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| 2024

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 57270 (02)
University: Mathematical Sciences
Department: Electrical Engineering
Advisor(s): Rastegar, Arash
Abstract:
Our goal is to reformulate MKR-dictionary for geodesic foliations on hyperbolic 3-manifolds. Where the closed geodesic is considered as a knot. In this new perspective, by topological interpretation at arithmetical objects and considering the etale fundamental group, the ground has been prepared to obtain an analogy between the theory of knots and the algebraic numbers theory. The analogy between knots and prime numbers was pointed out by B. Mazur and also by Y. Manin and developed by the work of Kapranov and Reznikov. Using this analogy, Mazur formulated the Chebotarev density theorem related to prime numbers in number fields for an infinite sequence of knot on a three-dimensional manifold, and Mc Mullen proved it in several cases. Formulating the Chebotarev density theorem with the help of a conceptual definition of the height for the knot using the hyperbolic metric is the goal of this thesis
Keywords:
Prime Numbers Theorem ; Closed Geodesics ; Height Function ; Three Dimentional Manifold