Is there a fundamental connection between the unimaginably large and the infinitesimally small? How is it possible that the behavior of elementary particles, these tiny actors on the stage of the quantum world, and the magnificent structure of the entire universe can be described by the same mathematical language? This question is not just a mere philosophical dilemma, but lies at the very heart of revolutionary research that combines the deepest secrets of physics with the abstract beauty of geometry. The latest works by mathematicians Claudia Fevola and Anne-Laure Sattelberger shed new light on this fascinating synthesis, revealing how algebraic and so-called positive geometry are becoming the key to a unified understanding of nature, from subatomic collisions to the echoes of the Big Bang.

The Symbiosis of Mathematics and Physics: A Language for Describing Reality

The relationship between mathematics and physics has always been deep and mutually beneficial. Mathematics provides a precise language and powerful tools with which physicists describe natural phenomena, while the real world of physics constantly poses new challenges and encourages the development of new, abstract mathematical ideas. This unbreakable bond is more alive today than ever, especially in frontier areas of science such as quantum field theory and cosmology. It is precisely here, where our intuitive understanding of reality breaks down, that advanced mathematical structures become the only guide through the unknown. At the center of these new insights is algebraic geometry, a discipline that studies geometric shapes defined by the solutions of polynomial equations. However, in recent years, an even newer and more potent idea has emerged from this field – positive geometry.

This is an interdisciplinary field that has been directly spurred by revolutionary ideas from particle physics and cosmology. The inspiration came from a concept that complements and extends the traditional approach to calculating particle interactions, known as Feynman diagrams, in a completely new way. Instead of summing countless possible interactions, positive geometry offers a more elegant alternative: interactions are represented as the volumes of abstract, multidimensional geometric objects. The most famous such object is the amplituhedron, introduced in 2013 by theoretical physicists Nima Arkani-Hamed and Jaroslav Trnka.

A Revolution in the World of Particles: More than Feynman Diagrams

To understand the depth of this change, we must briefly return to the classic approach. In quantum field theory, when two or more particles collide, they can scatter in many different ways. To calculate the probability of each specific outcome, physicists have for decades relied on a method developed by Richard Feynman. His famous Feynman diagrams vividly depict all possible paths and interactions that particles can undergo during a collision. Each diagram corresponds to a complex mathematical expression, and the final probability, known as the scattering amplitude, is obtained by summing the contributions of all possible diagrams.

This approach has been incredibly successful and is the foundation of the Standard Model of particle physics. However, it also has significant limitations. For more complex collisions, the number of relevant Feynman diagrams grows at an astronomical rate, turning calculations into an almost unsolvable task, even for the most powerful supercomputers. This is where the amplituhedron comes into play. This fascinating geometric object, a kind of multidimensional jewel, possesses an incredible property: its volume directly corresponds to the scattering amplitude. Instead of painstakingly summing thousands of diagrams, physicists can now solve the problem by calculating the volume of a single geometric shape. What is even more impressive is that fundamental physical principles, such as locality (the idea that interactions occur at a single point in space and time) and unitarity (the requirement that the sum of the probabilities of all possible outcomes must be 1), are not imposed from the outside but are subtly woven into the very geometry of the amplituhedron.

Echoes of Creation: Geometric Footprints in the Universe

The implications of this geometric approach extend far beyond the world of particle accelerators. Surprisingly, similar mathematical tools are proving crucial in studying the largest possible scales – the history and structure of the universe itself. Cosmologists today study the oldest light in the universe, the so-called Cosmic Microwave Background (CMB), and the distribution of galaxies to reconstruct the events that shaped our cosmos in its first moments.

The CMB is a kind of "fossil" remnant of the Big Bang, a picture of the universe when it was only 380,000 years old. Tiny temperature fluctuations in this radiation represent the seeds from which all the structures we see today – stars, galaxies, and galaxy clusters – later formed. Mathematical tools arising from positive geometry are now being used to describe the correlations between these ancient fluctuations. Objects known as cosmological polytopes, which are themselves examples of positive geometries, allow scientists to represent these correlations and work backward, trying to decipher the fundamental physical laws that governed at the moment of the universe's birth. The fact that the same type of mathematical structure – positive geometry – can describe both subatomic collisions and cosmic correlations strongly suggests a deep, hidden unity in the laws of nature.

The Mathematical Engine Driving Knowledge

The mathematics behind these discoveries is extremely sophisticated and connects several seemingly different disciplines. The authors rely on a wide range of tools in their work:

• Algebraic geometry: It provides the fundamental framework, defining shapes and spaces through systems of polynomial equations.


• Algebraic analysis: Through the study of mathematical objects called D-modules, this discipline allows for the analysis of complex differential equations satisfied by functions important to physics.


• Combinatorics: It deals with counting and describing the arrangements and interactions within these complex structures, which is crucial for understanding, for example, the structure of Feynman diagrams or the facets of polytopes.

The formal objects being studied, such as Feynman integrals or generalized Euler integrals, are not mere mathematical abstractions. They directly correspond to measurable, observable phenomena in high-energy physics and cosmology. Feynman's approach to studying scattering amplitudes boils down to analyzing intricate integrals associated with graphs. Algebraic geometry offers a systematic way to investigate these integrals. For example, a Feynman integral can be viewed as a pairing of so-called "twisted cycles and cocycles" of an algebraic manifold. Its geometric and homological properties reflect physical concepts, such as the number of "master integrals" that form a basis for all possible integrals in a given process.

A Field in Motion: International Collaboration and the Future

The work of Fevola and Sattelberger reflects a growing international effort that brings together leading minds from mathematics, particle physics, and cosmology. A significant role in this is played by the prestigious UNIVERSE+ project, funded by the European Research Council (ERC), which brings together pioneers of this field such as Nima Arkani-Hamed, Daniel Baumann, Johannes Henn, and Bernd Sturmfels. Their common goal is to precisely investigate these very connections between algebra, geometry, and theoretical physics.

The authors emphasize: "Positive geometry is still a young field, but it has enormous potential to significantly impact fundamental research in both physics and mathematics." The scientific community is now faced with the exciting task of elaborating on these new mathematical objects and theories in detail and verifying them experimentally. It is encouraging that numerous successful collaborations have already laid a solid foundation. The latest achievements not only advance our understanding of the physical world but also simultaneously push the boundaries of mathematics itself. Positive geometry is proving to be more than a tool; it is a potential universal language that could unite our understanding of nature at all its levels, from the quantum flicker to the cosmic expanse.