The main part the investigator's proposed research is the application of equivariant methods in two closely related areas, that of topology and of algebraic geometry. One of the investigator's main ongoing projects (joint with I. Kriz and K. Ormsby) is to apply the machinery of stable homotopy theory, such as the Adams and Adams-Novikov spectral sequences, to Morel and Voevodsky's motivic homotopy theory. Another of the investigator's projects in motivic homotopy theory, also joint with I. Kriz and K. Ormsby, is the study of equivariant motivic stable homotopy theory, which in turn also leads to new information in the world of equivariant topology. Together with I. Kriz, the investigator has a project studying equivariant spectra in topology arising from the realizations of motivic spectra, such as the so-called topological hermitian cobordism spectrum. The groups acting on such spectra contain Z/2, and the actions incorporate a Real (or complex conjugation) action. As shown by the recent work of Hill, Hopkins and Ravenel in solving the Kervaire invariant 1 problem, these objects are highly interesting sources of new homotopy theoretical information. In addition, the investigator also has a project in understanding string topology, as well as operad actions and deformation theory in both algebra and topology. This aspect of the investigator's work closely related to J. Lurie's recent notion of non-abelian Poincare duality on manifolds.
The overall theme of the investigator's research is in the interaction between two important areas of mathematics, that of algebraic topology and algebraic geometry. Algebraic topology can be thought of as "purely qualitative" geometry, where one is allowed to deform shapes or topological spaces, and associate to them certain algebraic and numerical invariants. On the other hand, algebraic geometry can be thought of as the study of certain much more rigid mathematical objects, built essentially from the solution sets of algebraic equations. Morel and Voevodsky have constructed a way of applying the methods of algebraic topology to the area of algebraic geometry, giving rise to a new field of mathematics, that of motivic homotopy theory. One of the investigator's projects (joint with I. Kriz and K. Ormsby) is to apply certain well-established machinery of algebraic topology to this world, which has the potential to answer long-standing questions. Another project is to gain an understanding of equivariant motivic homotopy theory, the goal of which is to shed light on structures in motivic homotopy theory by adding in the actions of groups to the story. In its turn, gaining an understanding of equivariant motivic homotopy theory will also lead to new information about objects in topology itself.
