Mathematics

This work releases curated 2020-2021 Genomes to Fields imagery from 356 drone flights across 19 environments covering 1,180 maize hybrids, and uses functional principal components of vegetation index and weather trajectories together with genomic information for yield prediction and QTL mapping, finding that combined genomic and phenomic kernels raised held-out hybrid prediction to r = 0.501 when environments were represented in training and 0.408 when environments were also withheld, that accumulated growing degree days offered no consistent advantage over days after planting, that weather contributed modest task-dependent gains, and that NGRDI functional principal components repeatedly mapped to quantitative trait loci on chromosomes 3 and 7.

According to a Nature news report, OpenAI claims its most advanced AI model proved that the Navier–Stokes equations produce a 'singularity' in some instances, predicting physically impossible infinite speeds; the report situates that claim by reviewing the equations' known limits for compressible fluids and rarefied gases, and alternative modelling routes such as the Boltzmann equation, molecular simulation, and a 'triple decker' multiscale coupling.

This is a competition announcement: the SAIR Foundation and Lean FRO have launched a multi-stage Lean Kernel Challenge whose Stage 1 asks participants to develop algorithms for eight fixed problems (Fibonacci, integer partitions, the Mertens function, prime counting, matrix permanent, Rule 110, SHA-256, and polynomial discriminant) and to prove in Lean that each algorithm matches the supplied specification for every input, with submissions due November 20, 2026, 23:59 AoE.

This work models iterative generative-model training as a closed-loop stochastic process on the probability simplex and analyzes its dynamics under the Fisher-Rao metric instead of the Euclidean metric, deriving contraction and invariance bounds that remain meaningful as dimension grows and giving a human-to-synthetic data ratio threshold that guarantees convergence to a Fisher-Rao ball, concluding that the effective required human-data ratio is higher than previously implied.

A Nature news report says OpenAI announced on 8 September that its AI model solved the Navier–Stokes problem in fluid dynamics and verified the proof using the Lean language, while mathematicians Tristan Buckmaster and Levent Alpöge say they had already been working on the problem with OpenAI and Anthropic tools and Andreas Thom says his discussions about non-sofic groups resembled OpenAI's later approach, prompting an open letter from 25 Fields Medal winners and wider debate about credit and training-data provenance in the AI era.

The paper establishes a degree-independent spectral independence bound for log-concave Holant problems on simple graphs, yielding relaxation-time bounds for Glauber dynamics of O_λ(m) for the monomer–dimer model at activity λ, O_{b,λ}(m) for b-matchings at fugacity λ>0, and O(bm) for uniform b-matchings, where m is the number of edges.

This guest post by the editors of the communal blog Proofs and Prompts describes why the blog was started and what it aims to do: as AI became a central topic of conversation in the mathematical community, the editors created a collective space for mathematicians who lack a natural platform, especially PhD students, and reports that about a month after launch contributors ranged from PhD students and undergrads to Fields medallists and hobbyists, with posts ranging from a call for a general moratorium on AI to the view that AI plays too small a role in mathematics, while noting that contributors remain concentrated in Western Europe and North America, few have experience developing or evaluating LLMs, and only a handful of the forty-or-so published posts were written by women, and inviting

This work develops a continuum-based reaction-diffusion model of XIST-mediated gene silencing spread on chromosomes, finding that XIST spread can be tuned by known negative feedback loops regulating its synthesis and degradation, while silencing spread is controlled by a wave-pinning mechanism driven by global regulation of the silencing complex together with local epigenetic regulators, and uses a 3D chromosome structure inferred from experimental data to show spatiotemporal regulation of silencing spread.

This guest post by Bryna Kra uses the Nivat conjecture to argue that AI has sharply lowered the cost of producing sophisticated proofs, as shown by several purported proofs she received this week, and that because a proof is more than a certificate of correctness—it is understanding, explanation, and collective knowledge—the mathematical community must redefine and reward discovery, proof, formalization, and exposition.