Paris, 04-06 December 2017

A valuation of an algebraic variety X over a field k is a map ν: k(X)* →Φ, where Φ is a totally ordered abelian group, which satisfies ν|k* = 0 and ν(f g) = ν(f) + ν(g), ν(f + g) ≥ min(ν(f), ν(g)). It can be thought of as a generalized vanishing order along ”something”. This ”something” can be for example a codimension one subvariety of a suitable birational model of X (divisorial valuation) or a transcendental curve traced on X.

Valuations and spaces of valuations (sets of valuations of k(X) endowed with a suitable topology) have become important tools in many problems of algebraic geometry, number theory and complex dynamics.

In this meeting we put the emphasis on two aspects of valuation theory:

1. The Poincaré series of a valuation or a bunch of valuations: to a valuation one can associate a graded algebra which encodes its properties.
The Poincaré series is the generating series of the dimensions of the homogeneous components of this algebra. Magical links between this series and topological invariants of singularities have been developed recently.

2. The behavior of certain functions defined on the space of valuations associated to a singular point reflects the geometry of the singularity.

An omnipresent tool for both aspects is the theory of key polynomials which gives a description of the graded algebra associated to a valuation. A course on this topic is also prepared.



Mini-courses:

Charles Favre (Palaiseau): Some functionals on the non-archimedean link of an isolated singularity.
Ann Lemahieu (Nice):
Introduction to the Poincaré series à la Campillo, Delgado and Gusein-Zade.
Michel Vaquié (Toulouse):
Extension of valuations, key polynomials and defect.


Talks:

Tamas Laszlo (Bilbao): On (topological) Poincaré series.
Wael Mahboub (Lebanon): Valuation over regular local rings of dimension 2.


Organizers:


Ana Belen de Felipe, Hussein Mourtada, Matteo Ruggiero, Bernard Teissier.


Administrator:

Elodie Destrebecq