Diese Arbeit entwickelt aufbauend auf bekannten Resultaten die Modultheorie über der äußeren Algebra in Teilen weiter, insbesondere werden die Tiefe eines Moduls und Moduln mit linearer injektiver Auflösung untersucht. Angewendet werden die Resultate auf die Orlik-Solomon Algebra eines Matroids.

We prove that solid closure and graded dagger closure agree for homogeneous ideals in two dimensional $\mathbb{N}$-graded domains of finite type over a field. We also prove that dagger closure is trivial for ideals in regular rings containing a field and that graded dagger closure is trivial for $\mathbb{N}$-graded regular rings of finite type over a field. Finally, we prove an inclusion result for graded dagger closure for homogeneous primary ideals in certain section rings of abelian varieties.

Lo scopo di questo lavoro è realizzare uno studio interdisciplinare sul rapporto tra Milano e i regimi succedutisi alla sua guida nel basso medioevo. Il punto di partenza è il principio, largamente diffuso nella storiografia, che sistemi di governo diversi plasmino lo spazio in maniera differente. L’analisi dello spazio urbano così ottenuta, però, può essere arricchita dall’uso di strumenti della geografia culturale. La città, tema tra i più importanti e complessi della cultura occidentale, è infatti il luogo di incontro naturale tra storia e geografia culturale: è il luogo in cui si svolge la storia, ma più di un semplice palcoscenico. Le caratteristiche del fenomeno urbano trovano massima espressione nel caso di Milano, che presenta una storia di primissimo piano tradottasi in una struttura urbana dalle forme assolutamente peculiari. Sia al suo interno sia verso l’esterno (per la sua capacità di plasmare il territorio, agendo sull’ambiente naturale e modellandolo in base alle proprie necessità, ribaltando le logiche del determinismo ambientale). Combinando gli strumenti propri della geografia culturale con quelli della ricerca storica, si ottiene la possibilità di usare la città stessa e le sue rappresentazioni come documento, affiancando all’analisi degli eventi storici lo studio della valenza simbolica di tali realizzazioni. / This work is aimed to create an interdisciplinary study on the relationship between Milan and its polical regimes in the late Middle Ages. The starting point is the principle, widely used in historiography, that different systems of government produce space differently. The analysis of urban space thus obtained, however, can be improved using instruments from cultural geography. The city - one of the most important and complex topics in Western culture - is the meeting place of the natural history and cultural geography: it is the place where the story takes place, but more than just a stage. Milan is the perfect example of city: it has a history of the first order which turned into an urban structure of absolutely peculiar shape. Both internally and externally (for its ability to shape the territory, by acting on the natural and modeling it to suit your needs, reversing the logic of environmental determinism). Combining the tools of cultural geography with those of historical research, you get the opportunity to use the city itself and its representations as a document, supporting the analysis of historical events, the study of the symbolic value of such achievements.

The two main parts of this doctoral thesis are a theorem that tight closure is contained in continuous closure via axes closure on the one hand and an algorithm to decide semistability of sheaves (or geometric vector bundles) via reduction to a linear algebra problem on the other hand. The sheaf stability algorithm was explicitly implemented by the author.

Ce travail part d'une interrogation : comment expliquer ce qu'est un camp de concentration nazi sans prendre en compte un de ces constituants, la pierre. Sans oublier pour cela l'Homme, la présente recherche s'est attachée à définir le camp de concentration comme espace architectural à travers l'étude de six camps : Dachau, Sachsenhausen, Buchenwald, Mauthausen, Auschwitz I et le camp de Natzweiler. Trois axes d'analyses furent choisis : le lieu, les formes et les matériaux employés dans le but de mettre en lumière une éventuelle continuité architecturale. A partir du constat qu'il existait bien une architecture véritable, l'auteur s'est demandé dans une dernière partie si des liens avec l'idéologie national-socialiste étaient possibles, comme cela est le cas avec l'architecture officielle du Troisième Reich. C'est ainsi qu'à partir des données collectées furent adjointes trois spécificités à l'architecture concentrationnaire : le camp de concentration nazi peut être perçu comme une cité idéale national-socialiste, une mise en place architectonique du Führerprinzip et le reflet de la Blut-und- Boden Doktrin. / This work is based on a questioning : how to explain what a nazi concentration camp is without taking into account one of its components, i.e. stone. Without overlooking the human dimension for all that, the following research aims at giving a definition of the concentration camp as an architectural space through the analysis of 6 camps : Dachau, Sachsenhausen, Buchenwald, Mauthausen, Auschwitz I and the Natzweiler concentration camp. Three main lines were chosen : the sites, the shapes and building materials, with a view to bringing into light a possible architectural continuity. Once the existence of an architectural structure was established, the author finally wondered whether links with the national-socialist ideology did exist as it is the case with the official architecture of the Third Reich. Thus, thanks to the collected data, three specificities of concentration camps architecture were added : the nazi concentration camp can be seen as a national-socialist perfect city, as the architectonical setting up of the Führerprinzip and as the reflection of the Blood and Soil ideology. / Die folgende Arbeit erwächst aus einer Fragestellung : Wie lässt sich ein Konzentrationslager erklären, ohne eines seiner Bestandteile - den Stein - in Betracht zu ziehen ? Ohne den Menschen außer Betracht zu lassen, hat sich die vorliegende Studie bemüht, ein Konzentrationslager als einen architektonischen Raum zu definieren. Es werden sechs Konzentrationslager analysiert : Dachau, Sachsenhausen, Buchenwald, Mauthausen, Auschwitz I und das KZ Natzweiler. Es werden drei Blickrichtungen gewählt : der Ort, die Formen und die verwendeten Materialien, um herauszufinden, ob es etwa eine architektonische Kontinuität gibt. Ausgehend von der Feststellung, dass tatsächlich von einer bestimmten Architektur die Rede sein konnte, hat sich der Autor in einem letzten Teil gefragt, ob Beziehungen mit der NS-Ideologie möglich waren - wie es eben der Fall bei der offiziellen Architektur des Dritten Reiches ist. Auf diese Weise wurden mit den vom Verfasser gesammelten Daten den Konzentrationslagern drei Eigenschaften zugeschrieben : das NS-Konzentrationslager kann als eine ideale NS-Stadt, als die architektonische Verwirklichung des Führerprinzips und als die Wiederspiegelung der Blut-und-Boden Doktrin betrachtet werden.

In der Dissertation habe ich versucht eine Theorie der globalen Gerechtigkeit darzustellen. Diese Theorie hat als ihre Basis das Denken von John Rawls. Rawls hat sich in seinem letzten Buch „Das Recht der Völker“ zu dem Thema geäußert. Ich erläutere seine Gedanken und zugleich kritisiere ich sie. Meine Kritikpunkte an Rawls: 1. Keine detaillierte Erläuterung der Aufteilung in ideale und nichtideale Theorie 2. Unvollkommenheit der Prinzipien für die ideale Theorie der globalen Gerechtigkeit Ich verweise auf die mögliche Verbesserung des Rawlschen Standpunktes über globale Gerechtigkeit durch: 1. eine Unterscheidung der normativen und deskriptiven idealen bzw. nichtidealen Theorie (damit wird die Struktur von Rawls‘s Arguments klarer) 2. Durch die Darstellung von drei Prinzipien die Rawlssche Prinzipien hätten sein könnten (damit bekommt seine Theorie inhaltlich mehr Wert) 3. Durch die Erläuterung des Prinzips der Solidarität als das, was an Rawls‘s Theorie besonders gewesen sein könnte. / In my disertation I have tried to present a theory of global justice. This theory has for its basis the thought of John Rawls. Rawls expressed his thaughts about the topic in his last book „The Law of Peoples.“ On the one hand I explain his theory while on the other I criticize it. My critique on Rawls goes accoring the following lines: 1. No elaborated disitiniction betwen the ideal and nonideal theory 2. Unfinished principles for the ideal theory of global justice I point to the possible amelioration of Rawls´ theory of global justice through: 1. Differentiation of the normative and descriptive ideal and nonideal theory (through this disticition is the stucture of Rawls´ argument clearer) 2. Introduction of three prinicples that should have been Rawls´s (through these three principles the content of his theory becomes more valuable) 3. Eplaning the principle of solidarity as the principle that could have been the specific for Rawls´ theory

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.

My PhD-project has two main research directions. The first direction is on partial regularities which we define as refinements of the Castelnuovo-Mumford regularity. Main results are: relationship of partial regularities and related invariants, like the a-invariants or the Castelnuovo-Mumford regularity of the syzygy modules; algebraic properties of partial regularities via a filter-regular sequence or a short exact sequence; generalizing a well-known result for the Castelnuovo-Mumford regularity to the case of partial regularities of stable and squarefree stable monomial ideals; finally extending an upper bound proven by Caviglia-Sbarra to partial regularities. The second direction of my project is to develop a theory on monomial preorders. Many interesting statements from the classical theory of monomial orders generalize to monomial preorders. Main results are: a characterization of monomial preorders by real matrices, which extends a result of Robbiano on monomial orders; secondly, leading term ideals with respect to monomial preorders can be studied via flat deformations of the given ideal; finally, comparing invariants of the given ideal and the leading term ideal with respect to a monomial preorder.

The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.