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Given a noetherian local integral domain R and a valuation ν of its field of fractions which is non negative on R, i.e., an inclusion R ⊂ R ν of R in a valuation ring, I study a geometric specialization of R to the graded ring gr ν R determined by the valuation. If R ν dominates R and the residue field extension k = R/m → R ν /m ν is trivial, this graded ring corresponds to an essentially toric variety, of Krull dimension ≤ dimR but possibly of infinite embedding dimension; it is of the form:ir . In order to apply this fact to a characteristic-blind proof of local uniformization by deformation of a partial resolution of singularities of Specgr ν R, in the case where R is equicharacteristic and excellent, with k algebraically closed, I explore the following strategy: 1) Extend the valuation ν to a valuationν of a suitable noetherian scalewise ν-adic completionR (ν) of R such that the natural map gr ν R → grνR (ν) is scalewise birational.2) Obtain a presentation, that is a surjective continuous morphism of k-algebraswhere the left hand term is a suitable scalewise completion of the polynomial ring, and the kernel is generated up to closure by elements whose initial forms for the term order t deduced fromν are the binomial equations defining grνR (ν) , that is, which are of the form:3) Show that all but finitely many of these equations serve only to express the images of all w j 's in the noetherian ringR (ν) in terms of finitely many of them. Then show that a toric map in the coordinates (w j ) j∈J which resolves the singularities of the finitely many binomial equations of grνR (ν) involving these finitely many variables will uniformizeν onR (ν) , and that such resolving toric maps exist. 4) Use the excellence of R and the birationality of the map gr ν R → grνR (ν) to lift this to a uniformization of ν on R.

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Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. W. P. Thurston, quoted in [21]Among the different ways of sharing ideas are discussions, letters, videos, blogs, platforms such as MathOverflow or Images des Mathématiques, and publications. Publications are subject to more precise rules, and require more prolonged effort because they are not only a means of communication but also the main repository of ideas and results.That is why the activity of Catriona over four decades has been so useful for the mathematical community. She has a unique way of imagining possible publications, encouraging without pushing, showing great patience and understanding adapted to each author (or editor). She possesses an amazingly rich perception of the mathematical community, knowing of so many mathematicians not only what they do, but also what they are. Catriona really cares about authors (or editors) as persons as well as about the quality of the texts. Adding to this an inexhaustible energy, Catriona plays a unique and very important role in the spreading of understanding, as Thurston writes, and thus for the progress of our science. As an expression of gratitude and friendship, I wish to dedicate to her an exposition of some of the problems I have come across and so illustrate the last sentence of Thurston's quote.