The goal of the school is to introduce PhD students and young researchers to new directions in real algebraic geometry, with special emphasis on novel techniques ranging from algebraic geometry and topology to tropical geometry. At the same time, the meeting will give the opportunity to early career mathematicians to present their results and to foster collaborations.
Organizers: Michele Ancona and Antonio Lerario
Mini courses
Abstract: I will present an important Conjecture in Sub-Riemannian geometry, called Sard Conjecture. I will start by discussing examples and giving some insight on why the difficulties behind the Conjecture. Later, I will rephrase the Conjecture in geometrical terms and point out how real-geometry is useful in addressing it. In particular, I will show why sub-analytic geometry can be useful in this context, and which are the open challenges.
Abstract: The celebrated Nash--Tognoli theorem states that any compact differentiable manifold is diffeomorphic to the real locus of a smooth projective real algebraic variety. After recalling the proof of this classical theorem, we will investigate to what extent smooth maps between compact differentiable manifolds can be approximated by algebraic maps. We will present positive and negative results, as well as open questions.
Abstract: How many components can have a smooth real surface of degree 5 in the real projective space ? How can be arranged the components of a real algebraic curve of a given degree in the real projective plane ? Such types of questions, going back to Hilbert 16th problem, are still widely open. I will present in this mini course classical obstructions / constructions results and also new directions relying on Viro’s patchworking method and tropical geometry.
Research Seminars
Anna Bot (Basel)
A smooth complex rational affine surface with uncountably many nonisomorphic real forms
Abstract: A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Many families of complex varieties have a finite number of nonisomorphic real forms, but up until recently no example with infinitely many had been found. In 2018, Lesieutre constructed a projective variety of dimension six with infinitely many nonisomorphic real forms, and last year, Dinh, Oguiso and Yu described projective rational surfaces with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms.
Abstract: A smooth projective variety over the complex numbers satisfies the integral Hodge conjecture if all its integral Hodge classes are algebraic. This property has a natural analogue over the real numbers, called the "real integral Hodge conjecture". In the case of real abelian varieties of dimension three, it remains an open question whether this property always holds. I will explain this, and prove the real integral Hodge conjecture for various classes of real abelian threefolds.
Miruna-Stefana Sorea (SISSA and ULBS)
Combinatorial study of morsifications of real univariate singularities
Abstract: We study a broad class of morsifications of germs of univariate real analytic functions. We characterize the combinatorial types of the resulting Morse functions, via planar contact trees constructed from Newton-Puiseux roots of the polar curves of the morsifications. This is based on joint work with Arnaud Bodin (University of Lille), Evelia Rosa García Barroso (Universidad de La Laguna) and Patrick Popescu-Pampu (University of Lille).
