Toric Topology was first identified 20 years ago and has developed rapidly, with remarkably varied input from cobordism and homotopy theory, algebraic and combinatorial geometry, commutative algebra, and symplectic geometry and integrable systems.

At its roots it is the study of topological spaces with well behaved toric symmetries. Central objects are toric, quasitoric and torus manifolds, and moment-angle complexes. Quasitoric and torus manifolds are topological generalizations of toric varieties. These manifolds are often algebraic or symplectic but need not be, instead having more flexibility in terms of studying topological and combinatorial properties. In particular, these properties give valuable information about the toric varieties themselves. Moment-angle complexes are spaces formed by gluing together products of discs and circles according to a recipe determined by an underlying simplicial complex. The construction of moment--angle complexes unifies several existing constructions from seemingly unrelated areas of mathematics, including: the intersection of special real and Hermitian quadrics studied in topology and holomorphic dynamics, level sets for moment maps in the construction of Hamiltonian toric manifolds via symplectic reduction, and complements of coordinate subspace arrangements.

Polyhedral products extend the unifying reach of moment-angle complexes. A polyhedral product is a functorial generalization of a moment-angle complex in which the discs and circles are replaced by CW-complexes and CW-subspaces. They appear in: the Whitehead filtration in homotopy theory, the study of asphericity in group cohomology, the study of right-angled Artin groups, right-angled Coxeter groups and graph products in geometric group theory, identities relating the Euler Phi Function to the Möbius function, and the study of robotics and arachnid mechanisms.

This programme is designed to give the subject a transformational push forward by bringing together experts from the many areas  currently using toric spaces, moment-angle complexes and polyhedral products. The program will  harness existing connections and create new ones. It will provide an environment favourable  for stimulating significant progress in a range of mathematical disciplines through synergy, and will be a focal point for elucidating new directions that will guide research for the next five to ten years.

Organised in partnership with the Clay Mathematics Institute