Abstract:

Motivic homotopy theory, introduced by Morel and Voevodsky, is the homotopy theory of smooth algebraic varieties with the affine line A1 playing the role of the interval [0,1]. This theory allows to systematically adapt methods of classical topology to the setting of algebraic geometry over general fields, providing new insights to geometric and cohomological properties of algebraic varieties. In my talk I will give a very brief introduction to the theory and discuss in more details some of the directions of the research, including (1) construction of the generalized cellular structures on some homogeneous varieties, (2) weakly oriented generalized motivic cohomology theories, (3) A1 Euler characteristic (4) vector fields on affine quadrics.