We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry an…
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…
Artificial neural networks (ANNs) have achieved significant success in tackling classical and modern machine learning problems. As learning problems grow in scale and complexity, and expand into multi-disciplinary territory, a more modular approach for scaling ANNs will be needed. Modular neural networks (MNNs) are neu…
PICLE uses probabilistic models to efficiently evaluate and compose modules for continual learning.
problem Challenging search space of module compositions in continual learning.
method Probabilistic framework to cheaply compute module compositions' fitness.
result First modular CL algorithm to achieve perceptual, few-shot, and latent transfer.
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.
Training a Neural Network (NN) with lots of parameters or intricate architectures creates undesired phenomena that complicate the optimization process. To address this issue we propose a first modular approach to NN design, wherein the NN is decomposed into a control module and several functional modules, implementing …
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.
For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.
Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect…
The paper develops a new approach to conditional risk measures using modular convex analysis.
problem Developing a new method for conditional risk measures.
method Random modular approach to conditional certainty equivalents and niveloids in the conditional L∞-space. result Retrieves a conditional variational formula for optimized certainty equivalents and applies it to the conditional entropic risk measure.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. Modular RL modules solve complex 3D Sokoban tasks.
problem Solving complex, integrated tasks combining visual, physical, and abstract reasoning.
method Compose RL modules in a sense-plan-act hierarchy, using only model-free methods.
result Modular RL outperforms state-of-the-art monolithic RL on Mujoban.
We present a novel clustering approach for moving object trajectories that are constrained by an underlying road network. The approach builds a similarity graph based on these trajectories then uses modularity-optimization hiearchical graph clustering to regroup trajectories with similar profiles. Our experimental stud…
Proposes a CL technique to improve accuracy and reduce forgetting.
problem Sequential task learners struggle with forgetting information from previous tasks.
method Extracts modular parts of neural networks and estimates task relatedness.
result Remarkable performance gain in robustness to forgetting for EWC and GEM methods.
New techniques prove quantum modularity for various functions.
problem Proving quantum modularity of false theta functions and related series.
method Developed techniques including Poisson summation formula and modular series framework.
result Unified approach to proving quantum modularity for various functions.
Modular deep learning framework using pairwise labels without backpropagation.
problem Efficiently training deep neural networks with limited supervision.
method Stacked linear models in feature spaces, provably optimal modular learning framework.
result High accuracy (94.88%) achieved with minimal labeled examples (1 per class).
The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.
problem Understanding fiber sum formulas for 4-manifolds.
method Using the connection between 4-manifolds and topological modular forms from 6d (1,0) SCFTs.
result Even free theories exhibit nontrivial fiber sum formulas, sensitive to individual theories and parameters.
A framework for modular training of robust generative models.
problem Training large generative models is resource-intensive and requires heuristic tuning.
method Modular training using a gating mechanism and a minimax game to find a robust gate.
result The modular approach can theoretically outperform monolithic baselines and is scalable.
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…
Unified approach to conformal and modular invariants on surfaces.
problem Constructing a general family of conformal invariants on surfaces.
method Using an identification of Teichmüller space and rigged moduli space, and analytic work on harmonic functions.
result Unified conformal and modular invariants can be viewed as generalized modular invariants and functions on the rigged moduli space.
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.
New algorithms detect changes in non-stationary MABs for better performance.
problem Non-stationary MAB environments where arm reward distributions change over time.
method Modular Detection Augmented Bandit (DAB) procedures with improved performance lower bounds.
result Modular DAB procedures achieve order-optimal regret bounds for various change detectors and bandit algorithms.
Modularity-aware GAE and VGAE improve community detection and link prediction.
problem Improving community detection with GAE and VGAE in the absence of node features.
method Introducing a modularity-aware message passing scheme and regularizer to GAE and VGAE encoders.
result Jointly addressing community detection and link prediction with high accuracy is possible.
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).
Clustering on hypergraphs has been garnering increased attention with potential applications in network analysis, VLSI design and computer vision, among others. In this work, we generalize the framework of modularity maximization for clustering on hypergraphs. To this end, we introduce a hypergraph null model, analogou…
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.
Modular curves X1(N) parametrize elliptic curves with a point of order N. They can be identified with connected components of projectivized strata PH(a,−a) of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …
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.
DynMSA detects market clusters for better portfolio allocation.
problem Identifying stable market clusters for effective portfolio management.
method Combining Random Matrix Theory with modularity optimization and spectral clustering.
result DynMSA outperforms baseline models in intra- and inter-cluster correlation differences.
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 …
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…
Constructs modular forms and proves divisibility results for odd-dimensional manifolds.
problem Constructing modular forms over specific groups and proving divisibility results.
method SL(2, Z) modular forms and Witten genus in odd dimensions.
result Obtained divisibility results of index of Toeplitz operators on spin and spin^c manifolds.
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.
New formulas derived for anomaly cancellation using modular forms and E8 bundles.
problem Anomaly cancellation in odd dimensions.
method Systematic twisting and generalization of SL(2,Z) modular forms to define new forms associated with E8 and E8*E8 bundles.
result Derivation of a new series of anomaly cancellation formulas.
ETQFTs created from non-semisimple modular categories.
problem Constructing ETQFTs from non-semisimple modular categories.
method Explicitly identify linear categories and functors in the image of ETQFTs constructed from modular categories.
result The circle category of ETQFTs is equivalent to the full subcategory of projective objects of the underlying modular category, which need not be semisimple.
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, includin…