The geometry of cochains on a space

Given a topological space one can associate to it a differential graded algebra (or structured ring spectrum) of cochains which is the algebraic avatar of the space (depending on a choice of commutative ground ring). This algebra refines the cohomology of the space and contains a wealth of information relevant to the homotopy type. The aim of this project is to explore the geometry of this algebra of cochains; given a differential graded algebra one can apply the machinery of derived noncommutative algebraic geometry to it, viewing it as a generalised noncommutative space in its own right. One can then ask when this space is smooth: for instance characteristic p cochains on the classifying space of a finite group is known to be smooth when the group is a p-group (and generally is not smooth otherwise). However, little is known for other classes of spaces, and one aim of the project is to investigate this and to understand how spaces with smooth cochain algebra are distinguished - this will be closely related to properties of the loop space and so again is topologically meaningful. In the non-smooth case one can form the singularity category, an invariant which is quite new in the topological context. Another aim is to investigate the structure of the singularity category. One can also associate another type of geometry to cochains on a space, namely the tensor-triangular geometry, abbreviated tt-geometry, of its category of perfect complexes. The starting point of tt-geometry is to associate to the algebra of cochains its tt-spectrum with is a space, with a sheaf of rings. The tt-spectrum is satisfies a universal property and measures how complicated the differential graded algebra is. This space has been computed in some examples, but little is known outside of very special spaces such as cochains on classifying spaces of groups, polynomial spaces, and spherically odd complete intersections. Another aspect of the project is to compute examples, e.g. for rational spaces, and try to understand for which spaces the natural map comparing the graded spectrum of the cohomology with the tt-spectrum is a homeomorphism. This connects to the noncommutative geometry aspect, via further comparison results, and additional tools that one can access in the smooth case. It will also be interesting to understand how the tt-spectrum and its geometry relates to the homotopy theory of the original space.