The principal investigator, Mark Hovey, is working on the chromatic splitting conjecture in stable homotopy theory. A standard procedure in algebra is to study an algebraic object by studying its localizations at various primes, then reassembling the localizations into a global picture. In stable homotopy theory, one can localize at an ordinary integer prime p, but after doing so, more primes become visible. These are known as Morava K-theories. Denote the localization of a spectrum X with respect to the nth Morava K-theory K(n) by L(K(n))X. Calculation of L(K(n))X can usually be reduced to a purely algebraic problem, but little is known about the reassembly of the L(K(n))X for different n into knowledge of X itself. The chromatic splitting conjecture, due to Mike Hopkins, is a conjecture about the composition L(K(m))L(K(n))X for finite spectra X, and it would allow one to recover X from the different L(K(n))X. The truth of this conjecture is known to have significant impact on several other conjectures in stable homotopy theory, and even partial results about it would be a major advance. One of the methods involved in the principal investigator's approach to this conjecture is the study of the derived functors of the inverse limit functor in a category of comodules. Homotopy theory is the study of curves, surfaces, and higher dimensional spaces up to continuous deformation. That is, in homotopy theory, the letter A and the letter O are considered equivalent, because one can gradually deform the letter A into a triangle by pushing in the legs, then gradually deform the triangle into a circle by pushing out on the corners. A difficulty with homotopy theory is that different dimensions behave differently. More precisely, given a space X, one can form its suspension by attaching it with wires to two points outside X. The suspension of a circle, for example, is a sphere, and the suspension of a sphere is a sphere of the next higher dimension. In stable homotopy theory, two spaces X and Y are considered equivalent if, after suspending X and Y repeatedly, one can be deformed into the other. For example, the surface of a donut is obtained by attaching a sphere to two circles that touch at one point (one circle goes around the hole, one circle goes around the filling, and the surface of the sphere gets spread out over the rest of the surface of the donut). But a donut cannot be deformed into two circles and a sphere, all touching at one point. However, the suspension of a donut can be deformed into two spheres and a 3-dimensional sphere, all touching at one point. The great advantage of stable homotopy theory is that problems become much more amenable to algebraic methods of solution. Many geometric problems turn out to depend only on the stable homotopy type of the space involved, so stable homotopy theory has been of considerable utility. ***
