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Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements^1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches³. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

 

Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group. 

Context. Terrestrial exoplanets in the habitable zone are likely a common occurrence. The long-term goal is to characterize the atmospheres of dozens of such objects. The Large Interferometer For Exoplanets (LIFE) initiative aims to develop a space-based mid-infrared (MIR) nulling interferometer to measure the thermal emission spectra of such exoplanets. Aims. We investigate how well LIFE could characterize a cloudy Venus-twin exoplanet. This allows us to: (1) test our atmospheric retrieval routine on a realistic non-Earth-like MIR emission spectrum of a known planet, (2) investigate how clouds impact retrievals, and (3) further refine the LIFE requirements derived in previous Earth-centered studies. Methods. We ran Bayesian atmospheric retrievals for simulated LIFE observations of a Venus-twin exoplanet orbiting a Sun-like star located 10 pc from the observer. The LIFE SIM noise model accounted for all major astrophysical noise sources. We ran retrievals using different models (cloudy and cloud-free) and analyzed the performance as a function of the quality of the LIFE observation. This allowed us to determine how well the atmosphere and clouds are characterizable depending on the quality of the spectrum. Results. At the current minimal resolution ( R = 50) and signal-to-noise ( S / N = 10 at 11.2 μ m) requirements for LIFE, all tested models suggest a CO 2 -rich atmosphere (≥30% in mass fraction). Further, we successfully constrain the atmospheric pressure-temperature ( P–T ) structure above the cloud deck ( P–T uncertainty ≤ ± 15 K). However, we struggle to infer the main cloud properties. Further, the retrieved planetary radius ( R pl ), equilibrium temperature ( T eq ), and Bond albedo ( A B ) depend on the model. Generally, a cloud-free model performs best at the current minimal quality and accurately estimates R pl , T eq , and A B . If we consider higher quality spectra (especially S / N = 20), we can infer the presence of clouds and pose first constraints on their structure. Conclusions. Our study shows that the minimal R and S/N requirements for LIFE suffice to characterize the structure and composition of a Venus-like atmosphere above the cloud deck if an adequate model is chosen. Crucially, the cloud-free model is preferred by the retrieval for low spectral qualities. We thus find no direct evidence for clouds at the minimal R and S / N requirements and cannot infer the thickness of the atmosphere. Clouds are only constrainable in MIR retrievals of spectra with S / N ≥ 20. The model dependence of our retrieval results emphasizes the importance of developing a community-wide best-practice for atmospheric retrieval studies. 

Abstract Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb {C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the number of such points of height $\leq B$ grows slower than any positive power of $B$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.