Homotopy Type Theory (HoTT) develops Martin-Lof Intuitionistic Type Theory (ITT) by adding axioms for equality that are directly inspired by analogies with Homotopy Theory in Mathematics. The key source of information is http://homotopytypetheory.org/ and we shall be loosely following the HoTT book which is freely available at http://homotopytypetheory.org/2013/06/20/the-hott-book/
I will aim to cover the material in Chapter 1. I'll summarise the foundations (as I understand them), remind people about the idea of dependent types, and explain the various basic constructs and the associated induction principles. I hope to be able to get as far as path induction, which is the point where HoTT starts to look different and interesting. I might be able to state (but not say anything about) univalence.
Dr Walter Dean
Department of Philosophy
University of Warwick
http://go.warwick.ac.uk/whdean