Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

TL;DR

The paper presents a vision of AI as an active research partner in theoretical science, demonstrated through Gemini Deep Think-driven case studies across information theory, cryptography, geometry, and graph theory. It articulates a practical playbook for human-AI collaboration, emphasizing iterative prompting, cross-domain idea transfer, counterexample generation, rigorous formalization, external validation, and neuro-symbolic loops that couple symbolic reasoning with automated testing. Key results include refuting an online SWM conjecture via autonomous counterexamples, exposing a fatal flaw in a SNARG construction through adversarial review, and solving geometry/graph problems by integrating harmonic analysis, Kirszbraun extensions, and Bethe approximations. The work also explores AI-enabled IDE workflows, autonomous verification pipelines, and improvements to coresets and streaming submodular optimization, underscoring the potential and limitations of AI in accelerating deep mathematical reasoning. Overall, the paper argues that with careful human orchestration and structured AI workflows, frontier models can significantly shorten discovery cycles and reveal novel connections across disciplines, albeit with rigorous verification and awareness of non-elementary tools required for certain proofs.

Abstract

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Figures (23)

• Figure 1: Overview of the reasoning architecture used in many testimonials: an extensive exploration of the solution space combined with deep reasoning and a long tail of automated and human verification and in several cases, guidance and iterative feedback.
• Figure 2: Visualizing Cross-Pollination: The AI model successfully resolved open problems by reframing them in entirely different mathematical domains, bringing advanced topological and geometric theorems to bear on algorithmic and combinatorial problems.
• Figure 3: Iterative self-correction prompt.
• Figure 4: Original prompt for model
• Figure 5: Early prompt for trying to understand the query complexity for two rounds.

Theorems & Definitions (162)

• Definition 3.1: Permutation Variants korula2015online
• Conjecture 3.2: Conjecture 15 of korula2015online
• Theorem 3.3
• Lemma 4.3
• proof
• Lemma 4.4: Compactness of $\mathcal{M}(S^{d-1})$
• proof
• Lemma 4.5: Portmanteau Lemma (Partial)
• proof
• Lemma 4.6