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We consider categories of rational maps to algebraic groups and study existence and construction of universal objects for such categories, using the duality theory of Laumon 1-motives. In particular, we obtain functorial descriptions of the Albanese of a singular projective variety X over an algebraically closed field of characteristic zero, which is the universal regular quotient of a relative Chow group of points on X, and of its dual., updated and shortened version

In SGA 1, A.Grothendieck developped the notion of etale fundamental group which is a generalization of the Galois group. And he further studied the homotopy sequence associated to a fibration of schemes. In this thesis we study the corresponding homotopy sequence for two generalizations of the etale fundamental group--- the Nori's fundamental group and the algebraic fundamental group. In SGA 1, A.Grothendieck developped the notion of etale fundamental group which is a generalization of the Galois group. And he further studied the homotopy sequence associated to a fibration of schemes. In this thesis we study the corresponding homotopy sequence for two generalizations of the etale fundamental group--- the Nori's fundamental group and the algebraic fundamental group.

Duisburg, Essen, Univ., Diss., 2012 In SGA 1, A.Grothendieck developped the notion of etale fundamental group which is a generalization of the Galois group. And he further studied the homotopy sequence associated to a fibration of schemes. In this thesis we study the corresponding homotopy sequence for two generalizations of the etale fundamental group--- the Nori's fundamental group and the algebraic fundamental group.

Duisburg, Essen, Univ., Diss., 2012 Diese Arbeit beschäftigt sich mit der Frage, ob gewisse glatte Varietäten X über einem vollständigen lokalen Körper K mit algebraisch abgeschlossenem Residualkörper K-ratinale Punkte haben. Wir betrachten zahme Galoiserweiterungen L/K von K und Modelle vom Basiswechsel von X nach L mit einer Wirkung der Galoisgruppe von L/K. Wir zeigen, dass wenn diese Wirkung einen Fixpunkt im glatten Ort hat, X einen K-rationalen Punkt hat. Unter der Annahme, dass das betrachtete Modell ein quasiprojektives schwaches Néronmodell ist, konstruieren wir ein schwaches Néronmodell von X mit einer universellen Eigenschaft. Mit Hilfe der Konstruktion dieses schwachen Néronmodelles zeigen wir Theoreme, die die motivische Serrinvariant und das rationale Volumen einer K-Varietät X mit den entsprechenden Invarianten ihres Basiswechsel nach L in Verbindung setzt. Als Anwendung beweisen wir die Existenz von K-rationalen Punkten einiger glatter K-Varietäten mit potentiell guter Reduktion.

To any Feynman graph (with 2n edges) we can associate a hypersurface $X\subset\PP^{2n-1}$. We study the middle cohomology $H^{2n-2}(X)$ of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs $WS_n$, $n\geq 3$. Using the same technique, we introduce the generalized zigzag graphs and prove that $W_5(H^{2n-2}(X))=\QQ(-2)$ for all of them (with $W_{*}$ the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs.

Duisburg, Essen, Univ., Diss., 2008 To any Feynman graph (with 2n edges) we can associate a hypersurface $X\subset\PP^{2n-1}$. We study the middle cohomology $H^{2n-2}(X)$ of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs $WS_n$, $n\geq 3$. Using the same technique, we introduce the generalized zigzag graphs and prove that $W_5(H^{2n-2}(X))=\QQ(-2)$ for all of them (with $W_{*}$ the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs.

We study a class of filtered flat vector bundles of rank 2 which we call pseudo-eeliptic bundles. We construct pseudo-elliptic bundles starting from a deformation datum defined over the function field of a curve over an algebraically closed field of positive characteristic. This leads us to study a variant of Dwork's accessary parameter problem. We give applications of our results to Galois theory.

In the first part of this thesis we study indecomposable vector bundles of degree zero over an elliptic curve. We show that every such bundle generates a ring and a Tannakian category, such that the ring and the group scheme associated to the Tannakian category are of the same dimension. Furthermore we show that this result does not extend to curves of genus 2. In the second part we consider those at bundles over a compact Kähler manifold, whose monodromy representation factors through the second power of the augmentation ideal. We show that they are ex-tensions of trivial at bundles, and give a condition on their de Rham cohomology group, such that the corresponding Higgs bundle via the Simpson correspondence has Higgs field equal to zero. Dissertation, Universität Duisburg-Essen, 2002