Colloquim talk; Kristin Shaw: Betti numbers of real algebraic hypersurfaces
Almost 150 years ago Harnack proved a tight upper bound on the number of connected components of a real planar algebraic curve of degree d. This led to Hilbert’s 16th problem which asked for a classification of the topological types of real algebraic curves in the plane. To this day this problem is only solved for curves up to degree 7. In higher dimensions we can ask analogous questions, and we know very little about the possible topologies of real algebraic hypersurfaces. For example, we do not have a tight upper bound on the number of connected components of a real quintic surface in projective space!
In any degree and any dimension there are two sides to the classification problem; one side is to provide constructions of real algebraic varieties having possible topological types, the other side is to provide obstructions to realizing impossible topological types. In this talk I will explain the formulation and proof of a conjecture of Itenberg which, for a particular class of real algebraic hypersurfaces, bounds their individual Betti numbers in terms of the Hodge numbers of their complexifications. The real hypersurfaces we consider arise from Viro’s patchworking, which is a combinatorial method for constructing topological types of real algebraic varieties. It is known that these bounds are not satisfied by all real algebraic hypersurfaces, so in a sense these bounds obstruct the method of construction offered by Viro. Although the bounds prove that the patchworking method is limited, the tools used in the proof offer complete control over the topology of these combinatorially constructed surfaces.
This talk is on joint work with Arthur Renaudineau.
NB! Coffee/Tea/Biscuits from 14.00.