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Moduli spaces are geometric versions of parameter spaces. That is, the points of the space are in bijective correspondence with the objects being parametrized. The revolution that the field of Algebraic Geometry has undergone with the introduction of moduli spaces has led - over the last few years - to a burst of activity in the area, resolving long-standing problems and generating new results and questions. Remarkably, the notion of moduli, which dates back to B. Riemann, has been applied to various areas. For instance, elliptic curves, and their moduli, are used in security theory for cryptographic encryption. More generally, moduli of Riemann surfaces of any genus are connected to topological field theory and string theory. Moreover, they have been recently related to shape matching. Furthermore, moduli of Calabi-Yau threefolds are the natural answer to conjectures in Theoretical Physics. In fact, such threefolds are supposed to compactify the conjectural ten dimensional universe. Finally, moduli of points moving in projective space, and their configurations, are related to problems in computer vision. Over fields different from the complex numbers, moduli spaces arise naturally in applications to number theory (rational numbers, p-adic numbers, etc.) and robotics (real numbers). In this project we intend to develop our skills in order to investigate geometric properties of some moduli spaces mentioned before. A better understanding of their intrinsic structure - from a theoretical point of view - may affect their applications.
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