Functors in Homotopy Theory

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My research program is concerned with functors in homotopy theory. Topology concerns itself with the shapes and global properties of physical (and theoretical) objects which we refer to as spaces. Simple examples are a donut (or its surface), the earth (or its surface), or more generally a manifold, which might model all the positions that a physical system can be in. Algebraic topology uses algebra to understand these objects. In homotopy theory, we consider the objects as if they are made out of rubber and can be continuously deformed. A functor is a way of assigning to some object, in our case often a space, another object which is often more algebraic such as a number or a set of numbers. The functor should be compatible with some other structures such as the deformations mentioned above. We will study some functors that arise in homotopy theory. First we wish to know the image of some functors. So if we start with an arbitrary space and apply one of our functors, in particular the cohomology functor, which corresponding algebraic objects can we get? This problem has a long history dating back at least to the 1960's. The target objects are always graded rings and this problem has been solved in some special cases such as polynomial rings and also when working over the rational numbers. We will study this problem modulo torsion. This simplifies the problem somewhat but not as much information is lost as when we work over the rational numbers. In particular if we have two elements a and b of our ring, we still want to know if their product is a multiple of some other elements. Over the rational numbers this is always true, but it is not always true modulo torsion. The second topic is known as Manifold Calculus. It looks at functors associated to manifolds. The problem is such a functor is often complicated, and so we resolve it into its so-called Taylor tower. This is analogous to taking a function in calculus and replacing it by its Taylor polynomials. A general function can be very complicated, but polynomials are much easier to understand. We are mostly interested in what possible towers can occur, in other words, what are the polynomials in this context.