There is a long history of finding and using deep algebraic structures in algebraic topology. A very classical example of this is Adams work on understanding the homology of a based loop spaces of simply connected spaces via the co-bar construction. This lead to Eilenberg and Moores work on understanding homology of homotopy pull backs, and is also an important underlying part of Sullivan’s theory of minimal models for finite rational homotopy types.

Another example building on this is that of Koszul duality which relates modules over the dgas defined by the based loop spaces to modules over the cochain dga of the original space, which geometrically are related to fibrations over the space.

However, the structures of a based loop space is just one of many interesting (applicable) examples. Indeed, many questions in algebra, topology and geometry involves additional structure. E.g. smooth structures, algebraic structures, group equivariant structure.

There are several different subfields of algebraic topology which tries to understand such deeper/higher algebraic structures and their applications to geometry. E.g.:

• Algebraic K-theory.
• Operads.
• Equivariant homotopy theory.
• Infinity categories.

The proposed program aims to bring together researchers working in these areas - especially those working in their overlap.