New Directions in Toric Topology

The general purpose of the workshop is to accelerate the development of the emerging field of toric topology. Toric topology is turning out to have many interesting applications within classical algebraic topology, as well as exciting interactions with algebraic geometry, combinatorics, and symplectic geometry; activity in the subject is growing rapidly. The fundamental objects of study are pairs (M,alpha), where M denotes a smooth manifold (or one of various natural generalisations) that admits an action alpha of the torus T as a group of well-behaved symmetries. Any such M is known as a quasitoric manifold.The proposal has three main aims.The first is to contribute to ongoing studies of the moment-angle complexes (and their quotients) that are associated to each pair (M,alpha); these complexes arise naturally in areas such as algebraic geometry and combinatorics, and are therefore of independent interest. We shall apply the most up-to-date techniques of category theory and homotopy theory to probe their deeper topological structure.The second is to study the function space of based maps from the circle to an arbitrary quasitoric manifold (otherwise known as the loop space on M); there is considerable recent evidence that the toroidal symmetries invest this loop space with beautiful algebraic and geometric properties, and we wish to remain at the forefront of these developments. We shall apply methods adapted from the theories of complex manifolds, configuration spaces, and partial multiplications. The third is to investigate the group of self-equivalences of M in various standard categories, and discover how their topological structure is controlled by the action of T; simultaneously, we propose to address the related issue of classifying M within these categories by means of appropriate algebraic invariants. In the latter case, we expect to proceed by testing several families of examples (such as Bott towers), many of whose properties have recently been revealed.