My main interest is algebraic topology, which studies topological spaces by means of algebraic invariants. The first examples of such invariants are homotopy groups, which are groups of homotopy classes of maps from a sphere into a space. While homotopy groups are easy to define, they are usually very difficult to calculate and other invariants must be studied in the process.
I am most interested in a class of invariants called (generalized) homology theories. Such theories include classical ('ordinary') homology theory, K-theory, cobordism theories, algebraic K-theory, and many more. A very intriguing example of a homology theory is stable homotopy theory, which is most closely related to homotopy groups. Many problems in various areas of mathematics, including algebraic geometry, analysis and geometric topology can be reduced to calculating stable homotopy groups.
I am involved in calculations of stable homotopy groups and other generalized homology theories. For example, right now I am calculating Morava K-theories of classifying spaces of finite groups, which is an interesting mixture of topology, representation theory and local class field theory. Recently, I wrote a paper on asymptotic estimates of families of elements in the stable homotopy groups of spheres.
I am also interested in operads and structures up to homotopy. This area originated in topology (in the study of infinite loop spaces), but turned out to be the right way of looking at certain concepts in differential geometry and physics (string theory). Jointly with J.P. May, I recently wrote a monograph on applications of operads in topology. The main point of these books is that operads can be used in constructing generalized cohomology theories outside the usual context of topological spaces. For example, in algebraic geometry, one can hope to understand 'integral mixed motives', which is conjectural notion of generalized (co)homology theories on algebraic varieties. The book gives the definition of integral mixed Tate motives over a field, thus carrying out the first step of a program by P. Deligne. It also gives certain explanations and addenda to my previous joint paper with S. Bloch, where we defined rational mixed Tate motives.