Stability of singular Fano varieties

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In geometry, shapes that are positively curved, such as sphere, are called Fano varieties. Their study is central in mathematics as they form basic building blocks of more complicated shapes. Fano varieties approximate any shape that can be parametrised, hence are applicable in design. However, their primary application is within pure mathematics as well as theoretical physics. Although Fano varieties have simple geometry, their deformation could be complicated. For instance, a family of circular conics can deform into a singular conic (heart shape), or into two lines crossing, or a double line. This is mathematically undesired. The theory of stability for Fano varieties has been recently developed and predicts the existence of a unique limit with good geometric properties for any family of Fano varieties. This project aims to examine the stability theory for Fano varieties in higher dimensions. In first steps, several examples will be studied to predict a general algorithm to compute the stable models. The next step would be an attempt to prove that the algorithm always provides the stable limit, and finally to generalise to all dimensions.