Systematic approach to higher structures

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During the last 25 years, tools originally coming from homotopy theory have appeared in many other areas of mathematics: in algebraic geometry [Voe03][TTV08][Lurb], in mathematical physics [BD95][SS11], in symplectic geometry [PTVV13], in logic and type theory [V+13], etc. Every time they brought either solutions to long standing open problems, or brand new perspectives. These new tools mostly revolve around the idea of higher categories or more generally of higher structures. They are a type of algebraic structures in which the axioms cannot be stated as mere equalities between the operations, but are new operations producing explicit "equivalence" between previously defined operations. These new operations also have to satisfy some 'axioms' again expressed as new operations and so on, encoding a very rich structure with operations of all dimensions. Despite the great success these ideas have met, they are also extremely difficult to use. Only experts in homotopy theory have been able to apply them, and this has considerably slowed down their development. Many mathematicians working in the fields where they have been used have also voiced complaints about this difficulty. The main long-term goal of my research is to develop new methods that will allow to work with higher structures in a simpler, more efficient and intuitive way. Typically, I aim to develop a framework to translate the intuitive way experts think about higher structures into precise and concrete mathematical statement, as automatically as possible. This would have a very significant impact on future uses of higher structures outside of homotopy theory, and potentially even outside of mathematics. I will use my grant to fund and train a group of HQP working on this problem, but also on the applications of these methods to the theory of higher categories itself and on applications of higher structures to other fields of mathematics. Those are new and active areas of mathematics, very rich in problems ideal for PhD or MSc students, which can teach them highly desired skill that are becoming increasingly useful in many area of mathematics. I also plan to apply these new methods to several long standing open problems in the area: - C.Simpson's strictification conjecture. - Hovey's problem of constructing a « model structures of model structures ». - The homotopy hypothesis for Grothendieck higher groupoids. I have already made significant progress on these problems, and have a concrete plan to solve the first two in the next 3 years. There are also many other problems in higher category theory that my HQP will work on.