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PLEASE NOTE: If a candidate gives a layman's talk, the livestream will start fifteen minutes earlier.

To classify curves based on their similarities, we introduce certain invariants. Topologically, the set of complex points of an elliptic curve forms a torus (a 'donut'), in other words a real surface with one hole. In fact, every smooth projective curve over the complex numbers is topologically a real surface with some number of holes, and this number is called the genus of the curve. This notion generalizes to curves over any field k. Informally, we say that a curve is smooth if it has no sharp points (cusps) or self-intersections.

In this thesis, we study families of curves of genus g > 1 over fields of positive characteristic p > 0, focusing on invariants that are specific to this setting, such as the p-rank, Newton polygon, and Ekedahl-Oort type. A significant portion of the thesis is devoted to supersingular curves of genus g > 3, which are characterized by having the most 'unusual' Newton polygon and exhibit many intriguing properties. 

We explore the existence of smooth curves of genus g with a prescribed Newton polygon (such as the supersingular one) or a given Ekedahl-Oort type, and examine the geometric properties of the corresponding families. Combining computational and theoretical methods, we address several questions and conjectures related to these families and contribute to a better understanding of the geometry of smooth curves in characteristic p > 0.