9901219 The proposer will continue to construct the homotopy theory of algebraic varieties. As of today we know how to define the stable and unstable homotopy categories for algebraic varieties and the fundamental cohomology theories such as the ordinary (=motivic) cohomology, K-theory and cobordisms. The analogs of other basic ingredients of theordinary homotopy theory, such as the Steenrod operations in ordinarycohomology, are also partly understood. During this stage of the project the main attack will be made on two other ingredients which are so far missing; these are the duality theory and the recognition principle for T-loop spaces.
Homotopy theory of algebraic varieties is an approach to the study of algebraic varieties (= systems of algebraic equations) based on an analogy. This analogy exists between the abstract categorical properties of the affine line (= the system of equations with one variable and no equations) and the toplogical space [0,1] called the unit interval. Using it one can translate methods of homotopy theory from topological spaces to algebraic varieties which gives new tools to deal with problems in algebraic geometry and number theory.
