Centre Emile Borel |
In cooperation with the European Research Training Network “Real Algebraic and Analytic Geometry”
About 10
years ago, Boris Shapiro and Michael Shapiro made a remarkable
conjecture about real solutions to geometric problems coming from the
classical Schubert calculus. While the conjecture remains open, there
is truly overwhelming computational evidence supporting it, and
Eremenko and Gabrielov proved it for Grassmannians of 2-planes, where
the conjecture is the appealing statement that a rational function with
only real critical points must be real. In my talk, I will introduce the Shapiro conjecture and discuss what we know about it. This includes a simple counterexample and a refinement which is supported by massive experimental evidence. This evidence includes tantalizing computations which suggest a strengthening: that a certain discriminant polynomial is a sum of squares, or more generally that it has such an algebraic certificate of positivity. |
A substantial part of the talk will be devoted to the geometry of definable sets in a very special class of weakly o-minimal structures, namely weakly o-minimal non-valuational expansions of ordered groups. A weakly o-minimal structure expanding an ordered group is said to be non-valuational iff for any cut which is definable in , we have that . A weakly o-minimal expansion of a real closed field is non-valuational iff has no non-trivial valuations definable in . The geometry of sets definable in weakly o-minimal non-valuational expansions of ordered groups (fields) is closely related to the geometry of sets definable is o-minimal expansions of ordered groups (fields). In the weakly o-minimal non-valuational case, I will discuss an o-minimal style cell decomposition theorem and some of its most important consequences generalizing a relevant part of o-minimal geometry.
Back to the
programme of the workshop