The Hidden Geometry That Could Explain the Universe

How can the tiniest particles and the vast structure of the universe be explained using the same kind of mathematics? This puzzle is the focus of recent research by mathematicians Claudia Fevola (Inria Saclay) and Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences), published in the Notices of the American Mathematical Society.

• Bringing math and physics together: The researchers show how algebraic methods, combined with the emerging field of positive geometry, can unify our understanding of phenomena ranging from subatomic particles to galaxies.
• Looking past Feynman diagrams: While Feynman diagrams remain central to quantum field theory, positive geometry provides an additional framework, using shapes and spaces to describe particle interactions.
• From collisions to the early universe: Algebraic geometry, D-module theory, and combinatorics are among the mathematical tools being applied to particle physics and cosmology, helping scientists unravel both the behavior of particles and the conditions that shaped the cosmos after the Big Bang.

The Symbiosis of Math and Physics

Mathematics and physics have always been deeply connected. Math provides the language and methods to describe how the physical world works, while physics often inspires the creation of new branches of mathematics. This back-and-forth relationship continues to be essential in areas like quantum field theory and cosmology, where cutting-edge math and physics evolve together.

Algebraic Geometry Meets Positive Geometry

In their work, the authors show how algebraic ideas and geometric forms can illuminate phenomena across vastly different scales, from particle collisions in accelerators to the structure of the universe itself. Their research centers on algebraic geometry but also reaches into a relatively new field called positive geometry. This area, influenced by discoveries in particle physics and cosmology, builds on the idea of representing interactions as higher-dimensional shapes rather than traditional Feynman diagrams.

One striking example is the amplituhedron, introduced in 2013 by physicists Nima Arkani-Hamed and Jaroslav Trnka, which encodes complex particle interactions as volumes of geometric objects. Positive geometry brings with it a rich combinatorial framework and may provide simpler ways to calculate scattering amplitudes, the probabilities that describe how particles scatter after colliding.

The potential applications extend far beyond particle physics. In cosmology, researchers study faint traces of the cosmic microwave background and the large-scale arrangement of galaxies to piece together the universe’s earliest history. Similar geometric methods are now being used in this work. For example, cosmological polytopes, themselves forms of positive geometry, can describe correlations within the first light of the universe and help reconstruct the physical rules that shaped the cosmos.

A Geometry for the Universe

According to the article, positive geometry should not be viewed as a niche mathematical oddity but rather as a possible unifying framework for multiple areas of theoretical physics. These structures provide a natural way to express the flow of information between physical systems. In doing so, they reflect how humans often grasp abstract ideas by connecting them to more tangible experiences.

The mathematics behind this is sophisticated and spans multiple disciplines. The authors draw on algebraic geometry, which defines shapes and spaces through solutions to systems of polynomial equations, algebraic analysis, which studies differential equations through mathematical objects called D-modules, and combinatorics, which describes the arrangements and interactions within these structures.

The formal objects under consideration, such as Feynman integrals, generalized Euler integrals, or canonical forms of positive geometries, are not merely mathematical abstractions. They correspond to observable phenomena in high-energy physics and cosmology, enabling precision computations of particle behavior and cosmic structures alike.

Bridging Scales with Mathematics

The study presents an approach with broad applicability and scalability. Scattering processes are often illustrated using Feynman diagrams. Feynman’s approach in the study of scattering amplitudes boils down to the study of intricate integrals associated to such diagrams. Algebraic geometry provides a range of tools for systematically investigating these integrals.

The graph polynomial of a Feynman diagram is defined in terms of the spanning trees and forests of the underlying graph. The associated Feynman integral can be expressed as a Mellin transform of a power of this graph polynomial, interpreted as a function of its coefficients. These coefficients, however, are constrained by the underlying physical conditions. Feynman integrals are therefore closely connected to generalized Euler integrals, specifically through restrictions to the relevant geometric subspaces. One way to study these holonomic functions is via the linear differential equations they satisfy, which are D-module inverse images of hypergeometric D-modules. Constructing these differential equations explicitly, however, remains challenging. In theoretical cosmology, correlation functions in toy models also take the form of such integrals, with integrands arising from hyperplane arrangements.

The complement of the algebraic variety defined by the graph polynomial in an algebraic torus is a very affine variety, and the Feynman integral can be viewed as the pairing of a twisted cycle and cocycle of this variety. Its geometric and (co-)homological properties reflect physical concepts such as the number of master integrals. These master integrals form a basis for the space of integrals when the kinematic parameters vary, and the size of this basis is, at least generically, equal to the signed topological Euler characteristic of the variety.

A Field in Motion

Fevola and Sattelberger’s work reflects a growing international effort, supported by the ERC synergy grant UNIVERSE+ of Nima Arkani-Hamed, Daniel Baumann, and Johannes Henn, Bernd Sturmfels. It brings together mathematics, particle physics, and cosmology focusing on precisely these connections between algebra, geometry, and theoretical physics. “Positive geometry is still a young field, but it has the potential to significantly influence fundamental research in both physics and mathematics,” the authors emphasize. “It is now up to the scientific community to work out the details of these emerging mathematical objects and theories and to validate them. Encouragingly, several successful collaborations have already laid important groundwork.”

The recent developments are not only advancing our understanding of the physical world but also pushing the boundaries of mathematics itself. Positive geometry is more than a tool. It is a language. One that might unify our understanding of nature at all scales.

Reference: “Algebraic and Positive Geometry of the Universe: From Particles to Galaxies” by Claudia Fevola and Anna-Laura Sattelberger, 2025, Notices of the American Mathematical Society.
DOI: 10.1090/noti3220

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