Daily Papers

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

We point out that there are two different chiral spin liquid states on the triangular lattice and discuss the conducting states that are expected on doping them. These states labeled CS1 and CS2 are associated with two distinct topological orders with different edge states, although they both spontaneously break time reversal symmetry and exhibit the same quantized spin Hall conductance. While CSL1 is related to the Kalmeyer-Laughlin state, CSL2 is the ν=4 member of Kitaev's 16 fold way classification. Both states are described within the Abrikosov fermion representation of spins, and the effect of doping can be accessed by introducing charged holons. On doping CSL2, condensation of charged holons leads to a topological d+id superconductor. However on doping CSL1 , in sharp contrast , two different scenarios can arise: first, if holons condense, a chiral metal with doubled unit cell and finite Hall conductivity is obtained. However, in a second novel scenario, the internal magnetic flux adjusts with doping and holons form a bosonic integer quantum Hall (BIQH) state. Remarkably, the latter phase is identical to a d+id superconductor. In this case the Mott insulator to superconductor transition is associated with a bosonic variant of the integer quantum Hall plateau transition for the holon. We connect the above two scenarios to two recent numerical studies of doped chiral spin liquids on triangular lattice. Our work clarifies the complex relation between topological superconductors, chiral spin liquids and quantum criticality .

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this work, we propose a new protocol that combines magic state preparation and code switching to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the Z_4 surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D_4 quantum double model. This process ultimately produces a magic state in a condensed Z_2 surface code, which enables the implementation of a logical T gate in the standard Z_2 surface code. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding code switching between topological codes.

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further, development is limited by the scarcity of viable candidates. Here we present and discuss machine learning-accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10times more efficient than the trial-and-error approach while few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

Micro-Aerial Vehicles (MAVs) have the advantage of moving freely in 3D space. However, creating compact and sparse map representations that can be efficiently used for planning for such robots is still an open problem. In this paper, we take maps built from noisy sensor data and construct a sparse graph containing topological information that can be used for 3D planning. We use a Euclidean Signed Distance Field, extract a 3D Generalized Voronoi Diagram (GVD), and obtain a thin skeleton diagram representing the topological structure of the environment. We then convert this skeleton diagram into a sparse graph, which we show is resistant to noise and changes in resolution. We demonstrate global planning over this graph, and the orders of magnitude speed-up it offers over other common planning methods. We validate our planning algorithm in real maps built onboard an MAV, using RGB-D sensing.

Verifying the reliability of complex, multi-step reasoning in Large Language Models (LLMs) remains a fundamental challenge, as existing methods often lack both faithfulness and precision. To address this issue, we propose the Graph of Verification (GoV) framework. GoV offers three key contributions: First, it explicitly models the underlying deductive process as a directed acyclic graph (DAG), whether this structure is implicit or explicitly constructed. Second, it enforces a topological order over the DAG to guide stepwise verification. Third, GoV introduces the notion of customizable node blocks, which flexibly define the verification granularity, from atomic propositions to full paragraphs, while ensuring that all requisite premises derived from the graph are provided as contextual input for each verification unit. We evaluate GoV on the Number Triangle Summation task and the ProcessBench benchmark with varying levels of reasoning complexity. Experimental results show that GoV substantially improves verification accuracy, faithfulness, and error localization when compared to conventional end-to-end verification approaches. Our code and data are available at https://github.com/Frevor/Graph-of-Verification.

Open-world survival games pose significant challenges for AI algorithms due to their multi-tasking, deep exploration, and goal prioritization requirements. Despite reinforcement learning (RL) being popular for solving games, its high sample complexity limits its effectiveness in complex open-world games like Crafter or Minecraft. We propose a novel approach, SPRING, to read the game's original academic paper and use the knowledge learned to reason and play the game through a large language model (LLM). Prompted with the LaTeX source as game context and a description of the agent's current observation, our SPRING framework employs a directed acyclic graph (DAG) with game-related questions as nodes and dependencies as edges. We identify the optimal action to take in the environment by traversing the DAG and calculating LLM responses for each node in topological order, with the LLM's answer to final node directly translating to environment actions. In our experiments, we study the quality of in-context "reasoning" induced by different forms of prompts under the setting of the Crafter open-world environment. Our experiments suggest that LLMs, when prompted with consistent chain-of-thought, have great potential in completing sophisticated high-level trajectories. Quantitatively, SPRING with GPT-4 outperforms all state-of-the-art RL baselines, trained for 1M steps, without any training. Finally, we show the potential of games as a test bed for LLMs.

Pioneering advancements in large language model-powered agents have underscored the design pattern of multi-agent collaboration, demonstrating that collective intelligence can surpass the capabilities of each individual. Inspired by the neural scaling law, which posits that increasing neurons leads to emergent abilities, this study investigates whether a similar principle applies to increasing agents in multi-agent collaboration. Technically, we propose multi-agent collaboration networks (MacNet), which utilize directed acyclic graphs to organize agents and streamline their interactive reasoning via topological ordering, with solutions derived from their dialogues. Extensive experiments show that MacNet consistently outperforms baseline models, enabling effective agent collaboration across various network topologies and supporting cooperation among more than a thousand agents. Notably, we observed a small-world collaboration phenomenon, where topologies resembling small-world properties achieved superior performance. Additionally, we identified a collaborative scaling law, indicating that normalized solution quality follows a logistic growth pattern as scaling agents, with collaborative emergence occurring much earlier than previously observed instances of neural emergence. The code and data will be available at https://github.com/OpenBMB/ChatDev.

Recent advances in molecular representation learning have produced highly effective encodings of molecules for numerous cheminformatics and bioinformatics tasks. However, extracting general chemical insight while balancing predictive accuracy, interpretability, and computational efficiency remains a major challenge. In this work, we introduce a novel Graph Neural Network (GNN) architecture that combines compressed higher-order topological signals with standard molecular features. Our approach captures global geometric information while preserving computational tractability and human-interpretable structure. We evaluate our model across a range of benchmarks, from small-molecule datasets to complex material datasets, and demonstrate superior performance using a parameter-efficient architecture. We achieve the best performing results in both accuracy and robustness across almost all benchmarks. We open source all code All code and results can be found on Github https://github.com/rahulkhorana/TFC-PACT-Net.

This work introduces TopoBenchmarkX, a modular open-source library designed to standardize benchmarking and accelerate research in Topological Deep Learning (TDL). TopoBenchmarkX maps the TDL pipeline into a sequence of independent and modular components for data loading and processing, as well as model training, optimization, and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBenchmarkX is that it allows for the transformation and lifting between topological domains. This enables, for example, to obtain richer data representations and more fine-grained analyses by mapping the topology and features of a graph to higher-order topological domains such as simplicial and cell complexes. The range of applicability of TopoBenchmarkX is demonstrated by benchmarking several TDL architectures for various tasks and datasets.

The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators, have directed fundamental research in solid-state materials. Topological quantum chemistry has enabled the understanding of and the search for paramagnetic topological materials. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC), here we perform a high-throughput search for magnetic topological materials based on first-principles calculations. We use as our starting point the Magnetic Materials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic compounds with magnetic structures deduced from neutron-scattering experiments, and identify 130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and topological insulators. For each compound, we perform complete electronic structure calculations, which include complete topological phase diagrams using different values of the Hubbard potential. Using a custom code to find the magnetic co-representations of all bands in all magnetic space groups, we generate data to be fed into the algorithm of MTQC to determine the topology of each magnetic material. Several of these materials display previously unknown topological phases, including symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators and higher-order magnetic semimetals. We analyse topological trends in the materials under varying interactions: 60 per cent of the 130 topological materials have topologies sensitive to interactions, and the others have stable topologies under varying interactions. We provide a materials database for future experimental studies and open-source code for diagnosing topologies of magnetic materials.

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

High-quality code documentation is crucial for software development especially in the era of AI. However, generating it automatically using Large Language Models (LLMs) remains challenging, as existing approaches often produce incomplete, unhelpful, or factually incorrect outputs. We introduce DocAgent, a novel multi-agent collaborative system using topological code processing for incremental context building. Specialized agents (Reader, Searcher, Writer, Verifier, Orchestrator) then collaboratively generate documentation. We also propose a multi-faceted evaluation framework assessing Completeness, Helpfulness, and Truthfulness. Comprehensive experiments show DocAgent significantly outperforms baselines consistently. Our ablation study confirms the vital role of the topological processing order. DocAgent offers a robust approach for reliable code documentation generation in complex and proprietary repositories.

Topological materials present unconventional electronic properties that make them attractive for both basic science and next-generation technological applications. The majority of currently known topological materials have been discovered using methods that involve symmetry-based analysis of the quantum wavefunction. Here we use machine learning to develop a simple-to-use heuristic chemical rule that diagnoses with a high accuracy whether a material is topological using only its chemical formula. This heuristic rule is based on a notion that we term topogivity, a machine-learned numerical value for each element that loosely captures its tendency to form topological materials. We next implement a high-throughput procedure for discovering topological materials based on the heuristic topogivity-rule prediction followed by ab initio validation. This way, we discover new topological materials that are not diagnosable using symmetry indicators, including several that may be promising for experimental observation.

Triggered by limitations of graph-based deep learning methods in terms of computational expressivity and model flexibility, recent years have seen a surge of interest in computational models that operate on higher-order topological domains such as hypergraphs and simplicial complexes. While the increased expressivity of these models can indeed lead to a better classification performance and a more faithful representation of the underlying system, the computational cost of these higher-order models can increase dramatically. To this end, we here explore a simplicial complex neural network learning architecture based on random walks and fast 1D convolutions (SCRaWl), in which we can adjust the increase in computational cost by varying the length and number of random walks considered while accounting for higher-order relationships. Importantly, due to the random walk-based design, the expressivity of the proposed architecture is provably incomparable to that of existing message-passing simplicial neural networks. We empirically evaluate SCRaWl on real-world datasets and show that it outperforms other simplicial neural networks.

When a topological insulator is incorporated into a Josephson junction, the system is predicted to reveal the fractional Josephson effect with a 4π-periodic current-phase relation. Here, we report the measurement of a 4π-periodic switching current through an asymmetric SQUID, formed by the higher-order topological insulator WTe_2. Contrary to the established opinion, we show that a high asymmetry in critical current and negligible loop inductance are not sufficient by themselves to reliably measure the current-phase relation. Instead, we find that our measurement is heavily influenced by additional inductances originating from the self-formed PdTe_{x} inside the junction. We therefore develop a method to numerically recover the current-phase relation of the system and find the 1.5,μm long junction to be best described in the short ballistic limit. Our results highlight the complexity of subtle inductance effects that can give rise to misleading topological signatures in transport measurements.

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases, such as fractional Chern insulators, yet their realization with both exact flatness and stable topology in local lattice models has been long hindered by fundamental no-go theorems. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the wavefunctions are non-analytic at isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatness and stable topology. We establish a general principle to construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in given space groups. Explicit examples exhibiting Chern numbers, strong Z_2 index, and crystalline-symmetry-protected invariants in two and three dimensions are provided. Furthermore, an automated algorithm identifies more than 50,000 robust, symmetry-indicated CTFBs. Filling such CTFBs yields short-range entangled topological states that exhibit power-law correlations. Crucially, all filled CTFB states admit exact tensor-network representations with finite bond dimensions, providing a tractable starting point for exploring strongly correlated topological matter.

The last few years have seen significant experimental progress in characterizing the copper-based hole-doped high temperature superconductors in the regime of low hole density, p. Quantum oscillations, NMR, X-ray, and STM experiments have shed much light on the nature of the ordering at low temperatures. We review evidence that the order parameter in the non-Lanthanum-based cuprates is a d-form factor density-wave. This novel order acts as an unexpected window into the electronic structure of the pseudogap phase at higher temperatures in zero field: we argue in favor of a `fractionalized Fermi liquid' (FL*) with 4 pockets of spin S=1/2, charge +e fermions enclosing an area specified by p.

In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near the Planck scale. The key idea is to investigate the Betti numbers of coarse-grained path integral histories, regularized in terms of dynamical triangulations, as a function of the coarse-graining scale. In two dimensions our analysis exhibits the well-known fractal structure of Euclidean quantum gravity.

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {Haldane bundle dataset}, which consists of synthetically generated complex line bundles on the 2-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

With their non-Abelian topological charges, real multi-bandgap systems challenge the conventional topological phase classifications. As the minimal sector of multi-bandgap systems, real triple degeneracies (RTPs), which serve as real 'Weyl points', lay the foundation for the research on real topological phases. However, experimental demonstration of physical systems with global band configurations consisting of multiple RTPs in crystals has not been reported. Here we present experimental evidence of RTPs in photonic meta-crystals, characterizing them using the Euler number, and establishing their connection with both Abelian and non-Abelian charges. By considering RTPs as the basic elements, we further propose the concept of a topological compound, akin to a chemical compound, where we find that certain phases are not topologically allowed. The topological classification of RTPs in crystals demonstrated in our work plays a similar role as the 'no-go' theorem in Weyl systems.

The higher Bruhat orders B(n,k) were introduced by Manin-Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected B(n,k) to oriented matroids. In this paper, we consider the enumeration of B(n,k) and improve upon Balko's asymptotic lower and upper bounds on |B(n,k)| by a factor exponential in k. A proof of Ziegler's formula for |B(n,n-3)| is given and a bijection between a certain subset of B(n,n-4) and totally symmetric plane partitions is proved. Central to our proofs are deletion and contraction operations for the higher Bruhat orders, defined in analogy with matroids. Dual higher Bruhat orders are also introduced, and we construct isomorphisms relating the higher Bruhat orders and their duals. Additionally, weaving functions are introduced to generalize Felsner's encoding of elements in B(n,2) to all higher Bruhat orders B(n,k).

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature has also led to a lack of unification in notation and language across Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL, and compare the recently published TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

We review theoretical and experimental highlights in transport in two-dimensional materials focussing on key developments over the last five years. Topological insulators are finding applications in magnetic devices, while Hall transport in doped samples and the general issue of topological protection remain controversial. In transition metal dichalcogenides valley-dependent electrical and optical phenomena continue to stimulate state-of-the-art experiments. In Weyl semimetals the properties of Fermi arcs are being actively investigated. A new field, expected to grow in the near future, focuses on the non-linear electrical and optical responses of topological materials, where fundamental questions are once more being asked about the intertwining roles of the Berry curvature and disorder scattering. In topological superconductors the quest for chiral superconductivity, Majorana fermions and topological quantum computing is continuing apace.

Topological magnons give rise to possibilities for engineering novel spintronics devices with critical applications in quantum information and computation, due to its symmetry-protected robustness and low dissipation. However, to make reliable and systematic predictions about material realization of topological magnons has been a major challenge, due to the lack of neutron scattering data for most materials. In this work, we significantly advance the symmetry-based approach for identifying topological magnons through developing a fully automated algorithm, utilizing the theory of symmetry indicators, that enables a highly efficient and large-scale search for candidate materials hosting field-induced topological magnons. This progress not only streamlines the discovery process but also expands the scope of materials exploration beyond previous manual or traditional methods, offering a powerful tool for uncovering novel topological phases in magnetic systems. Performing a large-scale search over all 1649 magnetic materials in Bilbao Crystallographic Server with a commensurate magnetic order, we discover 387 candidate materials for topological magnons, significantly expanding the pool of topological magnon materials. We further discuss examples and experimental accessibility of the candidate materials, shedding light on future experimental realizations of topological magnons in magnetic materials.

Vacancies are prevalent and versatile in solid-state physics and materials science. The role of vacancies in strongly correlated materials, however, remains uncultivated until now. Here, we report the discovery of an unprecedented vacancy state forming an extended buckled-honeycomb-vacancy (BHV) ordering in Ir_{16}Sb_{18}. Superconductivity emerges by suppressing the BHV ordering through squeezing of extra Ir atoms into the vacancies or isovalent Rh substitution. The phase diagram on vacancy ordering reveals the superconductivity competes with the BHV ordering. Further theoretical calculations suggest that this ordering originates from a synergistic effect of the vacancy formation energy and Fermi surface nesting with a wave vector of (1/3, 1/3, 0). The buckled structure breaks the crystal inversion symmetry and can mostly suppress the density of states near the Fermi level. The peculiarities of BHV ordering highlight the importance of "correlated vacancies" and may serve as a paradigm for exploring other non-trivial excitations and quantum criticality.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of models is that they are highly parameterized, which makes describing and interpreting the prediction strategies difficult. We use topological data analysis to transform these complex prediction models into pictures representing a topological view. The result is a map of the predictions that enables inspection. The methods scale up to large datasets across different domains and enable us to detect labeling errors in training data, understand generalization in image classification, and inspect predictions of likely pathogenic mutations in the BRCA1 gene.

The Markov numbers are positive integers appearing as solutions to the Diophantine equation x^2 + y^2 + z^2 = 3xyz. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Canakc{\i} and Schiffler showed that the Markov number m_{a{b}} is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from (0,0) to (a,b). Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path omega, associate a snake graph G(omega) and a continued fraction g(omega). The ordering <_M is given by the number of perfect matchings on G(omega), and the ordering <_L is given by the Lagrange number of g(omega). In this work, we settle two conjectures of Schiffler. First, we show that the path omega(a,b) = RRcdots R UU cdots U is the unique maximum over all lattice paths from (0,0) to (a,b) with respect to both orderings <_M and <_L. We then use this result to prove that sup L(omega) over all lattice paths is exactly 1+sqrt5.

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.

We conducted a high-throughput search for topological magnetic materials on 522 new, experimentally reported commensurate magnetic structures from MAGNDATA, doubling the number of available materials on the Topological Magnetic Materials database. This brings up to date the previous studies which had become incomplete due to the discovery of new materials. For each material, we performed first-principle electronic calculations and diagnosed the topology as a function of the Hubbard U parameter. Our high-throughput calculation led us to the prediction of 250 experimentally relevant topologically non-trivial materials, which represent 47.89% of the newly analyzed materials. We present five remarkable examples of these materials, each showcasing a different topological phase: Mn{}_2AlB{}_2 (BCSID 1.508), which exhibits a nodal line semimetal to topological insulator transition as a function of SOC, CaMnSi (BCSID 0.599), a narrow gap axion insulator, UAsS (BCSID 0.594) a 5f-orbital Weyl semimetal, CsMnF{}_4 (BCSID 0.327), a material presenting a new type of quasi-symmetry protected closed nodal surface and FeCr{}_2S{}_4 (BCSID 0.613), a symmetry-enforced semimetal with double Weyls and spin-polarised surface states.

The discovery of topological quantum states marks a new chapter in both condensed matter physics and materials sciences. By analogy to spin electronic system, topological concepts have been extended into phonons, boosting the birth of topological phononics (TPs). Here, we present a high-throughput screening and data-driven approach to compute and evaluate TPs among over 10,000 materials. We have clarified 5014 TP materials and classified them into single Weyl, high degenerate Weyl, and nodal-line (ring) TPs. Among them, three representative cases of TPs have been discussed in detail. Furthermore, we suggest 322 TP materials with potential clean nontrivial surface states, which are favorable for experimental characterizations. This work significantly increases the current library of TP materials, which enables an in-depth investigation of their structure-property relations and opens new avenues for future device design related to TPs.

Quantum materials hosting Weyl fermions have opened a new era of research in condensed matter physics. First proposed in 1929 in particle physics, Weyl fermions have yet to be observed as elementary particles. In 2015, Weyl fermions were detected as collective electronic excitations in the strong spin-orbit coupled material tantalum arsenide, TaAs. This discovery was followed by a flurry of experimental and theoretical explorations of Weyl phenomena in materials. Weyl materials naturally lend themselves to the exploration of the topological index associated with Weyl fermions and their divergent Berry curvature field, as well as the topological bulk-boundary correspondence giving rise to protected conducting surface states. Here, we review the broader class of Weyl topological phenomena in materials, starting with the observation of emergent Weyl fermions in the bulk and of Fermi arc states on the surface of the TaAs family of crystals by photoemission spectroscopy. We then discuss some of the exotic optical and magnetic responses observed in these materials, as well as the progress in developing some of the related chiral materials. We discuss the conceptual development of high-fold chiral fermions, which generalize Weyl fermions, and we review the observation of high-fold chiral fermion phases by taking the rhodium silicide, RhSi, family of crystals as a prime example. Lastly, we discuss recent advances in Weyl-line phases in magnetic topological materials. With this Review, we aim to provide an introduction to the basic concepts underlying Weyl physics in condensed matter, and to representative materials and their electronic structures and topology as revealed by spectroscopic studies. We hope this work serves as a guide for future theoretical and experimental explorations of chiral fermions and related topological quantum systems with potentially enhanced functionalities.

In this article, we introduce and study singular foliations of b^k-type. These singular foliations formalize the properties of vector fields that are tangent to order k along a submanifold W subset M. Our first result is a classification of these foliations, relating them to geometric structures defined in a formal neighborhood of the submanifold, such as jets of distributions that are involutive up to order k-1. When W is a hypersurface, singular foliations of b^k-type are Lie algebroids. In this particular case, they are generalizations of the b^k-tangent bundles introduced by Scott. Indeed, they are always locally isomorphic to b^k-tangent bundles, but globally such an isomorphism is obstructed by a holonomy invariant. Our second main result is a Riemann-Hilbert-style classification of singular foliations of b^k-type in terms of holonomy representations. In this paper, we study singular foliations of b^k-type from several different perspectives. In particular: (1) We study the problem of extending a k-th-order foliation to a (k+1)-th order foliation and prove that this is obstructed by a characteristic class. (2) When W is a hypersurface, we give a detailed study of algebroid differential forms and extend Scott's calculation of the cohomology. (3) We study algebroid symplectic forms in terms of the geometric structures induced on W. In particular, we find that there is a close relationship between the above obstruction class for extensions and the symplectic variation of the symplectic foliation induced on W.

Among a huge variety of known two-dimensional materials, some of them have anisotropic crystal structures; examples include so different systems as a few-layer black phoshphorus (phosphorene), beryllium nitride BeN_4, van der Waals magnet CrSBr, rhenium dichalgogenides ReX_2. As a consequence, their optical and electronic properties turn out to be highly anisotropic as well. In some cases, the anisotropy results not just in a smooth renormalization of observable properties in comparison with the isotropic case but in the appearance of dramatically new physics. The examples are hyperbolic plasmons and excitons, strongly anisotropic ordering of adatoms at the surface of two-dimensional or van der Waals materials, essential change of transport and superconducting properties. Here, we present a systematic review of electronic structure, transport and optical properties of several representative groups of anisotropic two-dimensional materials including semiconductors, anisotropic Dirac and semi-Dirac materials, as well as superconductors.

We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.

We identify seven new 192-periodic infinite families of elements in the 2-primary stable homotopy groups of spheres. Although their Hurewicz image is trivial for topological modular forms, they remain nontrivial after T(2)- as well as K(2)-localization. We also obtain new information about 2-torsion and 2-divisibility of some of the previously known 192-periodic infinite families in the stable stems.

We prove that for (compact) finite-rank median algebras the geometric rank equals the independence number of all (continuous) median-preserving functions to [0,1]. Combined with Rosenthal's dichotomy, this yields a generalized Helly selection principle: for finite-rank median algebras, the space of all median-preserving functions to [0,1] is sequentially compact in the pointwise topology. Generalizing joint results with E. Glasner on dendrons (rank-1), we establish that every continuous action of a topological group G by median automorphisms on a finite-rank compact median algebra is Rosenthal representable, hence dynamically tame. As an application, the Roller-Fioravanti compactification of finite-rank topological median G-algebras with compact intervals is often a dynamically tame G-system.

Given a finite poset mathcal P, the hypercube-height, denoted by h^*(mathcal P), is defined to be the largest h such that, for any natural number n, the subsets of [n] of size less than h do not contain an induced copy of mathcal P. The hypercube-width, denoted by w^*(mathcal P), is the smallest w such that the subsets of [w] of size at most h^*(mathcal P) contain an induced copy of mathcal P. In other words, h^*(mathcal P) asks how `low' can a poset be embedded, and w^*(mathcal P) asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that h^*(mathcal P)leq |mathcal P|-1 (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that w^*(mathcal P)leq|mathcal P|^2/4, but they conjectured that in fact w^*(mathcal P)leq |mathcal P| for any finite poset mathcal P. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifing an existing copy of our poset.

Topological materials provide a platform that utilizes the geometric characteristics of structured materials to control the flow of waves, enabling unidirectional and protected transmission that is immune to defects or impurities. The topologically designed photonic materials can carry quantum states and electromagnetic energy, benefiting nanolasers or quantum photonic systems. This article reviews recent advances in the topological applications of photonic materials for radiative heat transfer, especially in the near field. When the separation distance between media is considerably smaller than the thermal wavelength, the heat transfer exhibits super-Planckian behavior that surpasses Planck's blackbody predictions. Near-field thermal radiation in subwavelength systems supporting surface modes has various applications, including nanoscale thermal management and energy conversion. Photonic materials and structures that support topological surface states show immense potential for enhancing or suppressing near-field thermal radiation. We present various topological effects, such as periodic and quasi-periodic nanoparticle arrays, Dirac and Weyl semimetal-based materials, structures with broken global symmetries, and other topological insulators, on near-field heat transfer. Also, the possibility of realizing near-field thermal radiation in such topological materials for alternative thermal management and heat flux guiding in nano-scale systems is discussed based on the existing technology.

We explore the use of topological data analysis (TDA) combined with machine learning for discriminating standard model backgrounds from the invisible decay of the Z^prime boson associated with monophoton emission at a 3 TeV muon collider. Reconstructed events are mapped into a six-dimensional kinematic space and aggregated into bags of events, from which persistent homology is used to extract Betti number distributions. Within the Multiple Instance Learning paradigm, classifiers trained on these topological descriptors demonstrate significantly improved classification accuracy compared to the conventional ML approaches based on event-wise kinematic inputs. We also draw exclusion contours at 95\% CL in the (m_{Z^prime}, m_chi) parameter space, highlighting the potential of topological features to extend the discovery reach of future collider experiments.

Autoregressive models (ARMs) have become the workhorse for sequence generation tasks, since many problems can be modeled as next-token prediction. While there appears to be a natural ordering for text (i.e., left-to-right), for many data types, such as graphs, the canonical ordering is less obvious. To address this problem, we introduce a variant of ARM that generates high-dimensional data using a probabilistic ordering that is sequentially inferred from data. This model incorporates a trainable probability distribution, referred to as an order-policy, that dynamically decides the autoregressive order in a state-dependent manner. To train the model, we introduce a variational lower bound on the exact log-likelihood, which we optimize with stochastic gradient estimation. We demonstrate experimentally that our method can learn meaningful autoregressive orderings in image and graph generation. On the challenging domain of molecular graph generation, we achieve state-of-the-art results on the QM9 and ZINC250k benchmarks, evaluated using the Fr\'{e}chet ChemNet Distance (FCD).

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

We define and develop a homotopy invariant notion for the topological complexity of a map f:X to Y, denoted TC(f), that interacts with TC(X) and TC(Y) in the same way cat(f) interacts with cat(X) and cat(Y). Furthermore, TC(f) and cat(f) satisfy the same inequalities as TC(X) and cat(X). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC(f) to studying group homomorphisms f:Hto G.

A Harris Graph is a tough, Eulerian, non-Hamiltonian graph. Several approaches to creating new Harris graphs from existing ones are explored, including creating families of Harris graphs and combining Harris graphs. Pictures of all Harris Graphs through order 9 and the number of Harris graphs through order 12 are included. We also prove a result about barnacle-free Harris graphs.

We present an extensive quantum Monte Carlo study of a nearest-neighbor, singlet-projection model on the pyrochlore lattice that exhibits SO(N) symmetry and is sign-problem-free. We find that in contrast to the previously studied two-dimensional variations of this model that harbor critical points between their ground state phases, the non-bipartite pyrochlore lattice in three spatial dimensions appears to exhibit a first-order transition between a magnetically-ordered phase and some, as yet uncharacterized, paramagnetic phase. We also observe that the magnetically-ordered phase survives to a relatively large value of N=8, and that it is gone for N=9.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Adversarially perturbed images of text can cause sophisticated OCR systems to produce misleading or incorrect transcriptions from seemingly invisible changes to humans. Some of these perturbations even survive physical capture, posing security risks to high-stakes applications such as document processing, license plate recognition, and automated compliance systems. Existing defenses, such as adversarial training, input preprocessing, or post-recognition correction, are often model-specific, computationally expensive, and affect performance on unperturbed inputs while remaining vulnerable to unseen or adaptive attacks. To address these challenges, TopoReformer is introduced, a model-agnostic reformation pipeline that mitigates adversarial perturbations while preserving the structural integrity of text images. Topology studies properties of shapes and spaces that remain unchanged under continuous deformations, focusing on global structures such as connectivity, holes, and loops rather than exact distance. Leveraging these topological features, TopoReformer employs a topological autoencoder to enforce manifold-level consistency in latent space and improve robustness without explicit gradient regularization. The proposed method is benchmarked on EMNIST, MNIST, against standard adversarial attacks (FGSM, PGD, Carlini-Wagner), adaptive attacks (EOT, BDPA), and an OCR-specific watermark attack (FAWA).

Using an evolutionary algorithm in combination with first-principles density functional theory calculations, we identify two-dimensional (2D) CaP_3 monolayer as a new Dirac semimetal due to inversion and nonsymmorphic spatial symmetries of the structure. This new topological material, composed of light elements, exhibits high structural stability (higher than the phase known in the literature), which is confirmed by thermodynamic and kinetic stability analysis. Moreover, it satisfies the electron filling criteria, so that its Dirac state is located near the Fermi level. The existence of the Dirac state predicted by the theoretical symmetry analysis is also confirmed by first-principles electronic band structure calculations. We find that the energy position of the Dirac state can be tuned by strain, while the Dirac state is unstable against an external electric field since it breaks the spatial inversion symmetry. Our findings should be instrumental in the development of 2D Dirac fermions based on light elements for their application in nanoelectronic devices and topological electronics.

We present a comprehensive numerical study of a six-state clock model with a long-range dipolar type interaction. This model is motivated by the ferroelectric orders in the multiferroic hexagonal manganites. At low temperatures, trimerization of local atomic structures leads to six distinct but energetically degenerate structural distortion, which can be modeled by a six-state clock model. Moreover, the atomic displacements in the trimerized state further produce a local electric polarization whose sign depends on whether the clock variable is even or odd. These induced electric dipoles, which can be modeled by emergent Ising degrees of freedom, interact with each other via long-range dipolar interactions. Extensive Monte Carlo simulations are carried out to investigate low temperature phases resulting from the competing interactions. Upon lowering temperature, the system undergoes two Berezinskii-Kosterlitz-Thouless (BKT) transitions, characteristic of the standard six-state clock model in two dimensions. The dipolar interaction between emergent Ising spins induces a first-order transition into a ground state characterized by a three-fold degenerate stripe order. The intermediate phase between the discontinuous and the second BKT transition corresponds to a maze-like hexagonal liquid with short-range stripe ordering. Moreover, this intermediate phase also exhibits an unusual ferromagnetic order with two adjacent clock variables occupying the two types of stripes of the labyrinthine pattern.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.