Collaborating with experts on 18 research problems, an advanced version of Gemini Deep Think helped solve long-standing bottlenecks in algorithms, machine learning and combinatorial optimization, information theory, and economics. Highlights from our article “Accelerating research with Gemini” include (corresponding section numbers in the article):
- Crossing mathematical boundaries for network puzzles: Progress on classic computer science problems such as “Max-Cut” (efficient network division) and the “Steiner Tree” (high-dimensional point connection) had slowed. Gemini broke both deadlocks by thinking outside the box. He solved these discrete algorithmic puzzles by extracting advanced tools (such as Kirszbraun’s theorem, measure theory, and the Stone-Weierstrass theorem) from unrelated branches of continuous mathematics. See Sections 4.1 and 4.2.
- Resolving a decade-old conjecture about online submodular optimization: A 2015 theoretical paper proposed a seemingly obvious rule for data flows: making a copy of an arriving item is always less valuable than simply moving the original. Experts struggled for a decade to prove it. Gemini designed a very specific three-element combinatorial counterexample, rigorously proving that long-standing human intuition was false. See Section 3.1.
- Machine Learning Optimization: Training AI to filter out noise typically requires engineers to manually adjust a mathematical “penalty.” The researchers created a new technique that did this automatically, but they couldn’t mathematically explain why. Gemini analyzed the equations and showed that the method is successful by secretly generating its own “adaptive penalty” on the fly. See Section 8.3.
- Updating economic theory for AI: A recent ‘Disclosure Principle’ for auctioning AI generation tokens only worked mathematically when bids were restricted to rational numbers. Expanding the domain to continuous real numbers invalidated the original proof. Gemini employed advanced topology and order theory to extend the theorem, accommodating real-world continuous auction dynamics. See Section 8.4.
- Cosmic String Physics: Calculating the gravitational radiation of cosmic strings requires finding analytical solutions to complicated integrals containing “singularities.” Gemini found a novel solution using Gegenbauer polynomials. This naturally absorbed the singularities, collapsing an infinite series into a closed form, a finite sum. See Section 6.1.
The results, which span diverse fields—from information and complexity theory to cryptography and mechanism design—demonstrate how AI is fundamentally changing research. For more details, see our article.
Given the fluid conference-driven publication portfolio of computer science, we describe these results by academic track record rather than a rigid taxonomy. About half are targeted for robust conferences, including ICLR ’26 acceptance, while most of the remaining findings will inform future journal submissions. Even when course-correcting the field by identifying errors (Section 3.2) or refuting conjectures (Section 3.1), these results highlight the value of AI as a high-level scientific collaborator.
The future of human-AI collaboration
Building on Google’s previous advances (1, 2, 3, 4, 5), this work demonstrates that general core models, leveraged with agent reasoning workflows, can act as a powerful scientific companion.
Under the guidance of mathematicians, physicists and computer experts, Gemini Deep Think mode is proving useful in fields where complex mathematics, logic and reasoning are essential.
We are witnessing a fundamental change in the scientific workflow. As Gemini evolves, it acts as a “force multiplier” for the human intellect, handling knowledge retrieval and rigorous verification so scientists can focus on conceptual depth and creative direction. Whether refining evidence, searching for counterexamples, or linking disconnected fields, AI is becoming a valuable collaborator in the next chapter of scientific progress.
Expressions of gratitude
We thank the community of mathematical, physical and computer experts for their help and advice on this project.
This project was a large-scale collaboration between Google and its success is due to the combined efforts of many people and teams. Thang Luong and Vahab Mirrokni led the overall research directions with the deep technical expertise of Tony Feng and David Woodruff.
The authors of the first article “Towards autonomous research in mathematics” include: Tony Feng, Trieu H. Trinh, Garrett Bingham, Dawsen Hwang, Yuri Chervonyi, Junehyuk Jung, Joonkyung Lee, Carlo Pagano, Sang-hyun Kim, Federico Pasqualotto, Sergei Gukov, Jonathan N. Lee, Junsu Kim, Kaiying Hou, Golnaz Ghiasi, Yi Tay, YaGuang Li, Chenkai Kuang, Yuan Liu, Hanzhao (Maggie) Lin, Evan Zheran Liu, Nigamaa Nayakanti, Xiaomeng Yang, Heng-Tze Cheng, Demis Hassabis, Koray Kavukcuoglu, Quoc V. Le, Thang Luong. We thank the following experts for their comments and discussions on the work: Jarod Alper, Kevin Barreto, Thomas Bloom, Sourav Chatterjee, Otis Chodosh, Michael Hutchings, Seongbin Jeon, Youngbeom Jin, Aiden Yuchan Jung, Jiwon Kang, Jimin Kim, Vjekoslav Kovač, Daniel Litt, Ciprian Manolescu, Mona Merling, Agustin Moreno, Carl Schildkraut, Johannes Schmitt, Insuk Seo, Jaehyeon Seo, Terence Tao, Cheng-Chiang Tsai, Ravi Vakil, Zhiwei Yun, Shengtong Zhang, Wei Zhang, Yufei Zhao.
The authors of the second article, “Accelerating Scientific Research with Gemini: Case Studies and Common Techniques,” include David P. Woodruff, Vincent Cohen-Addad, Lalit Jain, Jieming Mao, Song Zuo, MohammadHossein Bateni, Simina Branzei, Michael P. Brenner, Lin Chen, Ying Feng, Lance Fortnow, Gang Fu, Ziyi Guan, Zahra Hadizadeh, Mohammad T. Hajiaghayi, Mahdi JafariRaviz, Adel Javanmard, Karthik CS, Ken-ichi Kawarabayashi, Ravi Kumar, Silvio Lattanzi, Euiwoong Lee, Yi Li, Ioannis Panageas, Dimitris Paparas, Benjamin Przybocki, Bernardo Subercaseaux, Ola Svensson, Shayan Taherijam, Xuan Wu, Eylon Yogev, Morteza Zadimoghaddam, Samson Zhou, Yossi Matías, Jeff Dean, James Manyika, Vahab Mirrokni. This list includes Google researchers who build agent reasoning on top of Gemini and our expert academic collaborators who verify and collaborate on Gemini. We also thank Corinna Cortés for her careful review of the article.
We appreciate the critical support of the rest of the DeepThink team: Anirudh Baddepudi, Michael Brenner, Irene Cai, Kristen Chiafullo, Paul Covington, Rumen Dangovski, Chenjie Gu, Huan Gui, Vihan Jain, Rajesh Jayaram, Melvin Johnson, Rosemary Ke, Maciej Kula, Nate Kushman, Jane Labanowski, Steve Li, Pol Moreno, Sidharth Mudgal, William. Nelson, Ada Maksutaj Oflazer, Sahitya Potluri, Navneet Potti, Shubha Raghvendra, James Roggeveen, Siamak Shakeri, Archit Sharma, Xinying Song, Mukund Sundararajan, Qijun Tan, Zak Tsai, Erik Wang, Theophane Weber, Winnie Xu, Zicheng Xu, Junwen Yao, Shunyu Yao, Adams Yu, Lijun Yu and Honglei Zhuang.
We would like to thank the Gemini post-training team for building the foundational model for Deep Think: Arash Ahmadian, Ankesh Anand, Charles Chen, Yong Cheng, Kedar Dhamdhere, Philipp Fränken, Justin Gilmer, Elena Gribovskaya, Luheng He, Yangsibo Huang, Rishabh Joshi, Ajay Kannan, Arvind Kannan, Guangda Lai, Robert Leland, Hanzhao (Maggie) Lin, Yingjie Miao, Bryce Petrini, Corbin Quick, Vikash Sehwag, Yue Song, Pranav Talluri, Ankur Taly, George Tucker, Michael Voznesensky, Manish Reddy Vuyyuru, Yiming Wang, Jinliang Wei, Qiao Zhang, Yuan Zhang, Zizhao Zhang.
We thank Quoc Le, Koray Kavukcuoglu, Demis Hassabis, James Manyika, Yossi Matias and Jeff Dean for sponsoring this project.
Last but not least, we thank Divy Thakkar, Adam Brown, Vinay Ramasesh, Alex Davies, Thomas Hubert, Eugénie Rives, Pushmeet Kohli and Benoit Schillings for their comments and support of the project.