Univalent Foundations Project

While working on the completion of the proof of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. I have started to actively learn about the related subjects around 2003/04. A few years ago I have come up with an idea for a new semantics for dependent type theories the class of formal systems which are widely used in the theory of programming languages. Unlike the usual semantics which interpret types as sets this ”univalent semantics” interprets types as homotopy types. The key property of the univalent interpretation is that it satisfies the univalence axiom a new axiom which makes it possible to automatically transport constructions and proofs between types which are connected by appropriately defined weak equivalences. In 2009/2010 I made a number of presentations on the univalent interpretation which were received with great interest by the type theoretic community. As of today two special events have been planned for the further discussion of the related ideas a workshop in Oberwolfach in March 2011 and a year long special program at the Institute for Advanced Study in 2012-2013. Eventually it became clear to me that the univalent semantics is just a first step and that I am really working on new foundations of mathematics. The key features of these ”univalent foundations” are as follows: