Confluentes Mathematici

Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers …. These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties.

We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.

We study the class of real-valued functions on convex subsets of ℝⁿ which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xiao). We also prove that a function on ℝ² is a tropical polynomial if and only if its restriction to each translate of a generic tropical line is a tropical polynomial.

We make some attempts to define a general notion of groups and fields of dimension one, and to determine their algebraic properties.

A number of results connecting quantum and classical Markov semigroups, as well as their dilations is reported. The method presented here is based on the analysis of the structure of the semigroup generator. In particular, measure-valued processes appear as a combination of classical reduction and classical dilation of a given quantum Markov semigroup.