During the reporting period, the UNIVERSE+ collaboration made substantial progress at the intersection of mathematics, quantum field theory, and cosmology. This advances the collaboration's overarching goal of uncovering the geometric structures underlying fundamental physical concepts. A significant portion of the work involved developing algebraic, combinatorial, and geometric tools to better understand scattering amplitudes, Feynman integrals, and cosmological correlators. These efforts have led to the systematic characterization of algebraic varieties and semialgebraic sets that govern kinematic spaces. They have also provided new insights into convex and Grassmannian geometries that are relevant for amplitudes and have offered novel descriptions of limit objects arising in positive geometries. Simultaneously, the collaboration advanced the algebraic and analytic structures underlying cosmological correlators. This included developing D-module methods and new computational frameworks based on differential equations.
A central theme across these activities was unifying techniques originally developed for flat-space scattering amplitudes with problems in cosmology. New differential-equation approaches revealed hidden structural patterns in cosmological Feynman integrals. Previously unknown geometric objects were also identified as organizing principles for correlators in de Sitter space. This marks the first appearance of certain Grassmannian structures in a cosmological context. Parallel advances in quantum field theory have led to cutting-edge calculations of scattering amplitudes and Feynman integrals relevant to Quantum Chromodynamics and supersymmetric gauge theories. These calculations have been accompanied by foundational studies of positive geometry. These advances included the discovery of new sign patterns in amplitudes, the development of geometric formulations of Wilson loops, and the identification of novel positive geometries that govern both particle scattering and cosmological observables.
These scientific advances were enabled by strong collaborative efforts across the consortium. Regular interaction among researchers, facilitated through collaboration visits, focused workshops, and extended research stays, proved essential in aligning mathematical and physical perspectives, accelerating the transfer of ideas, and fostering joint work across institutions. This sustained interaction has been a key factor in achieving the integrated, conceptually unified results described above.
Building on these developments, the collaboration pursued deeper conceptual questions about the nature of amplitudes and correlators. New formalisms were developed that recast scattering and cosmological processes in geometric terms. Exploratory work opened the door to a fundamentally non-recursive, dual description of amplitudes and correlators. This framework makes it possible to access regimes, most notably the limit of infinitely many particles, that are inaccessible using traditional recursive or diagrammatic approaches. Together, these results demonstrate the strength of the collaboration's integrated approach and highlight the emergence of positive geometry as a unifying language across particle physics and cosmology.
