Stable Homotopy without Homotopy

Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel--Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.