The aim of the summer school is to understand two recent results in the domain of resolution of singularities in positive characteristics.

On the one hand, resolution of threefolds by Vincent Cossart and Olivier Piltant [CP] and on the other hand, local uniformization of Abhyankar valuations by toric morphisms which is due to Bernard Teissier [T].

Resolution of singularities is a very important tool in Algebraic Geometry. It is a modification of a singular variety X into a non-singular variety Y which shares many properties with X; in particular Y is birational to X, i.e. they are isomorphic on an open dense subset.

Resolution of singularities has made it possible to define invariants of singularities which were crucial to the classification of algebraic varieties in Birational Geometry. It can also be seen as an extremely useful change of variables in many integration theories (p-adic integration, motivic integration, ... ) which permits to apply it to problems in number theory and analysis.

The celebrated theorem of Hironaka, which asserts that a resolution of singularities exists for a variety which is defined over a field of characteristic zero, was published in 1964. But the problem of the existence of a resolution in characteristic p>0 and for arithmetical varieties remains widely open except in small dimensions. The goal of the meeting that we would like to organize is to understand two recent results, of different natures, about resolution of singularities in all characteristics.

The first result is the proof of the existence of a resolution of singularities for threefolds by Cossart and Piltant. This result is a breakthrough in this domain: previously, for arithmetical varieties, only the case of varieties of dimension less than or equal to two were known (in the sixties). The proof cleverly uses the whole arsenal of techniques that was introduced by Zariski, Abhyankar and Hironaka and solves the problem after reducing it to another difficult problem, local uniformization, a very local version of resolution of singularities.

The second result is the proof of the existence of a local uniformization of an Abhyankar valuation by one toric morphism, due to Teissier. The proof uses new geometric techniques in valuation theory. This result is part of an approach to resolution of singularities outlined by Teissier himself. He conjectures the following: given XAn there exists an embedding An AN, Nn (and a toric structure) such that XAN can be resolved by one toric map.

The plan is to do two lecture series by the authors of the two results, and a series of talks by the participants (including young researchers) about the notions and the building blocks of the proofs of the results.

Since both approaches are quite different, the hope is that bringing together people of both direction will yield a synergy effect where both directions can profit from the others knowledge.

Moreover, introducing younger people and interested people from other areas to the topic carries the chance for new perspectives.

On the one hand there might be new input on the problem coming from other areas and on the other hand we like to advertise these techniques and ideas with the hope that these might also be useful beyond local uniformization.

[CP] V. Cossart and O. Piltant, Resolution of Singularities of Arithmetical Threefolds II, arXiv:1412.0868v1 [math.AG], 2014

[T] B. Teissier, Overweight deformations of affine toric varieties and local uniformization, "Valuation theory in interaction", Proceedings of the second international conference on valuation theory, Segovia–El Escorial, 2011, European Math. Soc. Publishing House, Congress Reports Series, Sept. 2014, 474–565, (see also arXiv:1401.5204v3 [math.AG], 2014).