Algebraic and Positive Geometry of the Universe: from Particles to Galaxies

TL;DR

This paper explores the emerging field of positive geometry and algebraic methods in understanding particle interactions and cosmological phenomena, highlighting mathematical tools and their interdisciplinary connections.

Contribution

It introduces key developments and mathematical frameworks that unify algebraic geometry, combinatorics, and physics in the study of the universe.

Findings

Development of positive geometries in physics

Connections between algebraic geometry and cosmology

Mathematical tools for particle interaction analysis

Abstract

In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and observables in the universe: on the one hand, Feynman's approach reduces to the study of intricate integrals; on the other hand, one encounters the study of positive geometries. This article introduces key developments, mathematical tools, and the connections that drive progress at the frontier between algebraic geometry, the theory of $D$-modules, combinatorics, and physics. All these threads contribute to shaping the flourishing field of positive geometry, which aims to establish a unifying mathematical language for describing phenomena in cosmology and particle physics.