Matt Harvey

Dimitri Kanevsky, Julian Salazar, Matt Harvey

Let $V$ be a smooth cubic surface over a $p$-adic field $k$ with good reduction. Swinnerton-Dyer (1981) proved that $R$-equivalence is trivial on $V(k)$ except perhaps if $V$ is one of three special types--those whose $R$-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces $V$ currently known to have non-trivial universal equivalence. Beyond being intractable to Swinnerton-Dyer's approach, we observe that if these surfaces also had non-trivial $R$-equivalence, they would contradict Colliot-Thélène and Sansuc's conjecture regarding the $k$-rationality of universal torsors for geometrically rational surfaces. By devising new methods to study $R$-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions (the third special type, which contains every existing case of non-trivial universal equivalence), $R$-equivalence is trivial or of exponent 2. For the explicit cases, we confirm triviality: the diagonal cubic $X^3+Y^3+Z^3+ζ_3 T^3=0$ over $\mathbb{Q}_2(ζ_3)$--answering a long-standing question of Manin's (Cubic Forms, 1972)--and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982). This is the first in a series of works derived from a year of interactions with generative AI models such as AlphaEvolve and Gemini 3 Deep Think, with the latter proving many of our lemmas. We disclose the timeline and nature of their use towards this paper, and describe our broader AI-assisted research program in a companion report (in preparation).

Gemini Team, Petko Georgiev, Ving Ian Lei et al. · deepmind, mila

In this report, we introduce the Gemini 1.5 family of models, representing the next generation of highly compute-efficient multimodal models capable of recalling and reasoning over fine-grained information from millions of tokens of context, including multiple long documents and hours of video and audio. The family includes two new models: (1) an updated Gemini 1.5 Pro, which exceeds the February version on the great majority of capabilities and benchmarks; (2) Gemini 1.5 Flash, a more lightweight variant designed for efficiency with minimal regression in quality. Gemini 1.5 models achieve near-perfect recall on long-context retrieval tasks across modalities, improve the state-of-the-art in long-document QA, long-video QA and long-context ASR, and match or surpass Gemini 1.0 Ultra's state-of-the-art performance across a broad set of benchmarks. Studying the limits of Gemini 1.5's long-context ability, we find continued improvement in next-token prediction and near-perfect retrieval (>99%) up to at least 10M tokens, a generational leap over existing models such as Claude 3.0 (200k) and GPT-4 Turbo (128k). Finally, we highlight real-world use cases, such as Gemini 1.5 collaborating with professionals on completing their tasks achieving 26 to 75% time savings across 10 different job categories, as well as surprising new capabilities of large language models at the frontier; when given a grammar manual for Kalamang, a language with fewer than 200 speakers worldwide, the model learns to translate English to Kalamang at a similar level to a person who learned from the same content.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.