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Title: Generalized Geometry of Goncharow & Configuration Chain Complexes.
Authors: Khan, Javed
Keywords: Mathematics
Issue Date: 2020
Publisher: Federal Urdu University of Arts Sciences & Tech. Islamabad
Abstract: The goal of this thesis is to present generalized geometry of two famous chain complexes through generalized homomorphisms. First one is Grassmannian configuration chain complex of free abelian groups generated by all the projective configurations of m points in any n-dimensional vector space Vn(F) defined over some arbitrary field F, while other is Goncharov polylogarithmic group complex of classical polylog groups. Many researchers defined geometry of Grassmannian configuration with classical polylogarithmic groups only for lower weights, i.e. n = 2 and 3, to present commutative diagrams. Here geometry for lower weights is not only redefined in different ways but also it is generalized for higher weights, i.e. n = 4, 5, 6 up to any weight n ∈ N. Initially, geometry for special cases for weight n = 2 and n = 3 is introduced in detail. Bloch Suslin polylogarithmic group complex and Grassmannian configuration chain complexes are connected through morphisms for weight 2 such that the associated polygon is proven to be commutative and composition of morphisms is bi-complex. For weight 3, Goncharov classical poly-logarithmic and Grassmannian configuration chain complexes are connected to provide commutative and bi-complex diagram. Then geometry of Goncharov motivic complex and Grassmannian configuration complex is defined for weight 4 up to generalized weight n ∈ N through two types of generalized morphisms. All the associated diagrams are shown to be bi-complex and commutative. Lastly, and most importantly, extensions in geometry of Goncharov polylogarithmic and Grassmannian configuration chain complexes are introduced to generalize all morphisms between the above two chain complexes and also to generalize functional equations of polylogarithmic groups up to order n. For extensions of geometry, additional morphisms are introduced for weight 3 up to higher weight 6, to extend commutative and bi-complex diagrams. Then these extensions in geometry are generalized for any weight n to all morphisms between above two chain complexes. Associated generalized commutative and bi-complex diagrams are exhibited
Gov't Doc #: 20114
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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