Towards a unified approach to functor calculus

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Topology is the part of mathematics which provides a careful analysis of shapes, or topological spaces. Some of these spaces can be visually inspected because we can draw them or model them. But others have so many dimensions that they can hardly be imagined. The job of an algebraic topologist is to find ways to imagine the unimaginable by assigning mathematical values to topological spaces which appropriately represent the space. This assignment is accomplished using homotopy functors, a method of replacing something which is difficult to visualize, analyze or classify by something which is simpler to imagine or even compute. Unfortunately, many of the most valuable homotopy functors - those that are the most descriptive - are themselves very difficult to compute. In order to make this task easier, there is a kind of calculus for homotopy functors, which was originally pioneered by Goodwillie and which is known as Goodwillie calculus. The idea of Goodwillie calculus is to approximate valuable but complicated homotopy functors by simpler functors which are easier to compute. Some of the first applications of Goodwillie calculus were to famous functors like K-theory, which is important in many branches of mathematics including number theory, algebra, algebraic geometry, topology and analysis.******My main research goal is to make it easier to use Goodwillie calculus. In particular, one of my research goals is to formalize the relationship between Goodwillie's calculus and the calculus of Newton and Leibniz that we teach to first-year university students. These are not the same, but they have many of the same properties. Since mathematician have understood calculus of functions very well for several hundred years, if I can make this relationship precise then this will make it easier to use Goodwillie's calculus, too. Together with my colleagues Johnson, Osborne, Riehl and Tebbe, we have already found the precise relationship we want in a special case using the theory of differential categories.******Topology is used to model problems of all kinds. The positions of robots in an automated warehouse is modelled by a topological space, as is spacetime. In a world of data collected with an ever-increasing number of parameters, topology gives us one way of understanding the data as a whole, modelling the data, and using this to draw conclusions and make predictions from the information we have gained. For each application of topology in the real world, there are homotopy functors lurking which can simplify this information and make it easier to understand. Indeed, this is already being done: by using a simple homotopy functor to analyze scans of hepatic lesions, Carlsson et al were able to classify these lesions into a small number of disease types. As applications of topology become more prevalent, the tools we use will become more sophisticated. My work will simplify complicated homotopy functors when these tools are needed.