We give a deterministic $n^{O(\log n)}$-time algorithm for graph isomorphism. To our knowledge, this is the first proof of such a worst-case bound for general graphs. The algorithm uses the local-certificate and color-refinement steps of Babai's quasipolynomial algorithm, but retains the decomposition of each recursive instance when it solves an auxiliary coloring problem. There are two reasons for retaining this information. First, a local test can make progress without removing many vertices: it can eliminate a layer of permutation symmetry. A logarithmic bound on the total weight of these layers allows tests of constant size. Second, the auxiliary bipartite graphs arising in the symmetry-breaking step can be represented inside the current instance, after retaining a row decomposition when necessary. These recursive calls inherit the progress already made. The analysis gives a recurrence with quadratic logarithmic cost and a fixed power decrease in every fresh recursive call, improving the explicit cubic logarithmic exponent in the analysis of the corrected Luks--Babai algorithm. We present the algorithm through colorings, partitions, and bipartite graphs, and collect the permutation-group ingredients in one appendix. The result extends to string isomorphism and returns one isomorphism and generators for the full automorphism group.
David P. Woodruff, Vincent Cohen-Addad, Lalit Jain, Jieming Mao, Song Zuo, MohammadHossein Bateni, Simina Branzei, Michael P. Brenner 等 36 位作者
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.