The course will take place during the period of May 22 to June 16, at the BICMR. It is intended for students from year 3 to 5 of their studies in mathematics. Some knowledge of algebra (fields, polynomials) is assumed. Ideally, students should also know a little Galois theory. Familiarity with basic notions of model theory is also assumed (e.g., the course of René Cori). In any case, the level of the course will be adapted to the participants, and if necessary, reviews of definitions and important results will be made.

Valued fields play a very important role in mathematics, especially in number theory and algebraic geometry. The aim of this course is, after giving the definitions, to present their basic algebraic properties. We will also describe several examples, among which the field of $p$-adic numbers and the field of generalized power series. We will then study the model theory of valued fields, and how it can be applied to non-logical problems. In particular we will give quantifier elimination results for the theory of algebraically closed valued fields, and for the theory of $p$-adic fields. The course will conclude with applications. Possible  applications include: results of Ax and Kochen; introduction to $p$-adic integration and Denef’s results; another way of looking at Berkovich spaces.

Bibliography

• Antonio J.  Engler, Alexander Prestel, Valued fields, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.

• Lou van den Dries, Lectures on the model theory of valued fields, in Model theory in algebra, analysis and arithmetic, 55 — 157, Lecture Notes in Math., 2111, Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2014.

Timetable:

May 24  10:00—12:00

May 25  13:00—15:00

May 26  10:00—12:00

Application Form