**Title:** Differential equations of order one (positive characteristic)

**Abstract:** This is joint work with Marius van der Put. We recall the definition from Grothendieck’s [EGA4] of a certain non-commutative algebra *D* of differential operators in characteristic *p* > 0, serving as an analogue of the classical Weyl algebra generated by *z* and *d/dz*. The notion “stratified order one differential equation *f*(*y*^{1}, *y*, *z*) = 0″ in characteristic *p* > 0 is defined in terms of the algebra *D*. In the talk, relations of this definition with more classical notions such as super-singularity and the Cartier operator are given. The definition allows one to formulate an analogue of the Grothendieck-Katz conjecture, which in the present instance asserts that an order one equation in characteristic zero admits infinitely many algebraic solutions, precisely when for almost all primes *p*, a reduction modulo *p* of the equation is stratified. We can prove this conjecture in the special case of autonomous equations.