My main goal in this project is to carry out computations in motivic homotopy theory, i.e., the homotopy theory of algebraic varieties. Voevodsky's now famous proof of the Milnor Conjecture is based primarily on techniques in motivic homotopy theory. Specifically, I am interested in the motivic version of the Adams spectral sequence, with base field R or C. One reason to focus only on R or C is that thorough computations are possible over these fields, but the computations still exhibit non-classical phenomena. The motivic Adams spectral sequence has a major deficiency in that it is not known to converge. Even if the spectral sequence does not converge, it is nonetheless possible to draw many conclusions about motivic stable homotopy groups. I believe that these and other further computations will be an important guide to further study of motivic homotopy theory. There is also a chance that motivic computations will reveal something new about classical stable homotopy groups.
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. My goal is to carry out some fundamental computations in motivic homotopy theory. The goal is both a better understanding of the new algebraic theory as well as the classical geometric theory.