# Valuation Theory in Interaction

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*Antonio Campillo Lopez; Franz-Viktor Kuhlmann; Bernard Teissier*

A publication of the European Mathematical Society

For more than a century, valuation theory has had its
classical roots in algebraic number theory, algebraic geometry and the
theory of ordered fields and groups. In recent decades it has seen an
amazing expansion into many other areas. Moreover, having been
dormant for a while in algebraic geometry, it has now been
reintroduced as a tool to attack the open problem of resolution of
singularities in positive characteristic and to analyze the structure
of singularities. Driven by this topic, and by its many new
applications in other areas, the research in valuation theory itself
has also been intensified, with a particular emphasis on the deep open
problems in positive characteristic.

The multifaceted development of valuation theory has been monitored
by two International Conferences and Workshops: the first in 1999 in
Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial
in Spain. This book grew out of the second conference and presents
high quality papers on recent research together with survey papers
that illustrate the state of the art in several areas and applications
of valuation theory.

This book is addressed to researchers and graduate
students who work in valuation theory or the areas where it is
applied, as well as a general mathematical audience interested in the
expansion and usefulness of the valuation theoretical approach, which
has been called the “most analytic” form of algebraic
reasoning. For young mathematicians who want to enter these areas of
research, it provides a valuable source of up-to-date information.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Undergraduate students, research mathematicians, and the general mathematical audience interested in valuation theory.