arXiv subjects
Publications and source records attributed to Sergei Gukov.
We compute new lower bounds for determinants of $\{\pm 1\}$-matrices of orders $n=51$, $n=107$, $n=111$, $n=115$, and $n=119$, improving previous recorded bounds by $3.1\%$, $0.44\%$, $1.26\%$, $1.68\%$, and $2.12\%$, respectively. We provide the data necessary to construct these matrices.
Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.
The snake-in-the-box problem, introduced by Kautz in 1958, asks for the longest induced (chordless) path, called a snake, in the hypercube graph $Q_n$. The maximum length $a(n)$ is known in each dimension $n \leq 8$. We give snakes that are longer than the previous best-known in every dimension from $9$ to $13$, improving the lower bound on $a(n)$. All record-length paths are provided in a computer-verifiable dataset.
Applying machine learning techniques to solving long-standing mathematical conjectures can be particularly challenging due to their extreme reward sparsity. As an illustrative example, we consider Kalai's algebraic Hirsch conjecture and recast the construction of its counterexamples as a sparse-reward reinforcement learning problem on graphs. We propose a constrained options-based HRL framework with an equivariant graph neural network policy, which allows us to learn useful temporal abstractions for this task. We evaluate our approach over a wide range of degrees and demonstrate that it consistently outperforms classical RL algorithms as well as greedy search. By exploiting the hierarchical structure of the problem, we effectively provide a first-of-its-kind application of HRL to a problem in commutative algebra.
Mathematical search problems present a unique challenge for Reinforcement Learning (RL) due to vast search spaces and sparse rewards. In previous works, the Andrews-Curtis (AC) conjecture was established as an illustrative example of such problems. In this work, we identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible, with a scarcity of intermediate "hard-but-solvable" instances required for effective learning. We tackle this challenge through two primary avenues: novel data generation techniques to populate the difficulty gap, and significant algorithmic enhancements including the introduction of supermoves and Transformer-based architectures. We demonstrate substantial performance improvements over previous baselines, and release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.
We report the performance of Aletheia (Feng et al., 2026b), a mathematics research agent powered by Gemini 3 Deep Think, on the inaugural FirstProof challenge. Within the allowed timeframe of the challenge, Aletheia autonomously solved 6 problems (2, 5, 7, 8, 9, 10) out of 10 according to majority expert assessments; we note that experts were not unanimous on Problem 8 (only). For full transparency, we explain our interpretation of FirstProof and disclose details about our experiments as well as our evaluation. Raw prompts and outputs are available at https://github.com/google-deepmind/superhuman/tree/main/aletheia.
Recent advances in foundational models have yielded reasoning systems capable of achieving a gold-medal standard at the International Mathematical Olympiad. The transition from competition-level problem-solving to professional research, however, requires navigating vast literature and constructing long-horizon proofs. In this work, we introduce Aletheia, a math research agent that iteratively generates, verifies, and revises solutions end-to-end in natural language. Specifically, Aletheia is powered by an advanced version of Gemini Deep Think for challenging reasoning problems, a novel inference-time scaling law that extends beyond Olympiad-level problems, and intensive tool use to navigate the complexities of mathematical research. We demonstrate the capability of Aletheia from Olympiad problems to PhD-level exercises and most notably, through several distinct milestones in AI-assisted mathematics research: (a) a research paper (Feng26) generated by AI without any human intervention in calculating certain structure constants in arithmetic geometry called eigenweights; (b) a research paper (LeeSeo26) demonstrating human-AI collaboration in proving bounds on systems of interacting particles called independent sets; and (c) an extensive semi-autonomous evaluation (Feng et al., 2026a) of 700 open problems on Bloom's Erdos Conjectures database, including autonomous solutions to four open questions. In order to help the public better understand the developments pertaining to AI and mathematics, we suggest quantifying standard levels of autonomy and novelty of AI-assisted results, as well as propose a novel concept of human-AI interaction cards for transparency. We conclude with reflections on human-AI collaboration in mathematics and share all prompts as well as model outputs at https://github.com/google-deepmind/superhuman/tree/main/aletheia.
We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erd\H{o}s Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erd\H{o}s Problems.
We continue the investigation of symmetries and anomalies of $T[M]$ theories obtained by compactifying 6d SCFTs on an internal manifold $M$. We extend the notion of "polarizations on a manifold $M$" to cases where $M$ may have boundaries or defects. Through examples with $M$ of dimension two, three, and four, we illustrate recurring themes in compactifications -- for instance, the important roles played by Kaluza-Klein modes, and how the generalized symmetries (including higher-group and non-invertible ones) of $T[M]$, together with their anomalies, arise from non-trivial combinations of the parent 6d symmetries and the geometric structures of the internal manifold. For each dimension, we also focus on several topics that are especially interesting in that setting. These include: for 2-manifolds, the geometry of the "full moduli space" of $T[M_2]$ and its interaction with polarizations and symmetries; for 3-manifolds, the effect of torsion in homology on the spectrum of line operators in $T[M_3]$, together with applications to the study of quantum invariants such as $\hat Z_a(M_3, q)$; and for 4-manifolds, predictions for VOA$[M_4]$ following from symmetries of $T[M_4]$, as well as the construction of a new invariant of 4-manifolds that depends on two "$q$-parameters." Along the way, we discuss a range of topics that are of independent interest, such as how non-invertible symmetries in higher dimensions can become invertible under compactification, how to classify defects in quantum field theory via their response to a change of framing, and the interplay between $\hat Z_a$ and volume conjectures.
We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.
In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of ${\pi}_*({\rm TMF})$-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in ${\pi}_*({\rm TMF})$, conjecturally generalizing the theta function of a lattice. We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions.
This community paper developed out of the NSF Workshop on the Future of Artificial Intelligence (AI) and the Mathematical and Physics Sciences (MPS), which was held in March 2025 with the goal of understanding how the MPS domains (Astronomy, Chemistry, Materials Research, Mathematical Sciences, and Physics) can best capitalize on, and contribute to, the future of AI. We present here a summary and snapshot of the MPS community's perspective, as of Spring/Summer 2025, in a rapidly developing field. The link between AI and MPS is becoming increasingly inextricable; now is a crucial moment to strengthen the link between AI and Science by pursuing a strategy that proactively and thoughtfully leverages the potential of AI for scientific discovery and optimizes opportunities to impact the development of AI by applying concepts from fundamental science. To achieve this, we propose activities and strategic priorities that: (1) enable AI+MPS research in both directions; (2) build up an interdisciplinary community of AI+MPS researchers; and (3) foster education and workforce development in AI for MPS researchers and students. We conclude with a summary of suggested priorities for funding agencies, educational institutions, and individual researchers to help position the MPS community to be a leader in, and take full advantage of, the transformative potential of AI+MPS.
In recent papers [1,2], a new method to cross the natural boundary has been proposed, and applied to Mordell-Borel integrals arising in the study of Chern-Simons theory, based on decompositions into {\it resurgent cyclic orbits}. Resurgent analysis on the Stokes line leads to a unique transseries decomposition in terms of unary false theta functions, which can be continued across the natural boundary to produce dual $q$-series whose integer-valued coefficients enumerate BPS states. This constitutes a deeper new manifestation of resurgence in quantum field theoretic path integrals. In this paper we show that the algebraic structure of the {\it resurgent cyclic orbits}, combined with just the leading term of the $q$-series, completely determines the large order rate of growth of the dual $q$-series coefficients. The essential exponent of this asymptotic growth has a Cardy-like interpretation [10] of an effective central charge in a 3 dimensional quantum field theory with $\mathcal{N}=2$ supersymmetry related to the Chern-Simons theory through the $3d$-$3d$ correspondence.
We show that the fundamental property of preservation of relations, underlying resurgent analysis, provides a new perspective on crossing a natural boundary, an important general problem in theoretical and mathematical physics. This reveals a deeper rigidity of resurgence in a quantum field theory. We study the non-perturbative completion of complex Chern-Simons theory that associates to a 3-manifold a collection of $q$-series invariants labeled by Spin$^c$ structures, for which crossing the natural boundary corresponds to orientation reversal of the 3-manifold. Our new resurgent perspective leads to a practical numerical algorithm that generates $q$-series which are dual to unary $q$-series composed of false theta functions. Until recently, these duals were only known in a limited number of cases, essentially based on Ramanujan's mock theta functions, and the common belief was that the more general duals might not even exist. Resurgence analysis identifies as primary objects Mordell integrals: transforms of resurgent functions. Their unique Borel summed transseries decomposition on either side of the Stokes line is the unique decomposition into real and imaginary parts. The latter are combinations of unary $q$-series in terms of $q$ and its modular counterpart $\tilde{q}$, and are resurgent by construction. The Mordell integral is analytic across the natural boundary of the $q$ and $\tilde{q}$ series, and uniqueness of a similar decomposition which preserves algebraic relations on the other side of the boundary defines the unique boundary crossing of the $q$ series. This continuation can be efficiently implemented numerically. This identifies known unique mock modular identities, and extends well beyond. The resurgent approach reveals new aspects, and is very different from other approaches based on indefinite theta series, Appell-Lerch sums, and logarithmic vertex operator algebras.
In this paper we construct irregular representations of the affine Kac-Moody algebra $\widehat{sl}(2,\mathbb{C})$. We show how such irregular representations correspond to irregular Gaiotto-Teschner representations of the Virasoro algebra. The intertwiners for such representations satisfy a version of Knizhnik-Zamolodchikov (KZ) equations which we call irregular KZ equations. By connecting to 2d Liouville theory, we show how the conformal blocks governed by our irregular KZ equation correspond to 4d Argyres-Douglas theories with surface operator insertions. The corresponding flat connections describe braiding between such operators on the Gaiotto curve.
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.
In this paper, we present a geometric description of foams, which are prevalent in topological quantum field theories (TQFTs) based on quantum algebra, and reciprocally explore the geometry of Rozansky-Witten (RW) theory from an algebraic perspective. This approach illuminates various aspects of decorated TQFTs via geometry of the target space $X$ of RW theory. Through the formulation of the Knizhnik-Zamolodchikov (KZ) equation within this geometric framework, we derive the corresponding braiding and associator morphisms. We discuss applications where the target space of RW theory emerges as the Coulomb branch of a compactified 6d SCFT or Little String Theory, with the latter being particularly intriguing as it results in a compact $X$.
We explore statistical properties of BPS q-series for 3d N=2 strongly coupled supersymmetric theories that correspond to a particular family of 3-manifolds Y. We discover that gaps between exponents in the q-series are statistically more significant at the beginning of the q-series compared to gaps that appear in higher powers of q. Our observations are obtained by calculating saliencies of q-series features used as input data for principal component analysis, which is a standard example of an explainable machine learning technique that allows for a direct calculation and a better analysis of feature saliencies.