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Current set-theoretic research on infinity focuses on the following three broad areas: large Cardinals and inner model theory, descriptive set-theoretic methods and classification problems, and infinite combinatorics. The programme HIF will connect these three main strands of set-theoretic research and other fields of set theory to the wider scope of mathematics, to research in the foundations of mathematics, including some philosophical issues, and to research on computational issues of infinity, e.g. in theoretical computer science and constructive mathematics. Three workshops are planned during the programme: The first one (24-28 August 2015) will be the 5th European Set Theory Conference. The second workshop, entitled 'New challenges in iterated forcing' will be a Satellite Meeting held at the University of East Anglia in Norwich (November 2-6, 2015). A final workshop will take place on December 14-18, 2015. For full details please see http://www.newton.ac.uk/events.html .