Theory and applications of tangent categories

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Calculus is the most useful and practical area of mathematics.  It deals with how to calculate with changing quantities, and is thus a fundamental tool in sciences, economics, and engineering.  However, while it is extremely important in applied contexts, its study is but one part of a much wider world of mathematics, which includes many other topics, such as combinatorics (counting), topology (the study of abstract shapes) and computer science. On the other hand, in some of these other areas of mathematics, researchers have started to develop ideas that share many formal similarities with calculus.  These have gone by various names: polynomial functors (in combinatorics), the functor calculus (in topology) and differential linear logic (in computer science).   As these ideas have been developed, it has not been clear how they are related to each other, or if they truly share a direct similarity with ordinary calculus.   The research proposed for this grant offers a way to resolve this problem, by allowing one to view all these various notions as all aspects of one common idea, the notion of a tangent category.  By providing a common framework for these different ideas, tangent categories provide an essential language to translate ideas between different areas of mathematics.  Moreoever, one can develop much of differential geometry within the abstract setting of a tangent category, allowing the sophisticated ideas of differential geometry to be transferred into the study of polynomial functors, functor calculus, and differential linear logic.  This work thus allows us an entirely new way to advance our understanding of these important ideas.