Researchers found new spectral coordinates for the Calogero-Moser system.
problem The Rational Calogero-Moser system.
method Bi-Hamiltonian geometry and canonical spectral coordinates.
result Explicit construction of spectral canonical coordinates.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
Study periodic solutions using spectral data and Darboux coordinates.
problem Elliptic sinh-Gordon equation periodic solutions.
method Spectral data and Darboux coordinates.
result Existence of Darboux coordinates.
We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
Method constructs orthogonal curvilinear coordinates in constant curvature spaces.
problem Creating orthogonal coordinates in spaces of constant curvature.
method Modification of Krichever's method for Euclidean space, applied to constant curvature spaces.
result Examples of orthogonal coordinate systems on the sphere and hyperbolic plane constructed.
New spectral clustering method for graphs with uneven node degrees.
problem Challenges in community detection for graphs with heterogeneous degree distributions.
method Spectral clustering on spherical coordinates with degree correction.
result Improved performance in representing computer networks.
The paper extends a proposal for effective twisted superpotentials to higher rank.
problem Describing effective twisted superpotentials from class S theories geometrically.
method Introducing higher rank analogues of spectral networks and spectral coordinates, and finding generating functions.
result The generating functions of the effective twisted superpotentials agree with known results.
New Hermite approximations accelerate convergence with adaptive coordinate transformations.
problem Accelerating convergence of spectral approximations for Hermite expansions.
method Using normalizing flows for adaptive coordinate transformations and deriving error estimates.
result Error estimates for Hermite expansions under adaptive coordinate transformations.
The paper connects complexified coordinates to spectral networks and derives an index theorem.
problem Understanding the relationship between complexified Fenchel-Nielsen coordinates and spectral network coordinates.
method Comparison of exact expressions for 't Hooft defects in 4D SU(2) gauge theories. result Derivation of an index-like theorem for Dirac operators on singular monopole moduli spaces.
Identifies spectral curves for SU(3) coadjoint orbits.
problem Understanding the geometry of coadjoint orbits in SU(3).
method Using Hitchin pairs and spectral curves, identifies a Hamiltonian circle action and finds Darboux coordinates.
result Identifies a differential equation for the Hamiltonian.
Study of meromorphic connections and their spectral duals in gl3(C).
problem Exploring ℏ-deformed meromorphic connections and their spectral duals. method Using apparent singularities and their dual partners as Darboux coordinates, the Hamiltonian evolutions and reductions are derived.
result Spectral duality extends to Hamiltonian evolutions, tau-functions, and Hermitian matrix models on both sides.
Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measur…
The paper connects isomonodromic and isospectral deformations for sl2(C) connections.
problem Connecting isomonodromic and isospectral deformations for sl2(C) connections. method Explicitly constructing Lax pairs and Darboux coordinates to bridge isomonodromic and isospectral deformations.
result Explicit change of Darboux coordinates to match spectral invariants, solving an open issue.
Study shows spectral action coefficients are periods in specific spacetimes.
problem Understanding spectral action coefficients in Robertson-Walker spacetimes.
method Analyzes asymptotic expansion coefficients as periods of mixed Tate motives.
result Coefficients are periods involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces.
New method avoids redundancy in spectral embeddings.
problem Redundant coordinates in spectral dimensionality reduction.
method Introduces unpredictability constraints to avoid redundancy.
result Significantly more informative and compact representations.
Graph-based denoising framework for smooth manifolds.
problem Denoising of signals on smooth manifolds.
method Spectral Graph Wavelet transform applied to the graph Fourier frequency domain.
result Significantly outperforms state-of-the-art denoising methods.
Spheres' spectral structure converges to Gaussian space's as dimensions grow.
problem Understanding spectral convergence between high-dimensional spheres and Gaussian spaces.
method Proving spectral convergence using projections and eigenvalues.
result Spectral structure on high-dimensional spheres converges to Gaussian space's as dimensions increase.
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gauss…
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
COPT optimizes graph distances via simultaneous optimal transport.
problem Learning graph representations unsupervisedly.
method Simultaneous optimization of dual transport plans between vertices and graph signals.
result COPT preserves spectral information and outperforms state-of-the-art methods.
The paper improves alignment methods for deep neural networks using geometric and spectral analysis.
problem Improving alignment methods for deep neural networks.
method Geometric and spectral analysis of residual Jacobian chains.
result Deterministic and margin-verified results on the transport of dominant singular subspaces across layers.
We extend topological recursion to twisted Higgs bundles with singularities.
problem Computing Taylor expansions of period matrices for twisted Higgs bundles.
method We introduce a twisted topological recursion on the spectral curve of a twisted Higgs bundle, encoding singularities and performing the recursion explicitly.
result The g=0 twisted Eynard-Orantin differentials compute the Taylor expansion of the spectral curve's period matrix, independent of the ambient space. In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…
Study of WKB asymptotics of Stokes matrices and spectral curves, proving rhombus inequalities.
problem Analyzing WKB asymptotics of Stokes matrices and spectral curves.
method Using spectral network theory, Poisson geometry, and cluster structures.
result Real parts of leading WKB exponents satisfy rhombus inequalities.
New method deflates manifolds to visualize high-dimensional data.
problem Failure of nonlinear dimensionality reduction methods on simple manifolds.
method Iterative deflation of differential operators using single-coordinate estimates.
result Empirically, recovers novel embeddings on real-world and synthetic datasets.
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ωm with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to…
The paper proves geometric and spectral alignment for deep neural networks.
problem Understanding the singular spectra of deep neural network layers.
method Proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors.
result Exact power-law spectra form a trace-normalized Cartan orbit under Frobenius normalization.
Improved spectral embedding using wave simulation for data analysis.
problem Data dimensionality reduction and metric refinement.
method Wave simulation of eigenfunctions of the discrete Laplacian to refine metric for dimensionality reduction.
result Improved results for data embedding and dimensionality reduction.
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map φ:M→Rn, a …
Counting spheres in hyperbolic space with effective methods.
problem Counting spheres in Apollonian and Kleinian packings.
method Spectral methods and orbit counting, extending Kontorovich and Lax-Phillips techniques.
result Best-known effective error rate for sphere packing counting problems.
Geometrically computes superpotentials for certain 4D N=2 theories.
problem Computing effective twisted superpotentials for 4D N=2 theories.
method Spectral networks and abelianization to compute generating functions of brane opers.
result Geometric recipe for computing effective twisted superpotentials.
Paper solves investment problem with transaction costs using spectral method.
problem Optimal investment problem with transaction costs under potential utility.
method Spectral numerical method applied to a reformulated parabolic double obstacle problem.
result Spectral method proves more efficient for high precision solutions.
I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.
problem Identifying latent sub-manifolds from distance matrices in high-dimensional spaces.
method Coordinate-free inference using random distance matrix theory and generative noise models.
result Recovering latent geometry from integer-stable signatures of eigenvalues.
We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--…
Method constructs quantum holonomies for BPS states on line defects.
problem Determining spins of BPS states on line defects in 4d theories.
method Combines spectral networks and skein algebra.
result Confirms positivity conjectures in physics and math.
Fast algorithms for tensor decomposition robust to errors, with applications to dictionary learning.
problem Tensor decomposition with robustness to errors and sparsity constraints.
method Spectral algorithms with tensor-mode rearrangements, achieving guarantees similar to sum-of-squares (SOS) semidefinite programming.
result Efficient algorithms with running time n5 for decomposing tensors and learning sparse dictionaries, matching or surpassing previous polynomial-time methods. New framework assesses neural sensitivity to small perturbations.
problem Comparing neural representations' sensitivity to small changes.
method Local decodable information, Fisher information, and projected pullback/Fisher metric.
result Reveals differences in neural sensitivity not captured by activation alignment.
This paper explores the preference-based top-K rank aggregation problem. Suppose that a collection of items is repeatedly compared in pairs, and one wishes to recover a consistent ordering that emphasizes the top-K ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce (BTL) model…
The study analyzes spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
problem Analyzing spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
method Equivariant index theorem for Dirac operators on manifolds with φ-cusps under conditions on φ. result The cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.
A new method for unsupervised disentanglement in GANs.
problem Learning disentangled representations in generative models.
method Regularizing GANs by aligning Jacobian vectors with coordinate axes.
result Unsupervised disentanglement achieved in GANs through spectral regularization.
New algorithm proves deep networks can learn better than shallow ones.
problem Understanding the power difference between shallow and deep neural networks.
method Identifying a class of Boolean functions and proving that logarithmic-depth networks can learn them efficiently using hierarchical reconstruction.
result First algorithmic separation between constant-depth and logarithmic-depth neural networks.
The paper tackles sampling from Gibbs measures with constrained support, providing a sampling guarantee.
problem Sampling from Gibbs measures with constrained support, especially in the pre-asymptotic regime.
method Analyzing the spectral gap of Langevin dynamics to provide a non-asymptotic sampling guarantee.
result The low-temperature Gibbs distribution concentrates on a neighborhood of its mode in the pre-asymptotic regime.
AI agents on social networks rarely engage in extended conversations.
problem Understanding the persistence of interactions in AI-agent social networks.
method Analysis of Moltbook, a social network of AI agents, using interaction half-life and spectral tests.
result Most comments on Moltbook receive a direct reply within seconds, indicating a ``fast response or silence'' regime.
The paper proposes a new model for financial order books without assuming prices or quantities.
problem Understanding the geometry of financial order books without assuming prices or quantities.
method Modeling financial order books as an inflationary relational system without metric, temporal, or price coordinates. Observable quantities arise through spectral embeddings of the graph Laplacian.
result Projected supply and demand are constrained to gamma-like functional forms, which can be observed as integrated-gamma cumulative profiles in high-frequency data.
This paper introduces a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. This new model not only generalizes the commonly used linear mixing model, but also allows for possible nonlinear effects to be easily handled, relying on mild assumptions regarding…
We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topologic…
In X-ray binary star systems consisting of a compact object that accretes material from an orbiting secondary star, there is no straightforward means to decide if the compact object is a black hole or a neutron star. To assist this classification, we develop a Bayesian statistical model that makes use of the fact that …