Marc Levine Title: Motivic tubular neighborhoods and exponential maps Abstract: An algebraic version of of the tubular neighborhood construction for differentiable manifolds is possible in one of the Morel-Voevodsky homotopy categories. An algebraic version of the punctured tubular neighborhood is also possible, after a stabilization process. This latter construction leads to a stable algebraic version of the exponential map on the punctured normal bundle of an immersion. We apply this to give a provisional definition of the stable motivic cohomology of the moduli space of smooth curves.