arXiv research

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…

New Fuchsian groups found with special embedding properties.

New modularity function improves clustering of spatially embedded networks.

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $Γ_1$, $Γ_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colonΓ_1\toΓ_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automor…

Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.

New GCNs solve graph embedding problems efficiently and interpretably.

This paper compares hypernetworks and embedding methods for function approximation.

A new approach clusters data first, then embeds each cluster, improving transparency.

If $p : Y \to X$ is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to $p$, from the Teichmüller space ${\cal T}(X)$, for $X$, to ${\cal T}(Y)$ actually extends to an embedding between the Thurston compactification of the tw…

In this paper we take a problem of unsupervised nodes clustering on graphs and show how recent advances in attention models can be applied successfully in a "hard" regime of the problem. We propose an unsupervised algorithm that encodes Bethe Hessian embeddings by optimizing soft modularity loss and argue that our mode…

Inferencing with network data necessitates the mapping of its nodes into a vector space, where the relationships are preserved. However, with multi-layered networks, where multiple types of relationships exist for the same set of nodes, it is crucial to exploit the information shared between layers, in addition to the …

Modular RL modules solve complex 3D Sokoban tasks.

Next generation of embedded Information and Communication Technology (ICT) systems are collaborative systems able to perform autonomous tasks. The remarkable expansion of the embedded ICT market, together with the rise and breakthroughs of Artificial Intelligence (AI), have put the focus on the Edge as it stands as one…

Modularity-aware GAE and VGAE improve community detection and link prediction.

Geodesic patterns, shears, and Anosov representations of the modular group.

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led …

Criteria found for graph drawings on surfaces.

We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups admitting modular embeddings.

New mapping class group actions on Hochschild complexes for modular categories.

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …

We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expresse…

In likelihood-free settings where likelihood evaluations are intractable, approximate Bayesian computation (ABC) addresses the formidable inference task to discover plausible parameters of simulation programs that explain the observations. However, they demand large quantities of simulation calls. Critically, hyperpara…

The study constructs a dense orbit in the universal commensurability augmented Teichmüller space.

Pykg2vec is an open-source Python library for learning the representations of the entities and relations in knowledge graphs. Pykg2vec's flexible and modular software architecture currently implements 16 state-of-the-art knowledge graph embedding algorithms, and is designed to easily incorporate new algorithms. The goa…

This work proposes a hybrid method for error detection in noisy Knowledge Graphs.

SX-GeoTree improves spatially coherent explanations in geospatial regression trees.

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum mo…

Researchers found the global topology of the Eisenstein-Picard modular surface.

Improved community detection and link prediction with GAE and VGAE.

Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.

Modular neural networks generalize better with less data.

Study modular forms over Γ^0(2) and anomaly cancellation formulas.

Our aim is to introduce and advocate non-$Σ$ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-$Σ$ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…

Neural networks learn modular arithmetic but not all, extending known solutions to generalize.

Improved algorithm for modular links provides upper volume bounds.

Geodesics on modular surface yield arithmetic 3-manifolds.

Deep-embedding methods aim to discover representations of a domain that make explicit the domain's class structure and thereby support few-shot learning. Disentangling methods aim to make explicit compositional or factorial structure. We combine these two active but independent lines of research and propose a new parad…

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $Λ$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $Λ.$ We show that many known properties of th…

New modular forms for anomaly cancellation formulas on any dimensional manifolds.

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

Countable modular groups found on surfaces with infinite type.

Quantum modularity proven for specific theta series.

The paper introduces elliptic quasi-modular forms via moduli spaces.

In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis …

Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.