The fundamental belief upon which algebraic geometry is founded is that geometric questions can often be answered using algebraic techniques. This idea has shaped itself in countless forms during the years, but the fundamental strategy can be summarised thus: to every geometric object we attach an algebraic gadget that encodes the information we are interested in; then, we study the algebraic object rather than the geometric one. In the transformation process, some information will be inevitably lost, but this is not major issue: having less variables means having a simpler problem, and some of the information might have been useless to us anyway.
Depending on the question we want to answer, we might need to discard more or less information. Indeed, to distinguish between a line and a plane we just need to consider the number of directions in which we can move, that is, their dimension. However, if we wanted to tell apart a sphere from a doughnut, we would need a more refined invariant.
The central object of study in algebraic geometry are algebraic varieties. These objects are locally modelled by zero loci of polynomials functions and thus one might think that they are easily studied. However, it is their global structure that matters. Compare: a sphere and a doughnut are locally (topologically) the same, it is the hole, which one sees only when zooming out enough, that distinguishes them.
The invariant we consider to study algebraic varieties is their bounded derived category of coherent sheaves, and we look at it from from three different perspectives.
1. The flexibility of the derived category
The derived category is a more flexible object than the variety it comes from, and it is interesting to ask to what extent is this true. Namely, when does it happen that two different varieties have the same derived category? And, what is the relation between two varieties with the same derived category? There is a conjecture that answers this question, and one of the aims of the research project is to work towards a better understanding of this picture.
2. Symmetries of the derived category
A strategy that has proved to be extremely useful in studying geometric and algebraic objects is to look at their symmetries. The symmetries of the derived category are called autoequivalences and appear naturally in many different contexts. With the aim of broadening our knowledge regarding autoequivalences, in this research project we study those symmetries that arise as compositions of spherical twists around spherical objects, both from a grouptheoretic and a dynamical system pointofview.
3. Structures built from the derived category
Starting from the derived category, new invariants have been constructed. For example, Bridgeland's stability conditions and Hochschild cohomology. One of the aims of this research project is to widen our understanding of these invariants: we plan to use Bridgeland's stability conditions in the study of categorical dynamical systems, and to compute the Hochschild cohomology of some explicit varieties, so to enlarge the list of available examples.
The beauty of this whole story is that not only the three perspectives above are linked with each other  studying the symmetries of an object helps understanding the object itself  but they are also influenced by, and in turn influence, neighbouring areas of mathematical studies. Thus, interdisciplinarity is at the core of the proposed research project, and its completion will produce sensible advancements in many different research areas.
