Real, combinatorial and tropical aspects of the non-primitive combinatorial patchworking

• Applicant: Aloïs Demory
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554526864

Topology of real algebraic varieties is the study of the shapes formed by the zeroes of polynomials with real coefficients. Since the foundational works of A. Harnack and D. Hilbert in the second half of the XIXth century regarding real algebraic plane curves, research in the field is divided into two complementary directions: finding restrictions on the topology of real algebraic varieties defined by polynomials of a given degree and constructing real algebraic varieties with interesting topological properties. Combinatorial patchworking is a powerful method introduced by O. Viro in the 1980s for constructing real algebraic hypersurfaces with prescribed topological properties. It turned out to be a bridge between real or complex algebraic geometry and tropical geometry, a field concerned with combinatorial piecewise linear objects. The connections between real and tropical objects have been extensively studied in the case of primitive patchworking, a special case of combinatorial patchworking that is linked with the so-called smooth tropical hypersufaces. These developments have led to striking results concerning the topology of the real algebraic hypersurfaces that can be constructed using primitive patchworking. However, a lot less has been done for the non-primitive combinatorial patchworking. It is harder to understand than primitive patchworking, but also offers more possibilities. The main achievement of my PhD is the construction via non-primitive patchworking of maximal three- and four-dimensional real algebraic hypersurfaces that cannot be obtained using primitive patchworking. The proposed research project aims to investigate the topological properties of the real algebraic varieties that can be obtained using non-primitive patchworking. This will involve continuing to construct interesting examples of varieties in diverse ambient spaces, while also trying to understand the limits of the construction method. To achieve this, I plan on searching for combinatorial restrictions and on adapting the tools that were developed for smooth tropical hypersurfaces to a wider range of objects.

• DFG Programme: WBP Position