Dualities and Correspondences in Algebraic Geometry via Derived Categories and Noncommutative Methods

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The study of curves and surfaces given by the common zeroes of a set of polynomials has been pursued by humanity for thousands of years. In modern algebraic geometry, we study such sets in any dimension: these are called algebraic varieties. There are a number of questions that one can ask about one such variety: how "nice" is it? If we were standing on it, would it look to us like a curvy hill or like a rough mountain? If we are given two such varieties, can one tell if they are the same? Or if they are similar, for example if standing in most places on them they would look the same, and they only look different when looking at them from certain precise spots?Derived categories are a way to consider these geometric objects and translate much of the information about them into algebraic notions. While the derived category of a variety retains much of the information about the variety we started with, at the same time it allows us extra flexibility to work in an algebraic context. In the past two decades the field of derived categories has experienced an outpouring of activity as many classical algebraic geometry problems are solved passing through derived categories techniques.One fundamental question about derived categories is about how the derived categories of two different geometric objects are related. Some of these relations might come from relations and symmetries between the two varieties, but there are also other kinds of relations between them, which are deeper and harder to understand:1. First of all, it is important to understand what the maps (functors) between two derived categories are like. Many of these - but not all, as people used to think! - have a very pleasant and useful geometric description as "Fourier-Mukai functors". Part of my project will consist in analyzing and describing the "bad" maps that are not Fourier-Mukai functors, and how these arise naturally by deforming the "good" maps we know about. 2. Another relation between two derived categories, which will be investigated as part of my project, is given by a concept of "duality" at the categorical level. Describing this duality gives us a way to understand deeper relations between derived categories that haven't yet been discovered, and that will shed more light on the symmetries and behavior both at the level of derived categories and at the level of the geometric objects.3. Finally, in some instances the relations between derived categories turn out to be equivalences and hence representable by Fourier-Mukai functors, and the analysis on the level of derived categories gives us back a big amount of geometric information. My project will tackle one such instance, namely the investigation of some quotient singularities that are a generalization of the Kleinian singularities, and their resolutions of singularities.