Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Spectral sequence analysis for Sobolev mappings in Carnot groups.
problem Analyzing spectral sequences for Sobolev mappings in Carnot groups.
method Showed Pansu pullback induces a spectral sequence mapping.
result Pansu pullback induces a spectral sequence mapping.
Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
problem Calculating spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
method Established an effective procedure to calculate all coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map.
result Explicitly provided the first four coefficients of the spectral asymptotic formula.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Improved spectral clustering algorithm for better performance.
problem Improving the performance of spectral clustering algorithms.
method Developed a new performance guarantee under a weaker assumption and evaluated using a different spectral embedding map.
result Better performance guarantee under a weaker assumption and evaluation of a new spectral embedding map.
The paper describes a parametrisation of harmonic maps from a 2-torus to the 3-sphere.
problem Understanding the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere.
method Explicit parametrisation using spectral data and line bundles.
result The space of spectral data is a fibre bundle over the space of spectral curves, with nontrivial structure for certain invariance groups.
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
Constructs Serre spectral sequence for bounded cohomology.
problem No specific problem stated; focuses on a new mathematical construction.
method Constructs the Serre spectral sequence for bounded cohomology.
result Obtains a non-isometric generalization of Gromov's mapping theorem and partial results on simplicial volume.
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.
Study on JNR monopoles, focusing on their spectral curves and energy density.
problem Understanding JNR monopoles and their spectral curves.
method Analysis of spectral curves, rational maps, and holomorphic spheres.
result Established conditions for a spectral curve to be a JNR monopole and derived a formula for energy density at infinity.
We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the…
The paper studies magnetic field effects on surface eigenvalues and spectral properties.
problem Understanding magnetic effects on surface eigenvalues and spectral properties.
method Provided precise spectral asymptotics expansion for the magnetic Dirichlet-to-Neumann map on surfaces.
result The spectrum of the magnetic Dirichlet-to-Neumann map uniquely determines the number and length of boundary components, parallel transport, and magnetic flux.
Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of σ-connections on finitely generated projective modules. This ma…
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
SG-NTF completes HDI tensors with spectral mapping and spatio-temporal gating.
problem High-dimensional and incomplete tensor completion.
method Spectra-Guided Neural Tucker Factorization (SG-NTF) with Spatio-Temporal Co-Gating (STCG).
result Maintains competitive completion accuracy with parameter efficiency.
Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.
problem Prove nullity of biharmonic maps from flat 2-torus to round 2-sphere.
method Use spectral geometry, construct polynomial isomorphism with elliptic curve, determine rational points on spectral curve.
result Nullity of every map in the family is 5, confirming conjecture.
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some mar…
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a d-dimensional compact submanifold M in RD, we establish the spectral convergence rate…
Sharp spectral estimates for negatively curved foliations.
problem Estimating the bottom of the spectrum of Riemannian foliations.
method Analyzing the normal exponential map and using it to derive spectral estimates.
result Sharp estimates for the bottom of the spectrum of Riemannian foliations.
In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as ∞-category-theoretic, as our framework is constructed in the …
DMPS uses diffusion maps and LAWGD for efficient generative modeling.
problem Efficiently modeling complex data distributions.
method Diffusion maps for manifold learning and LAWGD for sampling.
result DMPS outperforms other methods on moderate-dimensional data.
Computes minimal dilatation for Thurston maps on surfaces.
problem Finding the minimal dilatation of Thurston maps on surfaces.
method Explicit computation using spectral radius in a congruence subgroup of PSL2(Z).
result Explicitly computes minimal dilatation for Thurston maps.
Commutes Pansu pullback with spectral complexes in Carnot groups.
problem Understanding the relationship between Pansu pullback and spectral complexes in Carnot groups.
method Proving commutativity between Pansu pullback and differentials in spectral complexes.
result Commutes Pansu pullback with spectral complexes in Carnot groups.
In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic …
This article has two purposes. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to Lie groups and symmetric spaces, focusing on spectral curves for harmonic 2-tori. The most unwieldy aspect of the spectral curve description is the periodicity conditions and th…
We construct a covering of Culler-Vogtmann Outer space by the Teichmuller spaces of punctured surfaces. By considering the equivariant homology for the action of Out(F_n) on this covering, we construct a spectral sequence converging to the homology of Out(F_n) that has E^1 terms given by the homology of mapping class g…
PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.
problem Limitations of common Neural Sheaf Diffusion implementations, including scalability and stability issues.
method Introduces Polynomial Neural Sheaf Diffusion (PolyNSD) with a degree-K polynomial propagation operator and spectral rescaling.
result PolyNSD achieves state-of-the-art results on both homophilic and heterophilic benchmarks with reduced runtime and memory requirements.
Given an SL(3) spectral curve over a simply connected Riemann surface, we describe in detail the reduction steps necessary to construct the core of a pre-building with versal harmonic map whose differential is given by the spectral curve.
We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the sam…
Exact spectral norm regularization improves neural network generalization.
problem Improving neural network generalization while protecting against noise.
method Exact spectral norm regularization of the Jacobian.
result Improved generalization performance compared to previous methods.
New proof of Llarull's rigidity theorem in odd dimensions via spectral flow.
problem Rigidity of smooth maps from compact spin manifolds to spheres.
method Spectral flow argument for odd dimensions, generalization to convex hypersurfaces.
result Generalization of Llarull's theorem to arbitrary smooth strictly convex hypersurfaces.
Reinforcement learning (RL) in Markov decision processes (MDPs) with large state spaces is a challenging problem. The performance of standard RL algorithms degrades drastically with the dimensionality of state space. However, in practice, these large MDPs typically incorporate a latent or hidden low-dimensional structu…
Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.
problem Constructing solutions and structures for dBKP and MS systems.
method Map construction and spectral characterisation of reductions.
result Defines Einstein-Weyl structures for dBKP and BMS systems.
Cross-domain recommendation can alleviate the data sparsity problem in recommender systems. To transfer the knowledge from one domain to another, one can either utilize the neighborhood information or learn a direct mapping function. However, all existing methods ignore the high-order connectivity information in cross-…
Recent research connects Hörmander's old work to modern boundary Laplacian analysis.
problem How close is the Dirichlet-to-Neumann map to the boundary Laplacian?
method Investigates techniques from Hörmander's 1950s manuscript to solve modern boundary Laplacian problems.
result Obtained results for DtN maps on non-smooth boundaries, Helmholtz equation, and differential forms.
In this note, we show that, if a pseudo-Anosov map φ:S→S admits a finite cover whose action on the first homology has spectral radius greater than 1, then the monodromy of any fibered structure of any finite cover of the mapping torus Mφ has the same property.
Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisa…
Study connects spectral properties to frame flows on curved manifolds.
problem Spectral properties and frame flows on curved manifolds.
method Link between spectral properties, frame flows, and polynomial maps between spheres.
result Ergodicity of frame flows on low-rank bundles.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
In this paper, the elastic Dirichlet-to-Neumann map Ξg is studied for the stationary elasticity system in a compact Riemannian manifold (Ω,g) with smooth boundary ∂Ω. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map Ξg. We …
This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.
problem Understanding the mathematics behind spectral clustering and its equivalence to PCA.
method Dividing spectral clustering into two categories based on graph connectivity and proving the equivalence to PCA.
result Spectral clustering and PCA are equivalent, with specific proofs for fully connected and multi-connected graphs.
A new homomorphism connects group actions on circles to Euler classes.
problem Understanding group actions on circles and their implications.
method Using crossed homomorphisms and Poincaré translation numbers.
result Relates the Euler class of actions to a specific homomorphism.
We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a c…
New methods avoid spectral pollution in transfer operators for accurate analysis.
problem Spectral pollution in finite-dimensional approximations of transfer operators.
method Algorithms for computing spectral properties of transfer operators without spectral pollution.
result Accurate spectral estimation across various applications, including protein folding models.