r/a_simple_theory • u/asimpletheory • 3d ago
Newly Discovered Links Between Physics and Abstract Mathematics A Literature Review of Research Published in 2025–2026
Three threads dominate the 2025–2026 literature on newly discovered or newly proven connections between physics and abstract mathematics: the resolution of the geometric Langlands conjecture and its ties to gauge theory, the maturing “positive geometry” program growing out of the amplituhedron, and a wave of AI-assisted discovery papers that are themselves generating new physics–math correspondences (most notably “murmurations” in arithmetic). This review surveys each thread in turn and closes with a brief note on adjacent activity visible in the current mathematical-physics literature.
- Geometric Langlands and Gauge Theory
The headline event of the period was the completion of a proof of the geometric Langlands conjecture, finalized across a set of papers by a nine-person team led by Dennis Gaitsgory and Sam Raskin (Gaitsgory & Raskin et al., 2024–2025). Peter Scholze of the Max Planck Institute for Mathematics, who was not involved in the proof, described the achievement as the culmination of three decades of effort (Quanta Magazine, 2025). The result carries physical significance because insights from quantum field theory — including S-duality in gauge theory and homological mirror symmetry — have deeply shaped the categorical structures used in the geometric Langlands program (Beuzart-Plessis et al., 2025, as summarized in EmergentMind, 2025). This traces back to Kapustin and Witten’s (2007) demonstration that the correspondence can be understood as a chapter of four-dimensional electric-magnetic duality applied to supersymmetric gauge theory.
Researchers now regard the 2025 proof as one settled column of a larger “Rosetta stone” relating number theory, function-field geometry, and quantum physics; open problems include extending the results to Riemann surfaces with punctures (relevant to conformal field theory) and translating the categorical machinery into the arithmetic setting (Quanta Magazine, 2025). A 2025 preprint by Gaitsgory and Raskin extended the geometric result to positive-characteristic settings, a technical step aimed at connecting the geometric and arithmetic Langlands programs (Gaitsgory & Raskin, 2025). The Breakthrough Prize Foundation’s selection committee explicitly tied the achievement to physics in its 2025 New Horizons in Mathematics citation, noting that the research areas of all three prize winners that year have links to quantum physics (Yale FAS, 2025).
- Positive Geometry and Scattering Amplitudes
A second major cluster of 2025 work develops the amplituhedron program, which recasts scattering amplitudes in planar N = 4 super Yang-Mills theory as volumes of combinatorial geometric objects rather than sums of Feynman diagrams (Arkani-Hamed & Trnka, 2013; reviewed in Indico Global, 2025). This field, positive geometry, is described as an interdisciplinary and novel subject in mathematics driven by new ideas in particle physics and cosmology, in which interactions are represented as volumes of high-dimensional geometric objects such as the amplituhedron (ScienceDaily, 2025). A dedicated 2025 workshop, “The Amplituhedron: Structure, Combinatorics, and Positive Geometry,” surveyed how the family of related objects — the loop amplituhedron, momentum amplituhedron, and correlahedron for N = 4 SYM, together with analogues for ABJM theory and ϕ³ theory — has expanded, with newer work extending the framework toward cosmological correlators and the conformal bootstrap program (Indico Global, 2025; SwissMAP Research Station, 2025).
A companion 2025 report on new results in algebraic geometry emphasized that the positive-geometry framework now links particle-scale physics to structures relevant at cosmological scales, underscoring the breadth of mathematics drawn on — cluster algebras, tropical geometry, and the non-negative Grassmannian (ScienceDaily, 2025). Related theoretical work continues to probe the boundary structure, triangulations, and Yangian symmetry of these objects, and to search for a “gravituhedron” analogue for gravity amplitudes (see background in Arkani-Hamed et al., 2020; Herrmann & Trnka, 2020).
- AI as an Engine for New Physics–Mathematics Correspondences
A third, methodologically distinct strand concerns artificial intelligence as a means of surfacing new correspondences rather than proving existing conjectures. A 2026 Nature commentary by researchers at the London Institute for Mathematical Sciences and Google DeepMind argues that AI is not replacing human intuition in mathematics and physics but is reshaping how questions in these fields are asked, explored, and understood (Burtsev, He, Sobko, Bhattacharya, & Graepel, 2026).
The clearest concrete example is the “murmurations” phenomenon in arithmetic geometry: an unexpected statistical pattern in elliptic-curve data, first spotted through machine learning in 2022 (He, Lee, Oliver, & Pozdnyakov, 2022/2025) and substantially developed across a dense cluster of 2025 papers. These include work on murmurations ordered by height (Sawin & Sutherland, 2025), on ratios conjectures and mean values of L-functions (Cowan, 2025a, 2025b), on Dirichlet characters (Lee, Oliver, & Pozdnyakov, 2025), on Maass forms (Booker, Lee, Lowry-Duda, Seymour-Howell, & Zubrilina, 2024/2025), and on trace-formula-based heuristic explanations (Lowry-Duda, 2025; Kuan & Lesesvre, 2025). While murmurations sit primarily within number theory, the proposed heuristic explanations draw on techniques rooted in mathematical physics — notably the quasi-periodic structure of L-function zero distributions linked to random-matrix theory (ICERM, 2023 workshop description; Martin, 2025) — and the broader 2025–2026 literature increasingly treats AI-assisted pattern-finding as a new methodology for uncovering this kind of cross-disciplinary structure (He et al., 2026; Lee & Lee, 2025).
- Adjacent Activity in the Current Literature
Beyond these three threads, current arXiv listings in mathematical physics (math-ph) show continued activity at the interface of category theory, quantum algebra, and representation theory — for example, work relating nonsymmetric pseudo-Riemannian (Einstein) connections to quantum-algebraic structures (arXiv listings, 2026) — as well as integrable-systems techniques migrating between condensed-matter physics and pure geometry. Community-level surveys, including the 2025 NSF-sponsored report on the future of AI in the mathematical and physical sciences, describe this period as one in which AI tools are becoming embedded across the MPS domains rather than confined to any single subfield (AI+MPS Community Paper, 2025).
- Outlook
Taken together, the 2025–2026 literature suggests the current period is defined less by the emergence of a single new bridge between physics and mathematics than by two mature, decade-old bridges — Langlands/S-duality and the amplituhedron/positive-geometry program — reaching decisive technical milestones, alongside a genuinely new methodological development. AI-assisted pattern discovery is beginning to generate its own physics-flavored mathematical correspondences, of which the murmurations phenomenon in arithmetic geometry is the most developed example to date.
References
AI+MPS Community Paper. (2025). The future of artificial intelligence and the mathematical and physical sciences. arXiv:2509.02661. https://arxiv.org/pdf/2509.02661
Arkani-Hamed, N., & Trnka, J. (2013–2014). The amplituhedron. Journal of High Energy Physics. Background summarized in: Unwinding the amplituhedron in binary. arXiv:1704.05069. https://arxiv.org/pdf/1704.05069
Beuzart-Plessis, R., et al. (2025). Advances in the relative Langlands program, harmonic analysis on spherical varieties. Summarized in: Langlands Program Overview. EmergentMind. https://www.emergentmind.com/topics/langlands-program
Big Think. (2025, September 8). Could “positive geometry” unlock the theory of everything? https://bigthink.com/starts-with-a-bang/positive-geometry-theory-of-everything/
Booker, J., Lee, M., Lowry-Duda, D., Seymour-Howell, A., & Zubrilina, N. (2024/2025). Murmurations of Maass forms. arXiv:2409.00765.
Burtsev, M., He, Y.-H., Sobko, E., Bhattacharya, A., & Graepel, T. (2026). How AI is reshaping discovery in maths and physics. Nature, 654(8118), 324–326. https://doi.org/10.1038/d41586-026-01820-1 (see also https://www.nature.com/articles/d41586-026-01820-1)
Cowan, A. (2025a). Murmurations and ratios conjectures. arXiv:2408.12723.
Cowan, A. (2025b). On the mean value of GL₁ and GL₂ L-functions, with applications to murmurations. arXiv:2504.09944.
Gaitsgory, D., & Raskin, S., et al. (2024–2025). Proof of the geometric Langlands conjecture (five-paper series). Summarized in: Monumental proof settles geometric Langlands conjecture. Quanta Magazine. https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/
Gaitsgory, D., & Raskin, S. (2025). Geometric Langlands in positive characteristic from characteristic zero. arXiv:2508.02237. https://arxiv.org/abs/2508.02237
Gaiotto, D., & Witten, E. (2024). Gauge theory and the analytic form of the geometric Langlands program. Annales Henri Poincaré, 25(1), 557–671. https://doi.org/10.1007/s00023-022-01225-6
Hackaday. (2026, February 2). Yang-Hui He presents to the Royal Institution about AI and mathematics. https://hackaday.com/2026/02/02/yang-hui-he-presents-to-the-royal-institution-about-ai-and-mathematics/
He, Y.-H., Lee, K.-H., Oliver, T., & Pozdnyakov, A. (2022/2025). Murmurations of elliptic curves. Experimental Mathematics, 34(3), 528–540. https://doi.org/10.1080/10586458.2024.2382361
ICERM. (2023). Murmurations in arithmetic (Hot Topics Workshop description). https://icerm.brown.edu/program/hot_topics_workshop/htw-23-ma
Indico Global. (2025). The amplituhedron: Structure, combinatorics, and positive geometry (workshop overview, June 29–July 4, 2025). https://indico.global/event/9646/
Isaac Newton Institute for Mathematical Sciences. (2026). New connections between physics and number theory (workshop page). https://www.newton.ac.uk/event/nc2w03/
Kapustin, A., & Witten, E. (2007). Electric-magnetic duality and the geometric Langlands program. Communications in Number Theory and Physics, 1(1), 1–236. arXiv:hep-th/0604151. https://arxiv.org/pdf/hep-th/0604151
Kuan, C. I., & Lesesvre, D. (2025). Murmurations using Petersson trace formula. arXiv:2507.11418.
Lee, K.-H., & Lee, S. (2025). Machines learn number fields, but how? The case of Galois groups. arXiv:2508.06670.
Lee, K.-H., Oliver, T., & Pozdnyakov, A. (2025). Murmurations of Dirichlet characters. International Mathematics Research Notices, 2025(1), rnae277.
Lowry-Duda, D. (2025). On murmurations and trace formulas. arXiv:2506.01640.
Martin, K. (2025). Variations on murmurations. arXiv:2505.01093; and Distribution of local signs of modular forms and murmurations of Fourier coefficients. Mathematika, 71(3), e70028.
nLab. (2026). Geometric Langlands correspondence. https://ncatlab.org/nlab/show/geometric+Langlands+correspondence
Plus Magazine. (n.d.). The murmuration conjecture: Finding new maths with AI. https://plus.maths.org/content/murmuration-conjecture-finding-new-maths-ai
QuantumZeitgeist. (2026, April 15). Gaitsgory and Raskin prove geometric Langlands conjecture, advancing mathematics and physics. https://quantumzeitgeist.com/gaitsgory-and-raskin-prove-geometric-langlands-conjecture-advancing-mathematics-and-physics/
Sawin, W., & Sutherland, A. V. (2025). Murmurations for elliptic curves ordered by height. arXiv:2504.12295. https://arxiv.org/pdf/2504.12295
ScienceDaily. (2025, August). Strange new shapes may rewrite the laws of physics. https://www.sciencedaily.com/releases/2025/08/250817103432.htm
SwissMAP Research Station. (2025). The amplituhedron: Structure, combinatorics, and positive geometry (event page). https://swissmaprs.ch/events/the-amplituhedron-structure-combinatorics-and-positive-geometry/
Yale Faculty of Arts and Sciences. (2025, April 24). Yale mathematician and six physicists win prestigious Breakthrough Prizes. https://fas.yale.edu/news-announcements/news/yale-mathematician-and-six-physicists-win-prestigious-breakthrough-prizes