Formalisation of Constructive Univalent Type Theory

There has been in the past 15-years remarkable achievements in the field of interactive theorem proving, both forchecking complex software and checking non trivial mathematical proofs.For software correctness, X.

Leroy (INRIA and College de France)has been leading since 2006 the CompCert project, with a fully verified C compiler.For mathematical proofs, these systems could handle complex arguments,such as the proof of the 4 color theorem or the formal proof of Feit-Thompson TheoremMore recently, the Xena project, lead by K.

Buzzard, is developing a large library of mathematical facts, andhas been able to help the mathematician P.

Scholze (field medalist 2018) to check a highly non trivial proof.All these examples have been carried out in systems based on the formalism of dependent type theory, andon early work of the PI.

In parallel to these works, also around 15-years ago,a remarkable and unexpected correspondance was discovered between this formalismand the abstract study of homotopy theory and higher categorical structures.A special year 2012-2013 at the Institute of Advance Study (Princeton) was organised bythe late V.

Voevodsky (field medalist 2002, Princeton), S.

Awodey (CMU) and the PI.Preliminary results indicate that this research direction is productive,both for the understanding of dependent type systems and higher category theory, and suggest several crucialopen questions.

The objective of this proposal is to analyse these questions, with the ultimate goalof formulating a new way to look at mathematical objects and potentially a new foundation of mathematics.This could in turn be crucial for the design of future proof systems able to handle complex highly modularsoftware systems and mathematical proofs.