Systematic approach to higher structures

在过去的25年里，最初来自同伦理论的工具已经出现在许多其他数学领域：代数几何[Voe03][TTV08][Lurb]，数学物理[BD95][SS11]，辛几何[PTVV13]，逻辑和类型论[V+13]等。每次他们要么为长期悬而未决的问题带来解决方案，要么带来全新的视角。这些新工具主要围绕更高的类别或更普遍的更高的结构的思想。它们是一种代数结构，其中的公理不能仅仅被表述为运算之间的等式，而是在先前定义的运算之间产生显式“等价”的新运算。这些新的操作还必须满足一些“公理”，再次表达为新的操作等等，编码一个包含所有维度操作的非常丰富的结构。尽管这些想法取得了巨大的成功，但它们也极其难以使用。只有同伦理论的专家才能够应用它们，这大大减缓了它们的发展。许多在使用它们的领域工作的数学家也对这种困难提出了抱怨。我研究的主要长期目标是开发新的方法，以更简单、更有效和直观的方式处理更高的结构。通常，我的目标是开发一个框架，将专家对高级结构的直观思考方式尽可能自动地转化为精确而具体的数学陈述。这将对同伦理论之外的高级结构的未来应用产生非常重大的影响，甚至可能在数学之外。我将用我的拨款资助和培训一组HQP研究这个问题，同时也研究这些方法在高等类别理论本身的应用以及高等结构在其他数学领域的应用。这些都是新的和活跃的数学领域，非常丰富的问题，非常适合博士或硕士学生，这可以教会他们非常需要的技能，这些技能在许多数学领域变得越来越有用。我还计划将这些新方法应用于该领域几个长期存在的开放性问题：c .辛普森的严格化猜想。- Hovey构建“模型结构的模型结构”的问题。- Grothendieck高群拟的同伦假设。我已经在这些问题上取得了重大进展，并在未来3年内有了解决前两个问题的具体计划。在高等范畴理论中还有许多其他问题是我的HQP将要研究的。

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.