such a grading making it into an (n+1)-preprojective algebra. We study the case when n=2 and G is abelian. We give an explicit classification of groups such that R*G is 3-preprojective by constructing such gradings. This is possible as long as G is not a subgroup of SL2(C) and not C2 x C2. For a fixed G, the algebra R*G admits different 3-preprojective gradings, so we associate a type to a grading and classify all types. Then we show that gradings of the same type are related by a certain kind of mutation. This gives a classification of 2-representation infinite algebras of type Ã. The involved quivers are those arising from hexagonal dimer models on the torus, and the gradings we consider correspond to perfect matchings on the dimer, or equivalently to periodic lozenge tilings of the plane. Consequently, we classify these tilings up to flips, which correspond to the mutation we consider. / <p>Oleksandra Gasanova, Universität Duisburg-Essen, is co-author of the included work</p>

In this thesis we discuss a moduli space of projective curves with a map to a given projective space. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to the projective space that is an isomorphism onto its image away from a finite set of closed points. We proof that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map from W to CM. / QC 20120229

This thesis consists of three articles. The first two concern a generalization of Rees algebras of ideals to modules. Paper A shows that the definition of the Rees algebra due to Eisenbud, Huneke and Ulrich has an equivalent, intrinsic, definition in terms of divided powers. In Paper B, we use coherent functors to describe properties of the Rees algebra. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors. In Paper C, we prove a generalization of Gotzmann's persistence theorem to finite modules. As a consequence, we show that the embedding of the Quot scheme of points into a Grassmannian is given by a single Fitting ideal. / <p>QC 20141218</p>

This thesis explores the topic of Galois theory at a relatively introductory level with the goal of proving the Abel Ruffini theorem. In the first part algebraic structures are considered: groups, ring, fields, etc. Following this, polynomial rings are introduced and the attention is then turned to finite field-extensions. In the final section of the main text solvable extensions are studied and the Abel-Ruffini theorem is proved. The discussion section gives a brief overview of analytic methods of solving polynomial-equations. / Den här uppsatsen utforskar Galoisteorin för att bevisa Abel-Ruffinis sats. I den första delen är algebraiska strukturer i fokus: Grupper, ringar, kroppar, etc. Efter detta intrduceras polynom-ringar, och fokuset vänds sedan till ändliga kropps-utvidgningar. I den sista delen av huvudtexten så studeras lösbara förvidgningar och Abel-Ruffini's sats bevisas. Diskusionen ger en översikt över analytiska lösningar av polynom-ekvationer.

The notion of fulfilment of a formula by a sequence of numbers, an approximation of truth due to Kripke, is presented and subsequently formalised in the weak arithmetic theory IΣ1, in some detail. After a number of technical results connecting the formalised notion to the meta-theoretical one a version of Gödel’s Incompleteness Theorem, that no consistent, recursively axiomatisable, Σ2-sound extension T of Peano arithmetic is complete, is shown by construction of a true Π2-sentence and a model of T where it is false, yielding its independence from T. These results are then generalised to a more general notion of fulfilment, proving that IΣ1 has no complete, consistent, recursively axiomatisable, Σ2-sound extensions by a similar construction of an independent sentence. This generalisation comes at the cost of some naturality, however, and an explicit falsifying model will only be obtained under additional assumptions. The aim of the thesis is to reproduce in some detail the notions and results developed by Kripke and Quinsey and presented by Quinsey and Putnam. In particular no novel results are obtained.

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In this thesis, we consider hom-associative algebras, which is an algebra with multiplication that is not necessarily commutative nor associative, but obeys a twisted version of associativity by a linear homomorphism. We will give some conditions for associativity, which helps us determine commuting elements. Under other conditions, such as different types of unitality conditions, we can also state some results regarding commuting elements in the general, non-associative case.

I detta arbete har lösningar till ekvationen xn+yn= 1 i en ändlig kropp av primtalsordning studerats. Både konkreta lösningar för vissa givna förutsättningar samt resultat gällande antalet lösningar har varit målet för dessa studier. I uppsatsen redovisas även de metoder som använts för att nå nämnda resultat. Eftersom ekvationen kan studeras för en oändlig mängd kroppar presenteras inget heltäckande resultat. Vissa resultat är dock generella under premissen att antingen kroppen eller ekvationen fixeras för något primtal p respektive exponent n. För sådana specialfall presenteras flera exempel. Huvudresultatet i arbetet visar att det alltid finns många icke-triviala lösningar för ekvationen över stora ändliga kroppar.

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work. / <p>QC 20151117</p>