Olivier Benoist

Sums of squares in analytic geometry

Abstract: Pfister has obtained the following quantitative version of Hilbert's 17th problem: a nonnegative real polynomial in n variables is a sum of 2^n squares of rational functions. After a general introduction to sums of squares problems, I will explain an analytic analogue of Pfister's theorem, which in particular implies that a nonnegative real-analytic function on a compact real-analytic manifold of dimension n is a sum of 2^n squares of real-analytic meromorphic functions.

Stefan Schreieder 

Irrationality of hypersurfaces

Abstract: This mini course introduces some of the cycle-theoretic tools in the study of irrationality of rationally connected varieties, such as decompositions of the diagonal and unramified cohomology. The main goal is to explain the proof that a very general complex hypersurface of dimension n ≥ 3 and degree  d ≥ log2(n)+2 is not retract rational, hence not (stably) rational. Further applications will be given if time allows. 

Susanna Zimmermann 

Classifying real birational involutions of the plane

Abstract. The history of classifying finite groups of birational maps of the plane up to conjugation is very old. The first list of involutions was compiled by Bertini but it was incomplete. The work of many culminated in the classification of birational involutions of the plane by Bayle-Beauville from 2000. It turns out that the classification is much more involved when we work over a non-closed field, already over a nice field like the field of real numbers. In this lecture I will explain the classification the real numbers.