New Methods In Real Algebraic GEometry – NewMIRAGE

Real algebraic geometry focuses on zero sets of polynomials with real coefficients (algebraic sets) and on sets where such polynomials are of constant sign (semialgebraic sets).
Although real algebraic geometry shares common notions with complex algebraic geometry, real algebraic varieties behave quite differently from the complex ones (for instance, the real projective space is an affine real algebraic variety) and Grothendieck's scheme theory is less suited to study real algebraic varieties as Hilbert's Nullstellensatz fails over the real numbers.
Therefore real algebraic geometry has followed a rather different path from the one of complex algebraic geometry: it relies less on commutative algebra and more on analytic tools.
This branch of mathematics is at the intersection of several areas such as algebraic geometry, commutative algebra, analytic geometry, differential topology, and model theory.
More recently, effective real algebraic geometry has grown rapidly with the development of algorithmic methods, in connection with more applied problems coming from robotics or computer-aided design.
The NewMIRAGE project is divided into two axes, each one having its own research problems.
Axis 1. A major problem in real algebraic geometry consists in defining a natural ring of functions on a real algebraic variety. This ring should have good algebraic properties and be rigid enough to encode information about the variety. The first axis of the NewMIRAGE project aims to focus on two classes of functions, namely continuous rational functions and regulous functions.
Axis 2. The Milnor fibre is a fundamental source of invariants to study singular analytic hypersurface in the complex case. Recently, a common generalisation of both the topological and motivic Milnor fibres has been introduced by J.-B. Campesato, G. Fichou and A. Parusinski. The second axis of the NewMIRAGE project consists in adapting and studying this framework to the real setting.