Mathematicians Claudia Fevola and Anna-Laura Sattelberger have introduced a novel framework where algebraic geometry acts as a bridge between particle physics and cosmology. Their research, published in the August 2025 issue of the Notices of the American Mathematical Society, is titled "Algebraic and Positive Geometry of the Universe: From Particles to Galaxies."
The work introduces positive geometry as a burgeoning field capable of representing cosmic phenomena and subatomic particle interactions through higher-dimensional geometric shapes. This paradigm offers an alternative to traditional Feynman diagrams, providing a new conceptual language for describing particle interactions. This research is a significant part of the international ERC synergy grant UNIVERSE+, which unites mathematicians and physicists exploring these connections.
The core objective of this interdisciplinary effort is to identify the fundamental geometric structures that underpin the universe, from the smallest particles to the largest galactic formations. The research suggests the existence of a universal geometric language that can elucidate diverse physical phenomena. Positive geometry, influenced by recent discoveries in particle physics and cosmology, represents interactions as geometric forms, offering a more intuitive visualization and calculation method compared to Feynman diagrams, which have long been central to quantum field theory.
The UNIVERSE+ project, a collaboration involving institutions such as the Institute for Advanced Study in Princeton and the Max Planck Institutes in Munich and Leipzig, aims to cultivate this new mathematical language. The project is funded by a €10 million ERC Synergy Grant. The research also touches upon the amplituhedron, a geometric object introduced in 2013 that encodes particle interactions as volumes of geometric objects, potentially simplifying calculations. This geometric framework is also being applied to cosmology, where structures like cosmological polytopes can describe correlations in the early universe and aid in reconstructing the physical laws that governed its formation.