Grant: Discovery Grants Program - Individual

What are the ways in which a long protein can fold? How can we program a robot to move from one configuration to another? What is the shape of a big data set? These are some of the problems we can tackle using topology, the study of spaces. In topology, a convenient model to describe spaces is to glue spheres and disks of different dimensions in certain ways. For example, the torus (i.e. the surface of a donut) can be built by gluing together two circles and one 2-dimensional disk. Describing all the spaces that can be built by gluing a disk to a sphere amounts to computing the ways a sphere can be mapped into another sphere, up to deformation. For example, the different ways to map a circle into another circle amount to the number of times the circle will be winding around. Maps between spheres of various dimensions are called the homotopy groups of spheres, which are notoriously hard to compute. A powerful tool to compute homotopy groups is given by the Adams spectral sequence. A spectral sequence is a device that breaks a hard problem into smaller pieces, computes the pieces separately, and helps us assemble the answer at the end. The Adams spectral sequence starts from information about cohomology, which is more tractable, and outputs information about homotopy, which is difficult. As an illustration, imagine a sphere, the surface of a ball. Its cohomology consists of a class in dimension 2, which detects that the sphere is 2-dimensional and is hollow inside. In contrast, the homotopy groups of the two-dimensional sphere are not all known, to this day. It turns out that the Adams spectral sequence is not limited to computing homotopy groups of spheres, in fact, not even limited to topology. There is a generalization of the Adams spectral sequence to triangulated categories, a structure that is used in topology, algebra, geometry, and other parts of mathematics. The goal of this research program is to explore features of the triangulated Adams spectral sequence and its computational consequences. In other words, to refine the tool itself, and to apply it to the various branches of mathematics where triangulated categories are used. The novelty of the program lies in the fact that the Adams spectral sequence is familiar to topologists, but not (yet) widely used by algebraists, geometers, and other users of triangulated categories.