The paper studies knots in modular flows using self-covers.
problem Understanding topological properties of modular knots.
method Constructing templates for Anosov flows and studying closed geodesics.
result Explicit construction of an infinite family of links with the trefoil.
Enhances multi-modular models by directing information flow between components.
problem Improving predictive performance in multi-modular models with misspecification.
method Introduces Semi-Modular Inference (SMI) with an influence parameter to control information flow between modules.
result SMI allows for tunable and directed information flow, improving prediction in some settings.
Lower bound found for volumes of modular link complements.
problem Finding a lower bound for the volumes of modular link complements.
method Analyzing the volumes of link complements associated with geodesics in the modular surface.
result First linear lower volume bound in terms of exponents of code words.
Study open orbits in causal flag manifolds with applications in AQFT.
problem Understanding open orbits in causal flag manifolds for applications in AQFT.
method Analyzing open orbits of symmetric subgroups on causal flag manifolds, focusing on invariant causal structures and modular flows.
result Determine the positivity regions of modular flows and their global hyperbolicity for different types of open orbits.
We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.
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 S3, and one is led …
Study modular geodesics and wedge domains in non-compactly causal symmetric spaces.
problem Understanding the geometric implementation of modular group in symmetric spaces.
method Analyzing the flow generated by Euler elements and their geometric properties.
result The wedge region W is connected and coincides with the observer domain under certain conditions.
Linking numbers of modular knots derived from geometric and algebraic properties.
problem Understanding linking numbers between modular knots and the trefoil.
method Geometric and algebraic properties of the modular group and its action on the hyperbolic plane.
result Derived several formulae for linking numbers with arithmetical, combinatorial, topological and group theoretical flavors.
A new neural network for efficient density estimation.
problem Efficient density estimation for high-dimensional data.
method Triangular neural network implementation of neural autoregressive flow (NAF).
result Achieves state-of-the-art bits-per-dimension indices on MNIST and CIFAR-10.
New methods for scalable inference in modular models with misspecified sub-models.
problem Model misspecification in multi-modular models complicates evidence combination.
method Variational methods for approximating Cut and SMI posteriors, and Variational Meta-Posterior.
result Feasibility of analysis with multiple cuts using a single set of variational parameters.
Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-det…
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
MOCA uses modular attention to estimate causal effects from complex data.
problem Estimating causal effects from observational data with complex, non-linear, and high-dimensional treatment and outcome mechanisms.
method MOCA is a transformer-based framework that separates treatment and outcome modeling through modular design and one-way attention mechanism, with cutting-feedback to prevent outcome influence on treatment representations.
result MOCA outperforms classical estimators and machine learning approaches across various simulated and real-world scenarios.
New method for scalable barycenter computation using Wasserstein gradient flows.
problem Scalability and integration of label information in barycenter computation.
method Gradient flows in Wasserstein space, time discretization, mini-batch optimal transport, modular regularization, task-aware functions, supervised information integration.
result Empirically validated new state-of-the-art barycenter solver with labeled barycenters outperforming unlabeled ones.
This paper extends foliation concepts to singular foliations using Lie ∞-algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie ∞-algebroids to define modular class. result Geometric meaning of modular class as an obstruction to universal Lie ∞-algebroids. Attention mechanism combines bottom-up and top-down signals in neural networks.
problem Combining robust perception with bottom-up and top-down signals.
method Attention mechanism over modulated recurrent neural networks.
result Bidirectional information flow leads to improved performance in various tasks.
Improved symbolic regression finds optimal formulas robust to noise.
problem Finding accurate formulas for noisy data.
method Exploits graph modularity, uses normalizing flows, and statistical hypothesis testing.
result Discoveres many formulas previously unattainable.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL2(Z)\PSL2(R). The complement of any finite number of orbits is a hyperbolic 3-manifold, which thus has a well-defined volume. We present strong nu…
We construct a Poincaré section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.
problem Counting ambiguous geodesics in curved spaces.
method Asymptotic formula for common perpendiculars in negatively curved spaces, applying to modular orbifolds and number fields.
result Confirms and extends Motohashi's conjecture on binary additive divisor problem.
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
Novel flows generate molecules without post-processing.
problem Generating new molecules efficiently and without post-processing issues.
method Continuous normalizing E(3)-equivariant flows based on node ODEs coupled as a graph PDE.
result Generated samples achieve state-of-the-art performance on QM9 and ZINC250K benchmarks.
We explain how neural networks learn to solve modular addition tasks.
problem How two-layer neural networks learn to solve modular addition tasks.
method Formalized a diversification condition during training, proving it allows the network to approximate the correct logic for modular addition.
result Neural networks can robustly identify the correct sum through phase symmetry and frequency diversification.
The integrability of the geodesic flow on the three-folds M3 admitting SL(2,R)-geometry in Thurston's sense is investigated. The main examples are the quotients MΓ3=Γ\PSL(2,R), where Γ⊂PSL(2,R) is a cofinite Fuchsian group. We show that the correspon…
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N=2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,Z) Weil representations, quantum mo…
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.
problem Efficiently propagating uncertainty in downstream Bayesian analysis without feedback.
method NeVI-Cut combines neural networks and normalizing flows for variational inference.
result NeVI-Cut achieves significant computational gains and higher accuracy than traditional methods.
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
Modular neural networks generalize better with less data.
problem Theoretical and practical understanding of how modularity improves neural network generalization.
method Theoretical analysis of sample complexity, development of a novel learning rule.
result Modular networks require fewer samples to generalize compared to nonmodular networks, especially in high-dimensional tasks.
Study modular forms over Γ^0(2) and anomaly cancellation formulas.
problem Anomaly cancellation formulas for modular forms over Γ^0(2).
method Study and analysis of modular forms over Γ^0(2).
result Anomaly cancellation formulas derived for modular forms over Γ^0(2).
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.
problem Neural networks struggle with modular arithmetic, especially for polynomials.
method Developed analytical solutions for MLP networks to learn modular addition and multiplication, then combined these solutions to generalize on arbitrary modular polynomials.
result Neural networks can learn and generalize solutions to modular polynomials, supporting the hypothesis that some polynomials are learnable.
MAGIC-Flow generates and classifies medical images with interpretability.
problem Challenges in generative modeling for medical imaging.
method Conditional multiscale normalizing flow architecture.
result MAGIC-Flow creates realistic, diverse samples and improves classification.
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…
Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
Geodesics on modular surface yield arithmetic 3-manifolds.
problem Understanding arithmetic properties of modular surfaces.
method Constructing geodesics and analyzing their lifts.
result Complements of canonical lifts are arithmetic 3-manifolds.
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.
problem Constructing new modular forms for anomaly cancellation formulas.
method Using E8 bundles, constructing modular forms on any dimensional manifolds. result Derived new anomaly cancellation formulas and applications.
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
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.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Quantum modularity proven for specific theta series.
problem Proving quantum modularity for partial theta series with periodic coefficients.
method Explicit proof using Kontsevich-Zagier series and colored Jones polynomials.
result Kontsevich-Zagier series is a weight 3/2 quantum modular form.
The paper introduces elliptic quasi-modular forms via moduli spaces.
problem Developing a theory of elliptic quasi-modular forms.
method Using moduli spaces and the Gauss-Manin connection.
result Presented a succinct theory of elliptic quasi-modular forms.
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.
problem Investigate Alexander polynomials of modular knots.
method Use Burau representation and geometric SL2(Z)-invariants. result Alexander polynomials of modular knots have both finite and infinite coefficient properties.
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a comp…