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L'existence de métrique canonique sur une variété projective était une conjecture de longue date et la majeure partie de cette conjecture est sur les variétés qui n'ont pas défini de première classe de Chern. Il existe un programme qui est connu comme le programme de Song-Tian, pour trouver une métrique canonique sur les modèles canoniques d'une variété projective avec la Programme de modèle Minimal analytique pour résoudre la partie restante de Calabi conjecture. Dans cette thèse, nous étendons le programme Song-Tian et donner une version logarithmiques de celui-ci. Nous étudions le flux de Kähler-Ricci conique qui peut être considéré comme la chirurgie analytique. Nous introduisons la notion de Weil-Petersson métrique logartithmique. Nous donnons une preuve courte de la formule de Gang Tian pour le potentiel Kähler de métrique Weil-Petersson logarithmique sur l'espace de modules des variétés de Log Calabi-Yau (si elle existe!) sur singularités coniques et Poincaré. / Existence of canonical metric on a projective variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class(most of the manifolds do not have definite first Chern class). Thereis a program which is known as SongTian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. The main aim of this thesis is better undrestanding of SongTian program on pair (X;D). In this thesis, we apply SongTian program for pair (X;D) via Log Minimal Model Program where D is a simple normal crossing divisor on X with conic singularities. We investigate conical Kähler Ricci flow on holomorphic fiber spaces (X;D) -→B whose generic fibers are log Calabi Yau pairs (Xs;Ds), c1(KB) < 0, and D is a simple normal crossing divisor on X (we consider the cases c1(KB) = 0, and c1(KB) > 0 also). We show that there is a unique conical Kähler Einstein metric on (X;D) which is twisted by logarithmic Weil Petersson metric and an additional term which we will find it explicitly. We consider the semipositivity of fiberwise singular Kahler Einstein metric via SongTian program. We consider a twisted Kähler Einstein metric along Mori fibre space. Moreover, we give an analogue version of SongTian program for Sasakian manifolds. We give an arithmetic version of SongTian program for arithmetic varieties. Also we give a short proof of Tian’s formula for Kähler potential of logarithmic WeilPetersson metric on moduli space of log CalabiYau varieties (if such moduli space exists!).

We study variants of P-frames and associated rings, which can be viewed as natural generalizations of the classical variants of P-spaces and associated rings. To be more precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either maximal or minimal. For a completely regular frame L, if the ring RL of real-valued continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented frame. These frames are less restricted than the cozero complemented frames. Using these frames we study some properties of what are called quasi m-spaces, and observe that the property of being a quasi m-space is inherited by cozero subspaces, dense z- embedded subspaces, and regular-closed subspaces among normal quasi m-space. M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal associated with I is either maximal or minimal. If all points of L are quasi P-points, we say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi P-space if and only if the frame OX is a quasi P-frame. We characterize these frames in terms of cozero elements, and, among cozero complemented frames, give a su cient condition for a frame to be a quasi P-frame. A Tychono space X is called a weak almost P-space if for every two zero-sets E and F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H. We present the pointfree version of weakly almost P-spaces. We de ne weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring RL of real-valued continuous functions on L is weakly regular. We introduce the notions of boundary frames and boundary rings, and use them to give another ring-theoretic characterization of boundary spaces. We show that X is a boundary space if and only if C(X) is a boundary ring. A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces each of which is an F-space is said to be nitely an F-space. Among normal spaces, S. Larson gave a characterization of these spaces in terms of properties of function rings C(X). By extending this notion to frames, we show that the normality restriction can actually be dropped, even in spaces, and thus we sharpen Larson's result. / Mathematics / D. Phil. (Mathematics)

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces. / Geography / Ph. D. (Mathematics)