Award: DMS 1202213, Principal Investigator: Daniel C. Isaksen
The PI will carry out computations in motivic homotopy theory, i.e., the homotopy theory of algebraic varieties. The main point of motivic homotopy theory is to import the powerful methods of homotopy theory from topology to algebraic geometry. The PI would like to turn this relationship around by using results from algebraic geometry to discover new information about classical homotopy theory. For example, the PI has used motivic homotopy theory over C to completely reconstruct the classical 2-complete Adams-Novikov spectral sequence in a large range. Similarly, motivic computations over R have revealed new information about equivariant stable homotopy theory. Specifically, the PI is interested in the motivic version of the Adams spectral sequence, with base field R or C, where thorough computations are possible.
Homotopy theory is a technique for studying geometric objects up to certain kinds of deformations. Homotopical approaches often allow for concrete calculations that are otherwise inaccessible. Computations of stable homotopy groups have been a major topic of research in topology since the middle of the 20th century. Although tremendous progress has been made, much remains unknown. In the 1990's, Fabien Morel and Vladimir Voevodsky developed motivic homotopy theory. In analogy to classical homotopy theory, motivic homotopy theory allows us to study algebraic objects up to certain kinds of deformations. Their goal was to import the techniques of homotopy theory into the realm of algebra. The PI's goal is the reverse: to use algebraic results to obtain new information about classical homotopy theory.
