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Positive Solutions in the Sciences

Project description

Uncovering hidden patterns in science

In fields from physics to biology, scientists rely on polynomial equations to model complex systems. However, it is often only the positive real solutions that matter. These solutions can reveal steady states in biochemical networks, strategies in game theory, or even the nature of particle collisions. Supported by the Marie Skłodowska-Curie Actions programme, the POSSIS project aims to develop new tools in real algebraic geometry to pinpoint positive solutions. Specifically, it will focus first on a classical problem in particle physics (Landau equations linked to Feynman integrals). These singularities signal observable events in particle scattering experiments. POSSIS methods could transform how scientists analyse a wide range of systems where positivity is not just preferred, it is essential.

Objective

Polynomial equations are fundamental across various scientific disciplines, serving as powerful tools for modeling and solving real-world problems. Often, only the positive real solutions of these equations are of interest. The goal of this project is to develop methods within the framework of real algebraic geometry, specifically aimed at solving problems related to the positive solutions of polynomials that arise in scientific applications.

The main focus of this project is a classical problem that arises from the scattering of elementary particles in physics. The primary objective is to develop a method for computing the positive solutions of the Landau equations. These positive solutions are crucial because they lead to singularities in the physical region of Feynman integrals, which correspond to observable phenomena in scattering experiments.

Beyond particle physics, this project aims to extend the applications of real algebraic geometry to the study of Nash equilibria in game theory, steady states of biochemical reaction networks, and statistical models in phylogenetics. In all these fields, the models are given by parametrized polynomial equation systems with parameters that share linear dependencies. While methods from applied algebraic geometry have already proven successful in studying complex solutions, investigating the positive solutions of these polynomials requires a paradigm shift toward real algebraic geometry.

Fields of science (EuroSciVoc)

Keywords

Programme(s)

Topic(s)

Funding Scheme

HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships

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Call for proposal

(opens in new window) HORIZON-MSCA-2024-PF-01

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Coordinator

UNIVERSITAET LEIPZIG
Net EU contribution
€ 202 125,12
Address
RITTERSTRASSE 26
04109 Leipzig
Germany
Region
Sachsen Leipzig Leipzig
Activity type
Higher or Secondary Education Establishments
Links
Total cost
No data