In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.

These are the notes of a talk in which we translate from the A1-homotopy theo- retic context an argument from (Mor99) showing that a homotopy invariant presheaf of spectra on the category of smooth schemes that satisfies Nisnevich descent auto- matically satisfies descent for abstract blow-ups. More concretely, Theorem 3.3.1 in (Mor99) roughly says that a Nisnevich distin- guished square of schemes is homotopy cartesian when seen in the stable A1-homotopy category. We complete the details of the proof of this theorem and write it in the lan- guage of presheaves of spectra. This is theorem 3.7 in this notes.

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem - to decide whether a word is an element of some basis of the free group - and the free factor problem can also be solved in polynomial time.